On certain supercuspidal representations of symplectic groups associated with
tamely ramified extensions :
the formal degree conjecture and
the root number conjecture

Koichi Takase The author is partially supported by JSPS KAKENHI Grant Number JP 16K05053

1 Introduction

1.1

Let $\displaystyle F/\mathbb{Q}_{p}$ be a finite extension with $\displaystyle p\neq 2$ whose integer ring $\displaystyle O_{F}$ has unique maximal ideal $\displaystyle\mathfrak{p}_{F}$ wich is generated by $\displaystyle\varpi_{F}$ . The residue class field $\displaystyle\mathbb{F}=O_{F}/\mathfrak{p}_{F}$ is a finite field of $\displaystyle q$ -elements. The Weil group of $\displaystyle F$ is denoted by $\displaystyle W_{F}$ which is a subgroup of the absolute Galois group $\displaystyle\text{\rm Gal}(\overline{F}/F)$ where $\displaystyle\overline{F}$ is a fixed algebraic closure of $\displaystyle F$ in which we will take the algebraic extensions of $\displaystyle F$ .

Let $\displaystyle G$ be a connected semi-simple linear algebraic group defined over $\displaystyle F$ . For the sake of simplicity, we will assume that $\displaystyle G$ splits over $\displaystyle F$ . Then the $\displaystyle L$ -group ${}^{L}G$ of $\displaystyle G$ is equal to the dual group $\displaystyle G^{^}$ of $\displaystyle G$ . An admissible representation

\displaystyle\varphi:W_{F}\times SL_{2}(\mathbb{C})\to\,^{L}G

of the Weil-Deligne group of $\displaystyle F$ is called a discrete parameter of $\displaystyle G$ over $\displaystyle F$ if the centralizer $\displaystyle\mathcal{A}_{\varphi}=Z_{\,{}^{L}G}(\text{\rm Im}\varphi)$ of the image of $\displaystyle\varphi$ in ${}^{L}G$ is a finite group. Let us denote by $\displaystyle\mathcal{D}_{F}(G)$ the $\displaystyle G^{^}$ -conjugacy classes of the discrete parameters of $\displaystyle G$ over $\displaystyle F$ . The conjectural parametrization of $\displaystyle\text{\rm Irr}_{2}(G)$ (resp. $\displaystyle\text{\rm Irr}_{s}(G)$ ), the set of the equivalence classes of the irreducible admissible square-integrable (resp. supercuspidal) representations of $\displaystyle G$ , by $\displaystyle\mathcal{D}_{F}(G)$ is (see [7, p.483, Conj.7.1] for the details)

Conjecture 1.1.1

For every $\displaystyle\varphi\in\mathcal{D}_{F}(G)$ , there exists a finite subset $\displaystyle\Pi_{\varphi}$ of $\displaystyle\text{\rm Irr}_{2}(G)$ such that

1)

$\displaystyle\text{\rm Irr}_{2}(G)=\bigsqcup_{\varphi\in\mathcal{D}_{F}(G)}\Pi_{\varphi}$ ,
2)

there exists a bijection of $\displaystyle\Pi_{\varphi}$ onto the equivalence classes $\displaystyle\mathcal{A}_{\varphi}^{^}$ of the irreducible complex linear representations of $\displaystyle\mathcal{A}_{\varphi}$ ,
3)

$\displaystyle\Pi_{\varphi}\subset\text{\rm Irr}_{s}(G)$ if $\displaystyle\varphi|_{SL_{2}(\mathbb{C})}=1$ .

The finite set $\displaystyle\Pi_{\varphi}$ is called a $\displaystyle L$ -packet of $\displaystyle\varphi$ .

According to this conjecture, any $\displaystyle\pi\in\text{\rm Irr}_{2}(G)$ is determined by $\displaystyle\varphi\in\mathcal{D}_{F}(G)$ and $\displaystyle\sigma\in\mathcal{A}_{\varphi}^{^}$ . So the formal degree of $\displaystyle\pi$ should be determined by these data. The formal degree conjecture due to Hiraga-Ichino-Ikeda [8] is (with the formulation of [7])

Conjecture 1.1.2

The formal degree $\displaystyle d_{\pi}$ of $\displaystyle\pi$ with respect to the absolute value of the Euler-Poincaré measure (see [12, $\displaystyle\S 3$ ] for the details) on $\displaystyle G(F)$ is equal to

\displaystyle\frac{\dim\sigma}{|\mathcal{A}_{\varphi}|}\cdot\left|\frac{\gamma(\varphi,\text{\rm Ad},\psi,d(x),0)}{\gamma(\varphi_{0},\text{\rm Ad},\psi,d(x),0)}\right|.

Here

\displaystyle\gamma(\varphi,\text{\rm Ad},\psi,d(x),s)=\varepsilon(\varphi,\text{\rm Ad},d(x),s)\cdot\frac{L(\varphi^{\vee},\text{\rm Ad},1-s)}{L(\varphi,\text{\rm Ad},s)}

is the gamma-factor associated with the $\displaystyle\varphi$ combined with the adjoint representation Ad of $\displaystyle G^{^}$ on its Lie algebra $\displaystyle\mathfrak{g}^{^}$ , and a continuous additive character $\displaystyle\psi$ of $\displaystyle F$ such that $\displaystyle\{x\in F\mid\psi(xO_{F})=1\}=O_{F}$ and the Haar measure $\displaystyle d(x)$ on the additive group $\displaystyle F$ such that $\displaystyle\int_{O_{F}}d(x)=1$ . See [7, pp.440-441] for the details.

\displaystyle\varphi_{0}:W_{F}\times SL_{2}(\mathbb{C})\xrightarrow{\text{\rm proj.}}SL_{2}(\mathbb{C})\to G^{^}

is the principal parameter (see [7, p.447] for the definition).

The formal degree conjecture concerns with the absolute value of the epsilon-factor

\displaystyle\varepsilon(\varphi,\text{\rm Ad},d(x),s)=w(\varphi,\text{\rm Ad})\cdot q^{a(\varphi,\text{\rm Ad})(\frac{1}{2}-s)}

where $\displaystyle a(\varphi,\text{\rm Ad})$ is the Artin-conductor and $\displaystyle w(\varphi,\text{\rm Ad})$ is the root number.

In order to state the root number conjecture, we need some notations. Let $\displaystyle T\subset G$ be a maximal torus split over $\displaystyle F$ with respect to which the root datum

\displaystyle(X(T),\Phi(T),X^{\vee}(T),\Phi^{\vee}(T))

is defined. Then the dual group $\displaystyle G^{^}$ is, by the definition, the connected reductive complex algebraic group with a maximal torus $\displaystyle T^{^}$ with which its root datum is isomorphic to

\displaystyle(X^{\vee}(T),\Phi^{\vee}(T),X(T),\Phi(T)).

Put $\displaystyle 2\cdot\rho=\sum_{0<\alpha\in\Phi^{\vee}(T)}\alpha$ , then $\displaystyle\epsilon=2\cdot\rho(-1)\in T$ is a central element of $\displaystyle G$ . Now the root number conjecture says that

Conjecture 1.1.3

[7, p.493, Conj.8.3]

\displaystyle\frac{w(\varphi,\text{\rm Ad})}{w(\varphi_{0},\text{\rm Ad})}=\pi(\epsilon)

where $\displaystyle\epsilon$ is the central element of $\displaystyle G$ defined above (see [7, p.492, (65)] for the details).

Since $\displaystyle G$ is assumed to be splits over $\displaystyle F$ , we have $\displaystyle w(\varphi_{0},\text{\rm Ad})=1$ (see [7, p.448]).

1.2

In this paper, we will construct quite explicitly supercuspidal representations of $\displaystyle G(F)=Sp_{2n}(F)$ associated with a tamely ramified extension $\displaystyle K/F$ of degree $\displaystyle 2n$ (Theorem 2.3.1). Here $\displaystyle K$ is a quadratic extension of over field $\displaystyle K_{+}$ of $\displaystyle F$ . When $\displaystyle K/F$ is normal, we will also give candidates of Langlands parameters of the supercuspidal representations (the section 3), and will verify the validity of the formal degree conjecture (Theorem 4.3.1) and the root number conjecture (Theorem 5.3.1) with them. Surprisingly the root number conjecture is valid only if $\displaystyle K/F$ is not totally ramified or $\displaystyle K/F$ is totally ramified and

\displaystyle\frac{q-1}{2}\cdot(n-1)\equiv 0\!\!\pmod{4}.

Our supercuspidal representations, denoted by $\displaystyle\pi_{\beta,\theta}$ , are given by the compact induction $\displaystyle\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$ from irreducible unitary representations $\displaystyle\delta_{\beta,\theta}$ of the hyperspecial compact subgroup $\displaystyle G(O_{F})=Sp_{2n}(O_{F})$ . Here $\displaystyle\pi_{\beta,\theta}$ and $\displaystyle\delta_{\beta,\theta}$ are characterized each other by the conditions

1)

$\displaystyle\delta_{\beta,\theta}$ factors through the canonical morphism $\displaystyle G(O_{F})\to G(O_{F}/\mathfrak{p}_{F}^{r})$ with $\displaystyle r\geq 2$ , and the multiplicity of $\displaystyle\delta_{\beta,\theta}$ in $\displaystyle\pi_{\beta,\theta}|_{G(O_{F})}$ is one,
2)

any irreducible unitary representation $\displaystyle\delta$ of $\displaystyle G(O_{F})$ which factors through the canonical morphism $\displaystyle G(O_{F})\to G(O_{F}/\mathfrak{p}_{F}^{r})$ , and a constituent of $\displaystyle\pi_{\beta,\theta}|_{G(O_{F})}$ , then $\displaystyle\delta=\delta_{\beta,\theta}$ .

The parameters $\displaystyle\beta$ and $\displaystyle\theta$ are associated with the tamely ramified extension $\displaystyle K/F$ , that is, $\displaystyle O_{K}=O_{F}[\beta]$ and $\displaystyle\theta$ is a certain continuous unitary character of

\displaystyle U_{K/K_{+}}=\{x\in K^{\times}\mid N_{K/K_{+}}(x)=\}

(see the subsection 2.2 for the precise definitions). We have the irreducible representation $\displaystyle\delta_{\beta,\theta}$ by the general theory given by [16].

The candidate of Langlands parameter is given by the method of Kaletha [10]. Regard the compact group $\displaystyle U_{K/K_{+}}$ as the group of $\displaystyle F$ -rational points of an elliptic torus of $\displaystyle Sp_{2n}$ . Then, by the local Langlands correspondence of tori (see [20]) and the Langlands-Schelstad procedure ([11]) gives a group homomorphism $\displaystyle\varphi$ of the Weil group $\displaystyle W_{F}$ of $\displaystyle F$ to the dual group $\displaystyle G^{^}=SO_{2n+1}(\mathbb{C})$ of $\displaystyle Sp_{2n}$ over $\displaystyle F$ .

Although a general theory of construction of the supercuspidal representation is given by [19, 10], that of ours is based upon a method of [13] which has an advantage of being more direct and explicit.

Note that the formal degree conjecture is proved by [15] between the supercuspidal representations of [10] and the Langlands parameters of Kaletha. In this paper, supercuspidal representations are constructed by a method different from that of [10], so it is of some interest.

1.3

The section 2 is devoted to the construction of the supercuspidal representation $\displaystyle\pi_{\beta,\theta}$ of $\displaystyle Sp_{2n}(F)$ . After recalling, in the subsection 2.1, the general theory of the regular irreducible representations of the finite group $\displaystyle G(O_{F}/\mathfrak{p}_{F}^{r})$ ( $\displaystyle r\geq 2$ ) given by [16], we will define the irreducible unitary representation $\displaystyle\delta_{\beta,\theta}$ of $\displaystyle Sp_{2n}(O_{F})$ in the subsection 2.2. The construction of the supercuspidal representation $\displaystyle\pi_{\beta,\theta}$ is given in the subsection 2.3.

The candidate of Langlands parameter is given in the section 3. The local Langlands correspondence of elliptic torus (Proposition 3.1.1) and the Langlands-Schelstad procedure (the subsection 3.2) are given quite explicitly. They give a candidate of Langlands parameter

\displaystyle\varphi:W_{F}\xrightarrow{\text{\rm canonical}}W_{K/F}\xrightarrow{\varphi_{1}\oplus\det\varphi_{1}}SO_{2n+1}(\mathbb{C})

where $\displaystyle\varphi_{1}=\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\vartheta}$ is the induced representation from a character $\displaystyle\widetilde{\vartheta}$ of $\displaystyle K^{\times}$ to the relative Weil group $\displaystyle W_{K/F}=W_{F}/\overline{[W_{K},W_{K}]}$ . The character $\displaystyle\widetilde{\vartheta}$ is defined by $\displaystyle\widetilde{\vartheta}(x)=\vartheta(x^{1-\tau})$ where $\displaystyle\text{\rm Gal}(K/K_{+})=\langle\tau\rangle$ and $\displaystyle\vartheta=c\cdot\theta$ with the character $\displaystyle c$ of $\displaystyle U_{K/K_{+}}$ which is generated by the Langlands-Schelstad procedure.

Using the explicit description of the parameter $\displaystyle\varphi$ , we will verify the formal degree conjecture in the section 4, and the root number conjecture in the section 5.

In section 6, we will discuss the case of $\displaystyle n=2$ where we can define another “natural” candidate for the Langlands parameter of $\displaystyle\pi_{\beta,\theta}$ . The representation space of $\displaystyle\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\theta}$ , with $\displaystyle\widetilde{\theta}(x)=\theta(x^{1-\tau})$ , has $\displaystyle W_{K/F}$ -quasi invariant symplectic form. Then the candidate is given by

W_{F}\xrightarrow{\text{\rm can.}}W_{K/F}\xrightarrow{\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\theta}}GSp_{4}(\mathbb{C})\xrightarrow{(\ast)}SO_{5}(\mathbb{C})

(1.1)

where $\displaystyle(\ast)$ is the accidental surjection. With respect to this parameter

1)

the formal degree conjecture is valid only if $\displaystyle K/F$ is unramified or totally ramified, and in this case

the root number cinjecture is valid only if

\displaystyle\theta(-1)=\begin{cases}1&:\text{\rm$\displaystyle K/F$ is unramified,}\\ (-1)^{\frac{q-1}{4}}&:\text{\rm$\displaystyle K/F$ is totally ramifed.}\end{cases}

This means that the parameter (1.1) is not the Langlands parameter of $\displaystyle\pi_{\beta,\theta}$ , in general.

Several basic facts on the local factor associated with representations of the Weil group are given in the appendix A.

The quasi-invariant symmetric or symplectic form in the induced representation on $\displaystyle W_{K/F}$ from the characters of $\displaystyle K^{\times}$ is discussed in the appendix B.

2 Supercuspidal representations of $\displaystyle Sp_{2n}(F)$

2.1 Regular irreducible characters of hyperspecial compact subgroup

Let us recall the main results of [16].

Fix a continuous unitary additive character $\displaystyle\psi:F\to\mathbb{C}^{1}$ such that

\displaystyle\{x\in F\mid\psi(xO_{F})=1\}=O_{F}.

Let $\displaystyle G=Sp_{2n}$ be the $\displaystyle O_{F}$ -group scheme such that, for any $\displaystyle O_{F}$ -algebra ¹¹1In this paper, an $\displaystyle O_{F}$ -algebra means an unital commutative $\displaystyle O_{F}$ -algebra. $\displaystyle R$ , the group of the $\displaystyle A$ -valued point $\displaystyle G(A)$ is a subgroup of $\displaystyle GL_{2n}(R)$ defined by

\displaystyle G(R)=\{g\in GL_{2n}(R)\mid gJ_{n}\,^{t}g=J_{n}\}

where

\displaystyle J_{n}=\begin{bmatrix}0&I_{n}\\ -I_{n}&0\end{bmatrix},\;\;\text{\rm where}\;\;I_{n}=\begin{bmatrix}&&1\\ &\mathinner{\mkern 1.0mu\raise 1.0pt\hbox{.}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise 7.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 1.0mu}&\\ 1&&\end{bmatrix}.

For a matrix $\displaystyle A\in M_{m,n}(R)$ , put ${}^{\mathfrak{t}}A=I_{n}\,^{t}AI_{m}\in M_{n,m}(R)$ . Let $\displaystyle\mathfrak{g}$ the Lie algebra scheme of $\displaystyle G$ which is a closed affine $\displaystyle O_{F}$ -subscheme of $\displaystyle\mathfrak{gl}_{n}$ the Lie algebra scheme of $\displaystyle GL_{n}$ defined by

\displaystyle\mathfrak{g}(R)=\{X\in\mathfrak{gl}_{2n}(R)\mid XJ_{n}+J_{n}\,^{t}X=0\}

for all $\displaystyle O_{F}$ -algebra $\displaystyle R$ . Let

\displaystyle B:\mathfrak{gl}_{2n}{\times}_{O_{F}}\mathfrak{gl}_{2n}\to\mathbb{A}_{O_{F}}^{1}

be the trace form on $\displaystyle\mathfrak{gl}_{2n}$ , that is $\displaystyle B(X,Y)=\text{\rm tr}(XY)$ for all $\displaystyle X,Y\in\mathfrak{gl}_{2n}(R)$ with any $\displaystyle O_{F}$ -algebra $\displaystyle R$ . Since $\displaystyle G$ is smooth $\displaystyle O_{F}$ -group scheme, we have a canonical isomorphism

\displaystyle\mathfrak{g}(O_{F})/\varpi^{r}\mathfrak{g}(O_{F})\,\tilde{\to}\,\mathfrak{g}(O_{F}/\mathfrak{p}^{r})=\mathfrak{g}(O_{F}){\otimes}_{O_{F}}O_{F}/\mathfrak{p}^{r}

([2, Chap.II, $\displaystyle\S 4$ , Prop.4.8]) and the canonical group homomorphism $\displaystyle G(O_{F})\to G(O_{F}/\mathfrak{p}^{r})$ is surjective, due to the formal smoothness [2, p.111, Cor. 4.6], whose kernel is denoted by $\displaystyle K_{r}(O_{F})$ . For any $\displaystyle 0<l<r$ , let us denote by $\displaystyle K_{l}(O_{F}/\mathfrak{p}^{r})$ the kernel of the canonical group homomorphism $\displaystyle G(O_{F}/\mathfrak{p}^{r})\to G(O_{F}/\mathfrak{p}^{l})$ which is surjective.

The following basic assumptions on $\displaystyle G$ are satisfied;

I)

$\displaystyle B:\mathfrak{g}(\mathbb{F})\times\mathfrak{g}(\mathbb{F})\to\mathbb{F}$ is non-degenerate,
II)

for any integers $\displaystyle r=l+l^{\prime}$ with $\displaystyle 0<l^{\prime}\leq l$ , we have a group isomorphism

$\displaystyle\mathfrak{g}(O_{F}/\mathfrak{p}^{l^{\prime}})\,\tilde{\to}\,K_{l}(O_{F}/\mathfrak{p}^{r})$

defined by $\displaystyle X\!\!\pmod{\mathfrak{p}^{l^{\prime}}}\mapsto 1+\varpi^{l}X\!\!\pmod{\mathfrak{p}^{r}}$ ,
III)

if $\displaystyle r=2l-1\geq 3$ is odd, then we have a mapping

$\displaystyle\mathfrak{g}(O_{F})\to K_{l-1}(O_{F}/\mathfrak{p}^{r})$

defined by $\displaystyle X\mapsto(1+\varpi^{l-1}X+2^{-1}\varpi^{2l-2}X^{2})\!\!\pmod{\mathfrak{p}^{r}}$ .

The condition I) implies that $\displaystyle B:\mathfrak{g}(O_{F}/\mathfrak{p}^{l})\times\mathfrak{g}(O_{F}/\mathfrak{p}^{l})\to O_{F}/\mathfrak{p}^{l}$ is non-degenerate for all $\displaystyle l>0$ , and so $\displaystyle B:\mathfrak{g}(O_{F})\times\mathfrak{g}(O_{F})\to O_{F}$ is also non-degenerate. By the condition II), $\displaystyle K_{l}(O_{F}/\mathfrak{p}^{r})$ is a commutative normal subgroup of $\displaystyle G(O_{F}/\mathfrak{p}^{r})$ , and its character is

\displaystyle\chi_{\beta}(1+\varpi^{l}X\!\!\pmod{\mathfrak{p}^{r}})=\psi\left(\varpi^{-l^{\prime}}B(X,\beta)\right)\quad(X\!\!\pmod{\mathfrak{p}^{l^{\prime}}}\in\mathfrak{g}(O_{F}/\mathfrak{p}^{l^{\prime}}))

with $\displaystyle\beta\!\!\pmod{\mathfrak{p}^{l^{\prime}}}\in\mathfrak{g}(O_{F}/\mathfrak{p}^{l^{\prime}})$ .

Since any finite dimensional complex continuous representation of the compact group $\displaystyle G(O_{F})$ factors through the canonical group homomorphism $\displaystyle G(O_{F})\to G(O_{F}/\mathfrak{p}^{r})$ for some $\displaystyle 0<r\in\mathbb{Z}$ , we want to know the irreducible complex representations of the finite group $\displaystyle G(O_{F}/\mathfrak{p}^{r})$ . Let us assume that $\displaystyle r>1$ and put $\displaystyle r=l+l^{\prime}$ with the minimal integer $\displaystyle l$ such that $\displaystyle 0<l^{\prime}\leq l$ , that is

\displaystyle l^{\prime}=\begin{cases}l&:\text{\rm if $\displaystyle r=2l$},\\ l-1&:\text{\rm if $\displaystyle r=2l-1$}.\end{cases}

Let $\displaystyle\delta$ be an irreducible complex representation of $\displaystyle G(O_{F}/\mathfrak{p}^{r})$ . The Clifford’s theorem says that the restriction $\displaystyle\delta|_{K_{l}(O_{F}/\mathfrak{p}^{r})}$ is a sum of the $\displaystyle G(O_{F}/\mathfrak{p}^{r})$ -conjugates of characters of $\displaystyle K_{l}(O_{F}/\mathfrak{p}^{r})$ :

\delta|_{K_{l}(O_{F}/\mathfrak{p}^{r})}=\left(\bigoplus_{\dot{\beta}\in\Omega}\chi_{\beta}\right)^{m}

(2.1)

with an adjoint $\displaystyle G(O_{F}/\mathfrak{p}^{l^{\prime}})$ -orbit $\displaystyle\Omega\subset\mathfrak{g}(O_{F}/\mathfrak{p}^{l^{\prime}})$ . In this way the irreducible complex representations of $\displaystyle G(O_{F}/\mathfrak{p}^{r})$ correspond to adjoint $\displaystyle G(O_{F}/\mathfrak{p}^{l^{\prime}})$ -orbits in $\displaystyle\mathfrak{g}(O_{F}/\mathfrak{p}^{l^{\prime}})$ .

Fix an adjoint $\displaystyle G(O_{F}/\mathfrak{p}^{l^{\prime}})$ -orbit $\displaystyle\Omega\subset\mathfrak{g}(O_{F}/\mathfrak{p}^{l^{\prime}})$ and let us denote by $\displaystyle\Omega^{^}$ the set of the equivalence classes of the irreducible complex representations of $\displaystyle G(O_{F}/\mathfrak{p}^{l^{\prime}})$ correspond to $\displaystyle\Omega$ . Then [16] gives a parametrization of $\displaystyle\Omega^{^}$ as follows:

Theorem 2.1.1

Take a representative $\displaystyle\beta\!\!\pmod{\mathfrak{p}^{l^{\prime}}}\in\Omega$ ( $\displaystyle\beta\in\mathfrak{g}O_{F})$ ) and assume that

1)

the centralizer $\displaystyle G_{\beta}=Z_{G}(\beta)$ of $\displaystyle\beta\in\mathfrak{g}(O_{F})$ in $\displaystyle G$ is smooth over $\displaystyle O_{F}$ ,
2)

the characteristic polynomial $\displaystyle\chi_{\overline{\beta}}(t)=\det(t\cdot 1_{2n}-\overline{\beta})$ of $\displaystyle\overline{\beta}=\beta\pmod{\mathfrak{p}}\in\mathfrak{g}(\mathbb{F})\subset\mathfrak{gl}_{2n}(\mathbb{F})$ is the minimal polynomial of $\displaystyle\overline{\beta}\in M_{2n}(\mathbb{F})$ .

Then there exists a bijection $\displaystyle\theta\mapsto\delta_{\beta,\theta}$ of the set

\displaystyle\left\{\theta\in G_{\beta}(O_{F}/\mathfrak{p}^{r})^{^}\;\;\;\text{\rm s.t. $\displaystyle\theta=\chi_{\beta}$ on $\displaystyle G_{\beta}(O_{F}/\mathfrak{p}^{r})\cap K_{l}(O_{F}/\mathfrak{p}^{r})$}\right\}

onto $\displaystyle\Omega^{^}$ .

The correspondence $\displaystyle\theta\mapsto\delta_{\beta,\theta}$ is given by the following procedure. The second condition in the theorem implies

\displaystyle G_{\beta}(O_{F}/\mathfrak{p}^{r})=G(O_{F}/\mathfrak{p}^{r})\cap\left(O_{F}/\mathfrak{p}^{r}\right)[\beta\!\!\pmod{\mathfrak{p}^{r}}],

in particular $\displaystyle G_{\beta}(O_{F}/\mathfrak{p}^{r})$ is commutative. So $\displaystyle G_{\beta}(O_{F}/\mathfrak{p}^{r})^{^}$ means the character group of $\displaystyle G_{\beta}(O_{F}/\mathfrak{p}^{r})$ .

$\displaystyle\Omega^{^}$ consists of the irreducible complex representations whose restriction to $\displaystyle K_{l}(O_{F}/\mathfrak{p}^{r})$ contains the character $\displaystyle\chi_{\beta}$ . Then the Clifford’s theory says the followings: put

	$\displaystyle\displaystyle G(O_{F}/\mathfrak{p}^{r};\beta)$	$\displaystyle\displaystyle=\left\{g\in G(O_{F}/\mathfrak{p}^{r})\mid\chi_{\beta}(g^{-1}hg)=\chi_{\beta}(h)\;\forall h\in K_{l}(O_{F}/\mathfrak{p}^{r})\right\}$
		$\displaystyle\displaystyle=\left\{g\in G(O_{F}/\mathfrak{p}^{r})\mid\text{\rm Ad}(g)\beta\equiv\beta\!\!\pmod{\mathfrak{p}^{l^{\prime}}}\right\}$

and let us denote by $\displaystyle\text{\rm Irr}(G(O_{F}/\mathfrak{p}^{r};\beta),\chi_{\beta})$ the set of the equivalence classes of the irreducible complex representations $\displaystyle\sigma$ of $\displaystyle G(O_{F}/\mathfrak{p}^{r};\beta)$ such that the restriction $\displaystyle\sigma|_{K_{l}(O_{F}/\mathfrak{p}^{r})}$ contains the character $\displaystyle\chi_{\beta}$ . Then $\displaystyle\sigma\mapsto\text{\rm Ind}_{G(O_{F}/\mathfrak{p}^{r};\beta)}^{G(O_{F}/\mathfrak{p}^{r})}\sigma$ gives a bijection of $\displaystyle\text{\rm Irr}(G(O_{F}/\mathfrak{p}^{r};\beta),\chi_{\beta})$ onto $\displaystyle\Omega^{^}$ .

Since $\displaystyle G_{\beta}$ is smooth over $\displaystyle O_{F}$ , the canonical homomorphism $\displaystyle G_{\beta}(O_{F}/\mathfrak{p}^{r})\to G_{\beta}(O_{F}/\mathfrak{p}^{l^{\prime}})$ is surjective. Hence we have

\displaystyle G(O_{F}/\mathfrak{p}^{r};\beta)=G_{\beta}(O_{F}/\mathfrak{p}^{r})\cdot K_{l^{\prime}}(O_{F}/\mathfrak{p}^{r}).

If $\displaystyle r=2l$ is even, then $\displaystyle l^{\prime}=l$ and, for any character $\displaystyle\theta\in G_{\beta}(O_{F}/\mathfrak{p}^{r})$ such that $\displaystyle\theta=\chi_{\beta}$ on $\displaystyle G_{\beta}(O_{F}/\mathfrak{p}^{r})\cap K_{l}(O_{F}/\mathfrak{p}^{r})$ , the character

\displaystyle\sigma_{\theta,\beta}(gh)=\theta(g)\cdot\chi_{\beta}(h)\quad(g\in G_{\beta}(O_{F}/\mathfrak{p}^{r}),h\in K_{l}(O_{F}/\mathfrak{p}^{r}))

of $\displaystyle G(O_{F}/\mathfrak{p}^{r};\beta)$ is well-defined, and $\displaystyle\theta\mapsto\sigma_{\theta,\beta}$ is a surjection onto $\displaystyle\text{\rm Irr}(G(O_{F}/\mathfrak{p}^{r};\beta),\chi_{\beta})$ . Hence

\displaystyle\theta\mapsto\delta_{\theta,\beta}=\text{\rm Ind}_{G(O_{F}/\mathfrak{p}^{r};\beta)}^{G(O_{F}/\mathfrak{p}^{r})}\sigma_{\theta,\beta}

is the bijection of Theorem 2.1.1.

If $\displaystyle r=2l-1$ is odd, then $\displaystyle l^{\prime}=l-1$ . Let us denote by $\displaystyle\mathfrak{g}_{\beta}=\text{\rm Lie}(G_{\beta})$ the Lie algebra $\displaystyle O_{F}$ -scheme of the smooth $\displaystyle O_{F}$ -group scheme $\displaystyle G_{\beta}$ . Then

\displaystyle\mathbb{V}_{\beta}=\mathfrak{g}(\mathbb{F})/\mathfrak{g}_{\beta}(\mathbb{F})

is a symplectic $\displaystyle\mathbb{F}$ -space with a symplectic $\displaystyle\mathbb{F}$ -form

\displaystyle D_{\beta}(\dot{X},\dot{Y})=B([X,Y],\overline{\beta})\in\mathbb{F}\quad(X,Y\in\mathfrak{g}(\mathbb{F})).

Let $\displaystyle H_{\beta}=\mathbb{V}_{\beta}\times\mathbb{C}^{1}$ be the Heisenberg group associated with $\displaystyle(\mathbb{V}_{\beta},D_{\beta})$ and $\displaystyle(\sigma^{\beta},L^{2}(\mathbb{W}^{\prime}))$ the Schrödinger representation of $\displaystyle H_{\beta}$ associated with a polarization $\displaystyle\mathbb{V}_{\beta}=\mathbb{W}^{\prime}\oplus\mathbb{W}$ . More explicitly the group operation of $\displaystyle H_{\beta}$ is defined by

\displaystyle(u,s)\cdot(v,t)=(u+v,st\cdot\widehat{\psi}(2^{-1}D_{\beta}u,v))

where $\displaystyle\widehat{\psi}(\overline{x})=\psi(\varpi^{-1}x)$ for $\displaystyle\overline{x}=x\!\!\pmod{\mathfrak{p}}\in\mathbb{F}$ , and the action of $\displaystyle h=(u,s)\in H_{\beta}$ on $\displaystyle f\in L^{2}(\mathbb{W}^{\prime})$ (a complex-valued function on $\displaystyle\mathbb{W}^{\prime}$ ) is defined by

\displaystyle(\sigma^{\beta}(h)f)(w)=s\cdot\widehat{\chi}\left(2^{-1}D_{\beta}(u_{-},u_{+})+D_{\beta}(w,u_{+})\right)\cdot f(w+u_{-})

where $\displaystyle u=u_{-}+u_{+}\in\mathbb{V}_{\beta}=\mathbb{W}^{\prime}\oplus\mathbb{W}$ .

Take a character $\displaystyle\theta:G_{\beta}(O_{F}/\mathfrak{p}^{r})\to\mathbb{C}^{\times}$ such that

\displaystyle\theta=\chi_{\beta}\;\;\text{\rm on}\;\;G_{\beta}(O_{F}/\mathfrak{p}^{r})\cap K_{l}(O_{F}/\mathfrak{p}^{r}).

Then an additive character $\displaystyle\rho_{\theta}:\mathfrak{g}_{\beta}(\mathbb{F})\to\mathbb{C}^{\times}$ is defined by

\displaystyle\rho_{\theta}(X\!\!\!\!\pmod{\mathfrak{p}})=\chi\left(-\varpi^{-l}B(X,\beta)\right)\cdot\theta\left(1+\varpi^{l-1}X+2^{-1}\varpi^{2l-2}X^{2}\!\!\!\!\pmod{\mathfrak{p}^{r}}\right)

with $\displaystyle X\in\mathfrak{g}_{\beta}(O_{F})$ . Fix a $\displaystyle\mathbb{F}$ -vector subspace $\displaystyle V\subset\mathfrak{g}(\mathbb{F})$ such that $\displaystyle\mathfrak{g}(\mathbb{F})=V\oplus\mathfrak{g}_{\beta}(\mathbb{F})$ . Then an irreducible representation $\displaystyle(\sigma^{\beta,\theta},L^{2}(\mathbb{W}^{\prime}))$ of $\displaystyle K_{l-1}(O_{F}/\mathfrak{p}^{r})$ is defined by the following proposition:

Proposition 2.1.2

Take a $\displaystyle g=1+\varpi^{l-1}T\!\!\pmod{\mathfrak{p}^{r}}\in K_{l-1}(O_{F}/\mathfrak{p}^{r})$ with $\displaystyle T\in\mathfrak{gl}_{n}(O_{F})$ . Then we have $\displaystyle T\!\!\pmod{\mathfrak{p}^{l-1}}\in\mathfrak{g}(O_{F}/\mathfrak{p}^{l-1})$ and

\displaystyle\sigma^{\beta,\rho}(g)=\tau\left(\varpi^{-l}B(T,\beta)-2^{-1}\varpi^{-1}B(T^{2},\beta)\right)\cdot\rho_{\theta}(Y)\cdot\sigma^{\beta}(v,1)

where $\displaystyle\overline{T}=[v]+Y\in\mathfrak{g}(\mathbb{F})$ with $\displaystyle v\in\mathbb{V}_{\beta}$ and $\displaystyle Y\in\mathfrak{g}_{\beta}(\mathbb{F})$ .

Then main result shown in [16], under the assumptions of Theorem 2.1.1, is that there exists a group homomorphism (not unique)

\displaystyle U:G_{\beta}(O_{F}/\mathfrak{p}^{r})\to GL_{\mathbb{C}}(L^{2}(\mathbb{W}^{\prime}))

such that

1)

$\displaystyle\sigma^{\beta,\theta}(h^{-1}gh)=U(h)^{-1}\circ\sigma^{\beta,\theta}(g)\circ U(h)$ for all $\displaystyle h\in G_{\beta}(O_{F}/\mathfrak{p}^{r})$ and $\displaystyle g\in K_{l-1}(O_{F}/\mathfrak{p}^{r})$ , and
2)

$\displaystyle U(h)=1$ for all $\displaystyle h\in G_{\beta}(O_{F}/\mathfrak{p}^{r})\cap K_{l-1}(O_{F}/\mathfrak{p}^{r})$ .

Now an irreducible representation $\displaystyle(\sigma_{\beta,\theta},L^{2}(\mathbb{W}^{\prime}))$ is defined by

\displaystyle\sigma_{\beta,\theta}(hg)=\theta(h)\cdot U(h)\circ\sigma^{\beta,\theta}(g)

for $\displaystyle hg\in G(O_{F}/\mathfrak{p}^{r};\beta)=G_{\beta}(O_{F}/\mathfrak{p}^{r})\cdot K_{l-1}(O_{F}/\mathfrak{p}^{r})$ with $\displaystyle h\in G_{\beta}(O_{F}/\mathfrak{p}^{r})$ and $\displaystyle g\in K_{l-1}(O_{F}/\mathfrak{p}^{r})$ , and $\displaystyle\theta\mapsto\sigma_{\beta,\theta}$ is a surjection onto $\displaystyle\text{\rm Irr}(G(O_{F}/\mathfrak{p}^{r};\beta),\chi_{\beta})$ . Then

\displaystyle\theta\mapsto\delta_{\beta,\theta}=\text{\rm Ind}_{G(O_{F}/\mathfrak{p}^{r};\beta)}^{G(O_{F}/\mathfrak{p}^{r})}\sigma_{\beta,\theta}

is the bijection of Theorem 2.1.1.

Because the connected $\displaystyle O_{F}$ -group scheme $\displaystyle G=Sp_{2n}$ is reductive, that is, the fibers $\displaystyle G{\otimes}_{O_{F}}K$ ( $\displaystyle K=F,\mathbb{F}$ ) are reductive $\displaystyle K$ -algebraic groups, the dimension of a maximal torus in $\displaystyle G{\otimes}_{O_{F}}K$ is independent of $\displaystyle K$ which is denoted by $\displaystyle\text{\rm rank}(G)$ . For any $\displaystyle\beta\in\mathfrak{g}(O_{F})$ we have

\dim_{K}\mathfrak{g}_{\beta}(K)=\dim\mathfrak{g}_{\beta}{\otimes}_{O_{F}}K\geq\dim G_{\beta}{\otimes}_{O_{F}}K\geq\text{\rm rank}(G).

(2.2)

We say $\displaystyle\beta$ is smoothly regular over $\displaystyle K$ if $\displaystyle\dim_{K}\mathfrak{g}_{\beta}(K)=\text{\rm rank}(G)$ (see [14, (5.7)]). In this case $\displaystyle G_{\beta}{\otimes}_{O_{F}}K$ is smooth over $\displaystyle K$ .

Let $\displaystyle G_{\beta}^{o}$ be the neutral component of $\displaystyle O_{F}$ -group scheme $\displaystyle G_{\beta}$ which is a group functor of the category of $\displaystyle O_{F}$ -scheme (see $\displaystyle\S 3$ of Exposé $\displaystyle\text{\rm VI}_{B}$ in [3]). The following statements are equivalent;

1)

$\displaystyle G_{\beta}^{o}$ is representable as an smooth open $\displaystyle O_{F}$ -group subscheme of $\displaystyle G_{\beta}$ ,
2)

$\displaystyle G_{\beta}$ is smooth at the points of unit section,
3)

each fibers $\displaystyle G_{\beta}{\otimes}_{O_{F}}K$ ( $\displaystyle K=F,\mathbb{F}$ ) are smooth over $\displaystyle K$ and their dimensions are constant

(see Th. 3.10 and Cor. 4.4 of [3]). So if $\displaystyle\beta$ is smoothly regular over $\displaystyle F$ and $\displaystyle\mathbb{F}$ , then $\displaystyle G_{\beta}^{o}$ is smooth open $\displaystyle O_{F}$ -group subscheme of $\displaystyle G_{\beta}$ . So we have

Proposition 2.1.3

The centralizer $\displaystyle G_{\beta}=Z_{G}(\beta)$ of $\displaystyle\beta$ in $\displaystyle G$ is smooth over $\displaystyle O_{F}$ if the following two conditions are fulfilled:

1)

$\displaystyle\beta\in\mathfrak{g}(O_{F})$ is smoothly regular over $\displaystyle F$ and $\displaystyle\mathbb{F}$ , and
2)

$\displaystyle G_{\beta}{\otimes}_{O_{F}}F$ and $\displaystyle G_{\beta}{\otimes}_{O_{F}}\mathbb{F}$ are connected.

Let us assume the two conditions of the preceding proposition. Since we have canonical isomorphisms

\displaystyle\mathfrak{g}(\mathbb{F})\,\tilde{\to}\,K_{m-1}(O_{F}/\mathfrak{p}^{m}),\qquad\mathfrak{g}_{\beta}(\mathbb{F})\,\tilde{\to}\,G_{\beta}(O_{F}/\mathfrak{p}^{m})\cap K_{m-1}(O_{F}/\mathfrak{p}^{m})

and the canonical morphism $\displaystyle G_{\beta}(O_{F})\to G_{\beta}(O_{F}/\mathfrak{p}^{m})$ is surjective for any $\displaystyle m>1$ , we have

\displaystyle|G(O_{F}/\mathfrak{p}^{m})|=|G(\mathbb{F})|\cdot q^{(m-1)\dim G},\qquad|G_{\beta}(O_{F}/\mathfrak{p}^{m})|=|G_{\beta}(\mathbb{F})|\cdot q^{(m-1)\text{\rm rank}\,G}

for all $\displaystyle m>0$ . Then we have

	$\displaystyle\displaystyle\sharp\Omega^{^}=$	$\displaystyle\displaystyle\sharp\left\{\theta\in G_{\beta}(O_{F}/\mathfrak{p}^{r})^{^}\;\;\;\text{\rm s.t. $\displaystyle\theta=\psi_{\beta}$ on $\displaystyle G_{\beta}(O_{F}/\mathfrak{p}^{r})\cap K_{l}(O_{F}/\mathfrak{p}^{r})$}\right\}$
	$\displaystyle\displaystyle=$	$\displaystyle\displaystyle\left(G_{\beta}(O_{F}/\mathfrak{p}^{r}):G_{\beta}(O_{F}/\mathfrak{p}^{r})\cap K_{l}(O_{F}/\mathfrak{p}^{r})\right)=\|G_{\beta}(O_{F}/\mathfrak{p}^{l})\|$
	$\displaystyle\displaystyle=$	$\displaystyle\displaystyle\|G_{\beta}(\mathbb{F})\|\cdot q^{(l-1)\text{\rm rank}\,G}=\frac{\|G(\mathbb{F})\|}{\sharp\overline{\Omega}}\cdot q^{(l-1)\text{\rm rank}\,G}$

where $\displaystyle\overline{\Omega}\subset\mathfrak{g}(\mathbb{F})$ is the image of $\displaystyle\Omega\subset\mathfrak{g}(O_{F}/\mathfrak{p}^{l^{\prime}})$ under the canonical morphism $\displaystyle\mathfrak{g}(O_{F}/\mathfrak{p}^{l^{\prime}})\to\mathfrak{g}(\mathbb{F})$ . On the other hand we have

\displaystyle\dim\sigma_{\beta,\theta}=\begin{cases}1&:\text{\rm$\displaystyle r$ is even},\\ q^{\frac{1}{2}\dim_{\mathbb{F}}(\mathfrak{g}(\mathbb{F})/\mathfrak{g}_{\beta}(\mathbb{F}))}=q^{(\dim G-\text{\rm rank}\,G)/2}&:\text{\rm$\displaystyle r$ is odd},\end{cases}

so we have

	$\displaystyle\displaystyle\dim\delta_{\beta,\theta}$	$\displaystyle\displaystyle=\left(G(O_{F}/\mathfrak{p}^{r}):G(O_{F}/\mathfrak{p}^{r};\beta)\right)\cdot\dim\sigma_{\beta,\theta}$
		$\displaystyle\displaystyle=\sharp\overline{\Omega}\cdot q^{(r-2)(\dim G-\text{\rm rank}\,G)/2}.$		(2.3)

In our case of $\displaystyle G=Sp_{2n}$ , the following two statements are equivalent for a $\displaystyle\beta\in\mathfrak{g}(O_{F})$ :

1)

$\displaystyle\overline{\beta}\in\mathfrak{g}(K)$ is smoothly regular over $\displaystyle K$ ,
2)

the characteristic polynomial of $\displaystyle\overline{\beta}\in\mathfrak{g}(K)\subset\mathfrak{gl}_{2n}(K)$ is equal to its minimal polynomial

where $\displaystyle\overline{\beta}\in\mathfrak{g}(K)$ is the image of $\displaystyle\beta\in\mathfrak{g}(O_{F})$ by the canonical morphism $\displaystyle\mathfrak{g}(O_{F})\to\mathfrak{g}(K)$ with $\displaystyle K=F$ or $\displaystyle\mathbb{F}$ . If further $\displaystyle\overline{\beta}\in\mathfrak{g}(K)\subset\mathfrak{gl}_{2n}(K)$ is nonsingular, then $\displaystyle G_{\beta}{\otimes}_{O_{F}}K$ is connected.

Now let $\displaystyle\Omega\subset\mathfrak{g}(O_{F}/\mathfrak{p}^{l^{\prime}})$ be a $\displaystyle G(O_{F}/\mathfrak{p}^{l^{\prime}})$ -adjoint orbit of $\displaystyle\beta\!\!\pmod{\mathfrak{p}^{l^{\prime}}}\in\mathfrak{g}(O_{F}/\mathfrak{p}^{l^{\prime}})$ with $\displaystyle\beta\in\mathfrak{g}(O_{F})$ such that $\displaystyle\beta\!\!\pmod{\mathfrak{p}}\in\mathfrak{g}(\mathbb{F})\subset\mathfrak{gl}_{2n}(\mathbb{F})$ is nonsingular and smoothly regular over $\displaystyle\mathbb{F}$ . Then Theorem 2.1.1 gives a parametrization of $\displaystyle\Omega^{^}$ by a subset of the character group $\displaystyle G_{\beta}(O_{F}/\mathfrak{p}^{r})$ .

Remark 2.1.4

The assumption in Theorem 2.1.1 that the centralizer $\displaystyle G_{\beta}$ to be smooth $\displaystyle O_{F}$ -group scheme can be replaced by the surjectivity of the canonical morphisms

\displaystyle G_{\beta}(O_{F})\to G_{\beta}(O_{F}/\mathfrak{p}^{l}),\quad\mathfrak{g}_{\beta}(O_{F})\to\mathfrak{g}_{\beta}(O_{F}/\mathfrak{p}^{l}),

for all $\displaystyle l>0$ .

2.2 Symplectic spaces associated with tamely ramified extensions

Let $\displaystyle K_{+}/F$ be a tamely ramified field extension of degree $\displaystyle n>1$ and $\displaystyle K/K_{+}$ a quadratic field extension with $\displaystyle\text{\rm Gal}(K/K_{+})=\langle\tau\rangle$ . Let

\displaystyle e=e(K/F),\qquad f=f(K/F)

be the ramification index and the inertial degree of $\displaystyle K/F$ respectively. Similarly put

\displaystyle e_{+}=e(K_{+}/F),\qquad f_{+}=f(K_{+}/F).

Then we have $\displaystyle ef=2n$ and $\displaystyle e_{+}f_{+}=n$ . There exists a $\displaystyle\omega\in O_{K}$ such that $\displaystyle\omega^{\tau}=-\omega$ and $\displaystyle O_{K}=O_{K_{+}}\oplus\omega\cdot O_{K_{+}}$ . Then we have

\displaystyle\text{\rm ord}_{K}(\omega)=e(K/K_{+})-1.

Let $\displaystyle K_{0}/F$ be the maximal unramified subextension of $\displaystyle K/F$ . Then $\displaystyle K_{0}/F$ is a cyclic Galois extension whose Galois group is generated by the geometric Frobenius automorphism Fr which induces the inverse of the Frobenius automorphism $\displaystyle[x\mapsto x^{q}]$ of the residue field $\displaystyle\mathbb{K}_{0}$ over $\displaystyle\mathbb{F}$ . Since $\displaystyle K/K_{0}$ is totally ramified, there exists a prime element $\displaystyle\varpi_{K}$ of $\displaystyle K$ such that $\displaystyle\varpi_{K}^{e}\in K_{0}$ . Then $\displaystyle\{1,\varpi_{K},\varpi_{K}^{2},\cdots,\varpi_{K}^{e-1}\}$ is an $\displaystyle O_{K_{0}}$ -basis of $\displaystyle O_{K}$ . The following two propositions are proved by Shintani [13, Lemma 4-7, Cor.1, Cor.2,pp.545-546]:

Proposition 2.2.1

Put $\displaystyle\beta=\sum_{i=0}^{e-1}a_{i}\varpi_{K}^{i}\in O_{K}$ ( $\displaystyle a_{i}\in O_{K_{0}}$ ). Then $\displaystyle O_{K}=O_{F}[\beta]$ if and only if the following two conditions are satisfied:

1)

$\displaystyle a_{0}^{\text{\rm Fr}}\not\equiv a_{0}\!\!\pmod{\mathfrak{p}_{K_{0}}}$ if $\displaystyle f>1$ ,
2)

$\displaystyle a_{1}\in O_{K_{0}}^{\times}$ if $\displaystyle e>1$ .

Proposition 2.2.2

Let $\displaystyle\chi_{\beta}(t)\in O_{F}[t]$ be the characteristic polynomial of $\displaystyle\beta\in O_{K}\subset M_{n}(O_{F})$ via the regular representation with respect to an $\displaystyle O_{F}$ -basis of $\displaystyle O_{K}$ . If $\displaystyle O_{K}=O_{F}[\beta]$ , then

1)

$\displaystyle\chi_{\beta}(t)\!\!\pmod{\mathfrak{p}_{F}}\in\mathbb{F}[t]$ is the minimal polynomial of $\displaystyle\overline{\beta}\in M_{n}(\mathbb{F})$ ,
2)

$\displaystyle\chi_{\beta}(t)\!\!\pmod{\mathfrak{p}_{F}}=p(t)^{e}$ with an irreducible polynomial $\displaystyle p(t)\in\mathbb{F}[t]$ ,
3)

if $\displaystyle e>1$ , then $\displaystyle\chi_{\beta}(t)\!\!\pmod{\mathfrak{p}_{F}^{2}}$ is irreducible over $\displaystyle O_{F}/\mathfrak{p}_{F}^{2}$ .

We can prove the following

Proposition 2.2.3

Take a $\displaystyle\beta\in M_{n}(O_{F})$ whose the characteristic polynomial be

\displaystyle\chi_{\beta}(t)=t^{n}-a_{n}t^{n-1}-\cdots-a_{2}t-a_{1}.

If $\displaystyle\chi_{\beta}(t)\!\!\pmod{\mathfrak{p}_{F}}\in\mathbb{F}[t]$ is the minimal polynomial of $\displaystyle\beta\!\!\pmod{\mathfrak{p}_{F}}\in M_{n}(\mathbb{F})$ , then

1)

$\displaystyle\{X\in M_{n}(O_{F})\mid[X,\beta]=0\}=O_{F}[\beta]$ ,
2)

for any $\displaystyle m>0$ , put $\displaystyle\overline{\beta}=\!\!\pmod{\mathfrak{p}_{F}^{m}}\in M_{n}(O_{F}/\mathfrak{p}_{F}^{m})$ , then

$\displaystyle\left\{X\in M_{n}(O_{F}/\mathfrak{p}_{F}^{m})\mid[X,\overline{\beta}]=0\right\}=O_{F}/\mathfrak{p}_{F}^{m}[\overline{\beta}],$

there exists a $\displaystyle inGL_{n}(O_{F})$ such that

\displaystyle g\beta g^{-1}=\begin{bmatrix}0&0&\cdots&0&a_{1}\\ 1&0&\cdots&0&a_{2}\\ &1&\ddots&\vdots&\vdots\\ &&\ddots&0&a_{n-1}\\ &&&1&a_{n}\end{bmatrix}.

Then we have

Proposition 2.2.4

There exists a $\displaystyle\beta\in O_{K}$ such that $\displaystyle O_{K}=O_{F}[\beta]$ and $\displaystyle\beta+\beta^{\tau}=0$ if and only if $\displaystyle K/K_{+}$ is unramified or $\displaystyle K/F$ is totally ramified.

[Proof] Assume that there exists a $\displaystyle\beta\in O_{K}$ such that $\displaystyle O_{K}=O_{F}[\beta]$ and $\displaystyle\beta+\beta^{\tau}=0$ . Then $\displaystyle K=K_{+}(\beta^{2})$ . If $\displaystyle K/F$ is not totally ramified, we have $\displaystyle\beta\in O_{K}^{\times}$ by Proposition 2.2.1, and hence $\displaystyle K/K_{+}$ is an unramified extension.

Assume that $\displaystyle K/F$ is totally ramified. Then $\displaystyle K_{0}=F$ and $\displaystyle\varpi_{K}^{e}\in O_{F}$ . Since the quadratic extension $\displaystyle K/K_{+}$ is ramified, there exists a prime element $\displaystyle\beta$ of $\displaystyle K$ such that $\displaystyle\beta^{2}\in K_{+}$ . Then $\displaystyle\beta=\varepsilon\cdot\varpi_{K}$ with $\displaystyle\varepsilon\in O_{K}^{\times}$ . Put $\displaystyle\varepsilon=\sum_{i=0}^{e-1}a_{i}\varpi_{K}^{i}$ with $\displaystyle a_{i}\in O_{F}$ . Then

\displaystyle\beta=a_{e-1}\varpi_{K}^{e}+\sum_{i=1}^{e-1}a_{i-1}\varpi_{K}^{i}

with $\displaystyle a_{0}\in O_{F}^{\times}$ . Now we have $\displaystyle\beta^{\tau}=-\beta$ and $\displaystyle O_{K}=O_{F}[\beta]$ by Proposition 2.2.1.

Assume that $\displaystyle K/K_{+}$ is unramified. Let $\displaystyle K_{+0}/F$ be the maximal unramified subextension of $\displaystyle K_{+}/F$ . Since $\displaystyle(K_{+0}:F)=f_{+}$ divides $\displaystyle(K_{0}:F)=f$ , we have $\displaystyle K_{+0}\subset K_{0}$ . We can chose $\displaystyle\varpi_{K}$ in $\displaystyle K_{+}$ so that $\displaystyle\varpi_{K}^{e}\in K_{+0}$ . For the residue fields, we have

\displaystyle(\mathbb{K}_{0}:\mathbb{K}_{+0})=\frac{(\mathbb{K}_{0}:\mathbb{F})}{(\mathbb{K}_{+0}:\mathbb{F})}=\frac{f}{f_{+}}=2.

Put $\displaystyle\mathbb{K}_{+0}=\mathbb{F}[\overline{\alpha}]$ with $\displaystyle\alpha\in O_{K_{+0}}^{\times}$ such that $\displaystyle\overline{\alpha}\not\in\left(\mathbb{K}_{+0}^{\times}\right)^{2}$ . Since $\displaystyle\mathbb{K}_{0}$ is the splitting filed of $\displaystyle f(X)=X^{2}-\overline{\alpha}\in\mathbb{K}_{+0}[X]$ , there exists $\displaystyle\gamma\in O_{K_{0}}^{\times}$ such that

\displaystyle f(\gamma)\equiv 0\!\!\pmod{\mathfrak{p}_{K_{0}}},\quad f^{\prime}(\gamma)\not\equiv 0\!\!\pmod{\mathfrak{p}_{K_{0}}}.

Hence there exists $\displaystyle a\in O_{K_{0}}^{\times}$ such that $\displaystyle f(a)=0$ and $\displaystyle a\equiv\gamma\!\!\pmod{\mathfrak{p}_{K_{0}}}$ . Since

\displaystyle\mathbb{K}_{0}=\mathbb{F}[\overline{\gamma}]=\mathbb{F}[\overline{a}],

we have $\displaystyle a^{\text{\rm Fr}}\not\equiv a\!\!\pmod{\mathfrak{p}_{K_{0}}}$ and $\displaystyle a^{\tau}=-a$ . Put $\displaystyle\beta=a(1+\varpi_{K})\in O_{K}$ , then $\displaystyle O_{K}=O_{F}[\beta]$ by Proposition 2.2.1 and $\displaystyle\beta^{\tau}=-\beta$ . $\displaystyle\blacksquare$

From now on let us assume that $\displaystyle K/K_{+}$ is unramified or $\displaystyle K/F$ is totally ramified, and take a $\displaystyle\beta\in O_{K}$ such that $\displaystyle O_{K}=O_{F}[\beta]$ and $\displaystyle\beta^{\tau}+\beta=0$ . Fix a prime element $\displaystyle\varpi_{K_{+}}$ of $\displaystyle K_{+}$ . Then a symplectic form on $\displaystyle F$ -vector space $\displaystyle K$ is defined by

\displaystyle D(x,y)=\frac{1}{2}T_{K/F}\left(\omega^{-1}\varpi_{K_{+}}^{1-e_{+}}x^{\tau}y\right)\quad(x,y\in K).

For any $\displaystyle a\in K$ , we have $\displaystyle D(xa,y)=D(x,y\cdot a^{\tau})$ for all $\displaystyle x,y\in K$ . Inparticular

\displaystyle\beta\in\mathfrak{sp}(K,D)=\{X\in\text{\rm End}_{F}(K)\mid D(xX,y)+D(x,yX)=0\;\forall x,y\in K\}

if we put $\displaystyle K\subset\text{\rm End}_{F}(K)$ by the regular representation.

Let $\displaystyle\{u_{i}\}_{1\leq i\leq n}$ be an $\displaystyle O_{F}$ -basis of $\displaystyle O_{K_{+}}$ . Then $\displaystyle x\mapsto\left(T_{K_{+}/F}(u_{1}x),\cdots,T_{K_{+}/F}(u_{n}x)\right)$ gives an isomorphism $\displaystyle\mathfrak{p}_{K_{+}}^{1-e_{+}}\,\tilde{\to}\,O_{F}^{n}$ of $\displaystyle O_{F}$ -module. Hence there exists an $\displaystyle O_{F}$ -basis $\displaystyle\{u^{\ast}_{i}\}_{1\leq i\leq n}$ of $\displaystyle\mathfrak{p}_{K_{+}}^{1-e_{+}}$ such that $\displaystyle T_{K_{+}/F}(u_{i}u^{\ast}_{j})=\delta_{ij}$ . Put $\displaystyle v_{i}=\omega\cdot\varpi_{K_{+}}^{e_{+}-1}\cdot u^{\ast}_{i}$ ( $\displaystyle 1\leq i\leq n$ ). Then $\displaystyle\{u_{1},\cdots,u_{n},v_{n},\cdots,v_{1}\}$ is a $\displaystyle O_{F}$ -basis of $\displaystyle O_{K}$ and a symplectic $\displaystyle F$ -basis of $\displaystyle K$ , that is

\displaystyle D(u_{i},u_{j})=D(v_{i},v_{j})=0,\quad D(u_{i},v_{j})=\delta_{ij}\quad(1\leq i,j\leq n).

This means that our $\displaystyle O_{F}$ -group scheme $\displaystyle G=Sp_{2n}$ is defined by the symplectic $\displaystyle F$ -space $\displaystyle(K,D)$ and the symplectic basis $\displaystyle\{u_{i},v_{j}\}_{1\leq i,j\leq n}$ .

By Proposition 2.2.2, the characteristic polynomial of $\displaystyle\overline{\beta}=\beta\!\!\pmod{\mathfrak{p}_{F}}\in M_{2n}(\mathbb{F})$ is equal to its minimal polynomial. Then, by Proposition 2.2.3, we have

\displaystyle\{X\in M_{2n}(O_{F})\mid[X,\beta]=0\}=O_{F}[\beta]=O_{K}

and

\displaystyle\{X\in M_{2n}(O_{F}/\mathfrak{p}_{F}^{l})\mid[X,\overline{\beta}]=0\}=O_{F}/\mathfrak{p}_{F}^{l}[\overline{\beta}]=O_{K}/\mathfrak{p}_{K}^{el}

for any $\displaystyle m>0$ . Put

\displaystyle U_{K/K_{+}}=\{\varepsilon\in O_{K}^{\times}\mid N_{K/K_{+}}(\varepsilon)=1\}.

Then we have

\displaystyle G_{\beta}(O_{F})=G(O_{F})\cap O_{K}=U_{K/K_{+}}.

We have also

\displaystyle\mathfrak{g}_{\beta}(O_{F})=\mathfrak{g}(O_{F})\cap O_{K}=\{X\in O_{K}\mid T_{K/K_{+}}(X)=0\}

and

	$\displaystyle\displaystyle G_{\beta}(O_{F}/\mathfrak{p}_{F}^{l})$	$\displaystyle\displaystyle=\{\overline{\varepsilon}\in\left(O_{K}/\mathfrak{p}_{K}^{el}\right)^{\times}\mid N_{K/K_{+}}(\varepsilon)\equiv 1\!\!\pmod{\mathfrak{p}_{K_{+}}^{e_{+}l}}\},$
	$\displaystyle\displaystyle\mathfrak{g}_{\beta}(O_{F}/\mathfrak{p}_{F}^{l})$	$\displaystyle\displaystyle=\{\overline{X}\in O_{K}/\mathfrak{p}_{K}^{el}\mid T_{K/K_{+}}(X)\equiv 0\!\!\pmod{\mathfrak{p}_{K_{+}}^{e_{+}l}}\}$

for all $\displaystyle l>0$ . Then the canonical morphisms

\displaystyle G_{\beta}(O_{F})\to G_{\beta}(O_{F}/\mathfrak{p}_{F}^{l}),\qquad\mathfrak{g}_{\beta}(O_{F})\to\mathfrak{g}_{\beta}(O_{F}/\mathfrak{p}_{F}^{l})

are surjective for all $\displaystyle l>0$ . In fact, Take a $\displaystyle\varepsilon\in O_{K}^{\times}$ such that $\displaystyle N_{K/K_{+}}(\varepsilon)\equiv 1\!\!\pmod{\mathfrak{p}_{K_{+}}^{e_{+}l}}$ . Because $\displaystyle K/K_{+}$ is tamely ramified, we have $\displaystyle N_{K/K_{+}}(1+\mathfrak{p}_{K}^{el})=1+\mathfrak{p}_{K_{+}}^{e_{+}l}$ . Hence there exists a $\displaystyle\eta\in 1+\mathfrak{p}_{K}^{el}$ such that $\displaystyle N_{K/K_{+}}(\eta)=\varepsilon$ . Then $\displaystyle\alpha=\varepsilon\eta^{-1}\in O_{K}^{\times}$ such that $\displaystyle N_{K/K_{+}}(\alpha)=1$ and $\displaystyle\alpha=\varepsilon\!\!\pmod{\mathfrak{p}_{K}^{el}}$ . Take a $\displaystyle X\in O_{K}$ such that $\displaystyle T_{K/K_{+}}(X)\equiv 0\!\!\pmod{\mathfrak{p}_{K_{+}}^{e_{+}l}}$ . If we put $\displaystyle X=s+\omega t$ with $\displaystyle s,t\in O_{K_{+}}$ , then $\displaystyle s\in\mathfrak{p}_{K_{+}}^{e_{+}l}\subset\mathfrak{p}_{K}^{el}$ . Hence we have $\displaystyle\omega t\in\mathfrak{g}_{\beta}(O_{F})$ and $\displaystyle\omega t\equiv X\!\!\pmod{\mathfrak{p}_{K}^{el}}$ .

Due to Remark 2.1.4, we can apply the general theory of subsection 2.1 to our $\displaystyle\beta\in\mathfrak{g}(O_{F})$ . Take an integer $\displaystyle r>1$ and put $\displaystyle r=l+l^{\prime}$ with minimal integer $\displaystyle l$ such that $\displaystyle 0<l^{\prime}\leq l$ . Let $\displaystyle\Omega\subset\mathfrak{g}(O_{F}/\mathfrak{p}_{F}^{l^{\prime}})$ be the adjoint $\displaystyle G(O_{F}/\mathfrak{p}_{F}^{l^{\prime}})$ -orbit of $\displaystyle\beta\!\!\pmod{\mathfrak{p}_{F}^{l^{\prime}}}\in\mathfrak{g}(O_{F}/\mathfrak{p}_{F}^{l^{\prime}})$ , and $\displaystyle\Omega^{^}$ the set of the equivalent classes of the irreducible representations of $\displaystyle G(O_{F}/\mathfrak{p}_{F}^{r})$ corresponding to $\displaystyle\Omega$ via Clifford’s theory described in subsection 2.1. Then we have a bijection $\displaystyle\theta\mapsto\delta_{\beta,\theta}$ of the continuous unitary character $\displaystyle\theta$ of $\displaystyle U_{K/K_{+}}$ such that

1)

$\displaystyle\theta$ factors through the canonical morphism $\displaystyle U_{K/K_{+}}\to\left(O_{K}/\mathfrak{p}_{K}^{er}\right)^{\times}$ ,
2)

for an $\displaystyle\alpha\in U_{K/K_{+}}$ such that $\displaystyle\alpha\equiv 1+\varpi_{F}^{l}x\!\!\pmod{\mathfrak{p}_{K}^{er}}$ with $\displaystyle x\in O_{K}$ such that $\displaystyle T_{K/K_{+}}(x)\equiv 0\!\!\pmod{\mathfrak{p}_{K_{+}}^{e_{+}l^{\prime}}}$ , we have $\displaystyle\theta(\alpha)=\psi\left(\varpi_{F}^{-l^{\prime}}T_{K/F}(x\beta)\right)$ .

onto $\displaystyle\Omega^{^}$ . Here $\displaystyle\psi:F\to\mathbb{C}^{\times}$ is a continuous unitary character of the additive group $\displaystyle F$ such that $\displaystyle\{x\in F\mid\psi(xO_{F})=1\}=O_{F}$ . Then we have

Proposition 2.2.5

\displaystyle\dim\delta_{\beta,\theta}=q^{n^{2}r}\cdot\prod_{k=1}^{n}\left(1-q^{-2k}\right)\times\begin{cases}\frac{1}{2}&:\text{\rm$\displaystyle K/F$ is totally ramified},\\ \frac{1}{1+q^{-f_{+}}}&:\text{\rm$\displaystyle K/K_{+}$ is unramified}.\end{cases}

[Proof] For the dimension formula (2.3), we have

\displaystyle\dim G=n(2n+1),\quad\text{\rm rank}\,G=n,\quad\sharp\overline{\Omega}=\frac{|G(\mathbb{F})|}{|G_{\beta}(\mathbb{F})|}

and

\displaystyle|G(\mathbb{F})|=|Sp_{2n}(\mathbb{F})|=q^{n(2n+1)}\cdot\prod_{k=1}^{n}\left(1-q^{-2k}\right).

On the other hand $\displaystyle G_{\beta}(\mathbb{F})$ is the kernel of

\displaystyle(\ast):\left(O_{K}/\mathfrak{p}_{K}^{e}\right)^{\times}\to\left(O_{K_{+}}/\mathfrak{p}_{K_{+}}^{e_{+}}\right)^{\times}\quad\left(\overline{\varepsilon}\mapsto\overline{N_{K/K_{+}}(\varepsilon)}\right).

Since $\displaystyle K/K_{+}$ is tamely ramified quadratic extension, we have

\displaystyle 1+\mathfrak{p}_{K_{+}}^{e_{+}}=N_{K/K_{+}}(1+\mathfrak{p}_{K}^{e})\subset N_{K/K_{+}}(O_{K}^{\times})\subset O_{K_{+}}^{\times},

and $\displaystyle(O_{K_{+}}^{\times}:N_{K/K_{+}}(O_{K}^{\times}))=e/e_{+}$ , hence

	$\displaystyle\displaystyle\|G_{\beta}(\mathbb{F})\|$	$\displaystyle\displaystyle=\frac{\left\|\left(O_{K}/\mathfrak{p}_{K}^{e}\right)^{\times}\right\|(O_{K_{+}}^{\times}:N_{K/K_{+}}(O_{K}^{\times}))}{\left\|\left(O_{K_{+}}/\mathfrak{p}_{K_{+}}^{e_{+}}\right)^{\times}\right\|}=\frac{e}{e_{+}}\cdot q^{n}\cdot\frac{1-q^{-f}}{1-q^{-f_{+}}}$
		$\displaystyle\displaystyle=q^{n}\times\begin{cases}2&:\text{\rm$\displaystyle K/F$ is totally ramified},\\ 1+q^{-f_{+}}&:\text{\rm$\displaystyle K/K_{+}$ is unramified}.\end{cases}$

$\displaystyle\blacksquare$

2.3 Construction of supercuspidal representations

We will keep the notations of the preceding subsection. The purpose of this subsection is to prove the following theorem:

Theorem 2.3.1

If $\displaystyle l^{\prime}=\left\lfloor\frac{r}{2}\right\rfloor\geq\text{\rm Max}\{2,2(e-1)\}$ , then the compactly induced representation $\displaystyle\pi_{\beta,\theta}=\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$ is an irreducible supercuspidal representation of $\displaystyle G(F)=Sp_{2n}(F)$ such that

1)

the multiplicity of $\displaystyle\delta_{\beta,\theta}$ in $\displaystyle\pi_{\beta,\theta}|_{G(O_{F})}$ is one,
2)

$\displaystyle\delta_{\beta,\theta}$ is the unique irreducible unitary constituent of $\displaystyle\pi_{\beta,\theta}|_{G(O_{F})}$ which factors through the canonical morphism $\displaystyle G(O_{F})\to G(O_{F}/\mathfrak{p}^{r})$ ,

with respect to the Haar measure on $\displaystyle G(F)$ such that the volume of $\displaystyle G(O_{F})$ is one, the formal degree of $\displaystyle\pi_{\beta,\theta}$ is equal to

\displaystyle\dim\delta_{\beta,\theta}=q^{n^{2}r}\cdot\prod_{k=1}^{n}\left(1-q^{-2k}\right)\times\begin{cases}\frac{1}{2}&:\text{\rm$\displaystyle K/F$ is totally ramified,}\\ \frac{1}{1+q^{-f_{+}}}&:\text{\rm$\displaystyle K/K_{+}$ is unramified}.\end{cases}

The rest of this subsection is devoted to the proof.

We have the Cartan decomposition

\displaystyle G(F)=\bigsqcup_{m\in\mathbb{M}}G(O_{F})t(m)G(O_{F})

where

\displaystyle\mathbb{M}=\{(m_{1},m_{2},\cdots,m_{n})\in\mathbb{Z}^{n}\mid m_{1}\geq m_{2}\geq\cdots\geq m_{n}\geq 0\}

and

\displaystyle t(m)=\begin{bmatrix}\varpi_{F}^{m}&0\\ 0&{}^{\mathfrak{t}}\varpi_{F}^{-m}\end{bmatrix}\;\;\text{\rm with}\;\;\varpi_{F}^{m}=\begin{bmatrix}\varpi_{F}^{m_{1}}&&\\ &\ddots&\\ &&\varpi_{F}^{m_{n}}\end{bmatrix}

for $\displaystyle m=(m_{1},\cdots,m_{n})\in\mathbb{M}$ .

For an integer $\displaystyle 1\leq i\leq n$ , let $\displaystyle L_{i}$ and $\displaystyle U_{i}$ be $\displaystyle O_{F}$ -group subscheme of $\displaystyle G=Sp_{2n}$ defined by

	$\displaystyle\displaystyle L_{i}$	$\displaystyle\displaystyle=\left\{\begin{bmatrix}a&&\\ &g&\\ &&{}^{\mathfrak{t}}a^{-1}\end{bmatrix}\biggm{\|}a\in GL_{i},g\in Sp_{2(n-i)}\right\},$
	$\displaystyle\displaystyle U_{i}$	$\displaystyle\displaystyle=\left\{\begin{bmatrix}1_{i}&A&B&C\\ &1_{n-i}&0&{}^{\mathfrak{t}}B\\ &&1_{n-i}&-\,^{\mathfrak{t}}A\\ &&&1_{i}\end{bmatrix}\in Sp_{2n}\right\}$

so that $\displaystyle P_{i}=L_{i}\cdot U_{i}$ is a maximal parabolic subgroup of $\displaystyle G=Sp_{2n}$ and $\displaystyle U_{i}$ (resp. $\displaystyle L_{i}$ ) is the unipotent (resp. Levi) part of $\displaystyle P_{i}$ . Put $\displaystyle U_{i}(\mathfrak{p}_{F}^{a})=U_{i}(O_{F})\cap K_{a}(O_{F})$ for a positive integer $\displaystyle a$ .

Proposition 2.3.2

If $\displaystyle K/F$ is unramified or $\displaystyle r\geq 4$ , then the compactly induced representation $\displaystyle\pi_{\beta,\theta}=\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$ is an admissible representation of $\displaystyle G(F)$ .

[Proof] It is enough to show that $\displaystyle\dim_{\mathbb{C}}\text{\rm Hom}_{K_{a}(O_{F})}(\text{\bf 1},\pi_{\beta,\theta})<\infty$ for all $\displaystyle a>0$ , where 1 is the trivial one-dimensional representation of $\displaystyle K_{a}(O_{F})$ . We have

\displaystyle G(F)=\bigsqcup_{s\in\mathbb{S}}K_{a}(O_{F})sG(O_{F})

where

\displaystyle\mathbb{S}=\{k\cdot t(m)\mid\dot{k}\in K_{a}(O_{F})\backslash G(O_{F}),m\in\mathbb{M}\}.

Then, by the restriction formula of induced representations and by the Frobenius reciprocity, we have

	$\displaystyle\displaystyle\text{\rm Hom}_{K_{a}(O_{F})}(\text{\bf 1},\pi_{\beta,\theta})$	$\displaystyle\displaystyle=\bigoplus_{s\in\mathbb{S}}\text{\rm Hom}_{K_{a}(O_{F})}\left(\text{\bf 1},\text{\rm ind}_{K_{a}(O_{F})\cap sF(O_{F})s^{-1}}^{G(F)}\delta_{\beta,\theta}^{s}\right)$
		$\displaystyle\displaystyle=\bigoplus_{s\in\mathbb{S}}\text{\rm Hom}_{K_{a}(O_{F})\cap sG(O_{F})s^{-1}}(\text{\bf 1},\delta_{\beta,\theta}^{s})$
		$\displaystyle\displaystyle=\bigoplus_{s\in\mathbb{S}}\text{\rm Hom}_{s^{-1}K_{a}(O_{F})s\cap G(O_{F})}(\text{\bf 1},\delta_{\beta,\theta}).$

So it is enough to show that the number of $\displaystyle s\in\mathbb{S}$ such that $\displaystyle\text{\rm Hom}_{s^{-1}K_{a}(O_{F})s\cap G(O_{F})}(\text{\bf 1},\delta_{\beta,\theta})\neq 0$ is finite. Take such a $\displaystyle s=k\cdot t\in\mathbb{S}$ with $\displaystyle k\in G(O_{F})$ and $\displaystyle t=t(m)$ ( $\displaystyle m\in\mathbb{M}$ ). Suppose

\displaystyle\text{\rm Max}\{m_{k}-m_{k+1}\mid 1\leq k<n\}=m_{i}-m_{i+1}\geq a.

Then we have $\displaystyle tU_{i}(\mathfrak{p}_{F}^{a})t^{-1}\subset K_{a}(O_{F})$ and hence

\displaystyle U_{i}(\mathfrak{p}_{F}^{l})\subset U_{i}(O_{F})\subset s^{-1}K_{a}(O_{F})s\cap G(O_{F})

and

\displaystyle\text{\rm Hom}_{U_{i}(\mathfrak{p}_{F}^{l})}(\text{\bf 1},\delta_{\beta,\theta})\supset\text{\rm Hom}_{s^{-1}K_{a}(O_{F})s\cap G(O_{F})}(\text{\bf 1},\delta_{\beta,\theta})\neq 0.

This means, by (2.1), that there exists a $\displaystyle g\in G(O_{F})$ such that $\displaystyle\chi_{\text{\rm Ad}(g)\beta}(h)=1$ for all $\displaystyle h\in U_{i}(\mathfrak{p}_{F}^{l})$ , that is $\displaystyle\psi\left(\varpi_{F}^{-l^{\prime}}\text{\rm tr}\left(g\beta g^{-1}X\right)\right)=1$ for all $\displaystyle X\in\text{\rm Lie}(U_{i})(O_{F})$ . Hence we have $\displaystyle\overline{g\beta g^{-1}}\in\text{\rm Lie}(P_{i})(O_{F}/\mathfrak{p}_{F}^{l^{\prime}})$ . Then the characteristic polynomial $\displaystyle\chi_{\beta}(t)\!\!\pmod{\mathfrak{p}_{F}}\in\mathbb{F}[t]$ is reducible. Hence $\displaystyle e>1$ by Proposition 2.2.2. Then $\displaystyle l^{\prime}=\left\lfloor\frac{r}{2}\right\rfloor\geq 2$ and $\displaystyle\chi_{\beta}(t)\!\!\pmod{\mathfrak{p}_{F}^{2}}$ is reducible over $\displaystyle O_{F}/\mathfrak{p}_{F}^{2}$ contradicting to Proposition 2.2.2. Hence we have

\displaystyle\text{\rm Max}\{m_{i}-m_{i+1}\mid 1\leq i<n\}<a.

Similar arguments using the parabolic subgroup $\displaystyle P_{n}$ shows that $\displaystyle 2m<a$ . This shows the required finiteness of $\displaystyle s\in\mathbb{S}$ . $\displaystyle\blacksquare$

Lemma 2.3.3

1)

If $\displaystyle\text{\rm Hom}_{U_{i}(\mathfrak{p}_{F}^{r-1})}(\text{\bf 1},\delta_{\beta,\theta})\neq 0$ for some $\displaystyle 1\leq i\leq n$ , then $\displaystyle i\equiv 0\!\!\pmod{f}$ and $\displaystyle e>1$ . If further $\displaystyle i<n$ , then $\displaystyle e\geq 3$ .
2)

If $\displaystyle\frac{r}{2}\geq 2$ , then $\displaystyle\text{\rm Hom}_{U_{i}(\mathfrak{p}_{F}^{r-2})}(\text{\bf 1},\delta_{\beta,\theta})=0$ for all $\displaystyle 1\leq i\leq n$ .

[Proof] Assume that $\displaystyle\text{\rm Hom}_{U_{i}(\mathfrak{p}_{F}^{k})}(\text{\bf 1},\delta_{\beta,\theta}\neq 0$ with some $\displaystyle 0<k\leq l^{\prime}$ . Then $\displaystyle U_{i}(\mathfrak{p}_{F}^{r-k})\subset K_{l}(O_{F})$ and (2.1) implies that there exists a $\displaystyle g\in G(O_{F})$ such that $\displaystyle\chi_{\text{\rm Ad}(G)\beta}(h)=1$ for all $\displaystyle h\in U_{i}(\mathfrak{p}_{F}^{r-k})$ , that is $\displaystyle\psi\left(\varpi_{F}^{-k}\text{\rm tr}\left(g\beta g^{-1}X\right)\right)=1$ for all $\displaystyle X\in\text{\rm Lie}(U_{i})(O_{F})$ . Then

\displaystyle g\beta g^{-1}\equiv\begin{bmatrix}A&\ast&\ast\\ 0&X&\ast\\ 0&0&-\,^{\mathfrak{t}}A\end{bmatrix}\!\!\pmod{\mathfrak{p}_{F}^{k}}

with $\displaystyle A\in\mathfrak{gl}_{i}(O_{F})$ and $\displaystyle X\in\mathfrak{sp}_{2(n-i)}(O_{F})$ . So the characteristic polynomial is

\displaystyle\chi_{\beta}(t)\equiv\det(t1_{i}-A)\det(t1_{2(n-i)}-X)\det(1_{i}+A)\!\!\pmod{\mathfrak{p}_{F}^{k}}.

If $\displaystyle k=1$ , then the first statement of Proposition 2.2.2 implies that

\displaystyle i=\deg\det(t1_{i}-A)\equiv 0\!\!\pmod{f}\;\;\text{\rm and}\;\;e>1.

If $\displaystyle l^{\prime}\geq 2$ and $\displaystyle k=2$ , then $\displaystyle\chi_{\beta}(t)\!\!\pmod{\mathfrak{p}_{F}^{2}}$ is reducible over $\displaystyle O_{F}/\mathfrak{p}_{F}^{2}$ contradicting to the third statement of Proposition 2.2.2. $\displaystyle\blacksquare$

Proposition 2.3.4

Assume that $\displaystyle l^{\prime}=\left\lfloor\frac{r}{2}\right\rfloor\geq\text{\rm Max}\{2,2(e-1)\}$ . Then

1)

$\displaystyle\dim_{\mathbb{C}}\text{\rm Hom}_{G(O_{F})}\left(\delta_{\beta,\theta},\pi_{\beta,\theta}\right)=1$ ,
2)

for any irreducible representation $\displaystyle(\delta,V_{\delta})$ of $\displaystyle G(O_{F})$ which factors through the canonical morphism $\displaystyle G(O_{F})\to G(O_{F}/\mathfrak{p}_{F}^{r})$ , if $\displaystyle\text{\rm Hom}_{G(O_{F})}(\delta,\pi_{\beta,\theta})\neq 0$ , then $\displaystyle\delta=\delta_{\beta,\theta}$ .

[Proof] Let $\displaystyle(\delta,V_{\delta})$ be an irreducible unitary representation of $\displaystyle G(O_{F})$ which factors through the canonical morphism $\displaystyle G(O_{F})\to G(O_{F}/\mathfrak{p}_{F}^{r})$ . Then we have

	$\displaystyle\displaystyle\text{\rm Hom}_{G(O_{F})}(\delta,\pi_{\beta,\theta})$	$\displaystyle\displaystyle=\bigoplus_{m\in\mathbb{M}}\text{\rm Hom}_{G(O_{F})}\left(\delta,\text{\rm ind}_{G(O_{F})\cap t(m)G(O_{F})t(m)^{-1}}^{G(O_{F})}\delta_{\beta,\theta}^{t(m)}\right)$
		$\displaystyle\displaystyle=\bigoplus_{m\in\mathbb{M}}\text{\rm Hom}_{G(O_{F})\cap t(m)G(O_{F})t(m)^{-1}}\left(\delta,\delta_{\beta,\theta}^{t(m)}\right)$
		$\displaystyle\displaystyle=\bigoplus_{m\in\mathbb{M}}\text{\rm Hom}_{t(m)^{-1}G(O_{F})t(m)\cap G(O_{F})}\left(\delta^{t(m)^{-1}},\delta_{\beta,\theta}\right).$

Assume that $\displaystyle\text{\rm Hom}_{t(m)^{-1}G(O_{F})t(m)\cap G(O_{F})}\left(\delta^{t(m)^{-1}},\delta_{\beta,\theta}\right)\neq 0$ for a $\displaystyle m=(m_{1},\cdots,m_{n})\in\mathbb{M}$ . If

\displaystyle\text{\rm Max}\{m_{k}-m_{k+1}\mid 1\leq k<n\}=m_{i}-m_{i+1}\geq 2

then we have $\displaystyle t(m)U_{i}(\mathfrak{p}_{F}^{r-2})t(m)^{-1}\subset U_{i}(\mathfrak{p}_{F}^{r})$ . Since $\displaystyle K_{r}(O_{F})\subset\text{\rm Ker}(\beta)$ , the restriction of $\displaystyle\delta^{t(m)^{-1}}$ to $\displaystyle U_{i}(\mathfrak{p}_{F}^{r-2})$ is trivial. On the other hand, we have

\displaystyle U_{i}(\mathfrak{p}_{F}^{r-2})\subset t(m)^{-1}U_{i}(\mathfrak{p}_{F}^{r})t(m)\cap U_{i}(O_{F})\subset t(m)^{-1}G(O_{F})t(m)\cap G(O_{F}).

Now we have $\displaystyle\text{\rm Hom}_{U_{i}(\mathfrak{p}_{F}^{r-2})}(\text{\bf 1},\delta_{\beta,\theta})\neq 0$ contradicting to the second statement of Lemma 2.3.3. Hence we have

\displaystyle\text{\rm Max}\{m_{k}-m_{k+1}\mid 1\leq k<n\}\leq 1.

Similarly we have $\displaystyle 2m_{n}\leq 1$ , that is $\displaystyle m_{n}=0$ . If there exists $\displaystyle 1\leq i<n$ such that $\displaystyle m_{i}-m_{i+1}=1$ . Then, with the similar arguments as above, we have $\displaystyle\text{\rm Hom}_{U_{i}(\mathfrak{p}_{F}^{r-1})}(\text{\bf 1},\delta_{\beta,\theta})\neq 0$ . The first statement of Lemma 2.3.3 implies that $\displaystyle i\equiv 0\!\!\pmod{f}$ . Since $\displaystyle ef=2n$ , this means $\displaystyle m_{1}<\frac{e}{2}$ , hence

4m_{1}\leq 2(e-1)\leq l^{\prime}.

(2.4)

Since $\displaystyle t(m)K_{l+2m_{1}}(O_{F})t(m)^{-1}\subset K_{l}(O_{F})$ and hence

\displaystyle K_{l+2m_{1}}(O_{F})\subset t(m)^{-1}G(O_{F})t(m)\cap G(O_{F}),

we have

\displaystyle\text{\rm Hom}_{K_{l+2m_{1}}}\left(\delta^{t(m)^{-1}},\delta_{\beta,\theta}\right)\supset\text{\rm Hom}_{t(m)^{-1}G(O_{F})t(m)\cap G(O_{F})}\left(\delta^{t(m)^{-1}},\delta_{\beta,\theta}\right)\neq 0.

Assume that $\displaystyle\delta$ corresponds, as explained in subsection 2.1, to an adjoint $\displaystyle G(O_{F}/\mathfrak{p}_{F}^{l^{\prime}})$ -orbit $\displaystyle\Omega^{\prime}\subset\mathfrak{g}(O_{F}/\mathfrak{p}_{F}^{l^{\prime}})$ of $\displaystyle\beta^{\prime}\!\!\pmod{\mathfrak{p}_{F}^{l^{\prime}}}$ ( $\displaystyle\beta^{\prime}\in\mathfrak{g}(O_{F})$ ). Then there exists $\displaystyle k,h\in G(O_{F})$ such that

\displaystyle\chi_{\text{\rm Ad}(k)\beta}(x)=\chi_{\text{\rm Ad}(h)\beta^{\prime}}(t(m)xt(m)^{-1})

for all $\displaystyle x\in K_{l+2m_{1}}(O_{F})$ . This means

\displaystyle k\beta k^{-1}\equiv t(m)^{-1}h\beta^{\prime}h^{-1}t(m)\!\!\pmod{\mathfrak{p}_{F}^{l^{\prime}-2m_{1}}}.

Then, because of (2.4), the matrix $\displaystyle t(m)k\beta k^{-1}t(m)^{-1}$ belongs to

\displaystyle h\beta^{\prime}h^{-1}+t(m)M_{2n}(\mathfrak{p}_{F}^{l^{\prime}-2m_{1}})t(m)^{-1}\subset h\beta^{\prime}h^{-1}+M_{2n}(\mathfrak{p}_{F}^{l^{\prime}-4m_{1}})\subset M_{2n}(O_{F}).

Since the characteristic polynomials of $\displaystyle t(m)k\beta k^{-1}t(m)^{-1}$ and $\displaystyle\beta$ are identical, there exists, by the third statement of Proposition 2.2.3, a $\displaystyle g\in GL_{2n}(O_{F})$ such that $\displaystyle t(m)k\beta k^{-1}t(m)^{-1}=g\beta g^{-1}$ . Then $\displaystyle g^{-1}t(m)k\in F[\beta]=K$ and

\displaystyle N_{K/F}(g^{-1}t(m)k)=\det(g^{-1}t(m)k)\in O_{F}^{\times}.

Hence $\displaystyle g^{-1}t(m)k\in O_{K}\subset M_{2n}(O_{F})$ and $\displaystyle t(m)\in M_{2n}(O_{F})$ , that is $\displaystyle m=(0,\cdots,0)$ . So we have proved

\displaystyle\text{\rm Hom}_{G(O_{F})}(\delta,\pi_{\beta,\theta})=\text{\rm Hom}_{G(O_{F})}(\delta,\delta_{\beta,\theta})

which clearly implies the statements of the proposition. $\displaystyle\blacksquare$

The admissible representation $\displaystyle\pi_{\beta,\theta}=\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$ of $\displaystyle G(F)$ is irreducible. In fact, if there exists a $\displaystyle G(F)$ -subspace $\displaystyle 0\lvertneqq W\lvertneqq\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$ , we have

	$\displaystyle\displaystyle 0\neq\text{\rm Hom}_{G(F)}(W,\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta})$	$\displaystyle\displaystyle\subset\text{\rm Hom}_{G(O_{F})}(W,\text{\rm Ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta})$
		$\displaystyle\displaystyle=\text{\rm Hom}_{G(O_{F})}(W,\delta)$

by Frobenius reciprocity. Hence $\displaystyle\delta\hookrightarrow W|_{G(O_{F})}$ . On the other hand, we have

	$\displaystyle\displaystyle 0$	$\displaystyle\displaystyle\neq\text{\rm Hom}_{G(F)}\left(\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta},\left(\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}\right)/W\right)$
		$\displaystyle\displaystyle=\text{\rm Hom}_{G(O_{F})}\left(\delta_{\beta,\theta},\left(\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}\right)/W\right),$

hence $\displaystyle\delta\hookrightarrow\left(\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}\right)/W$ . Now $\displaystyle\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$ is semi-simple $\displaystyle G(O_{F})$ -module, we have

\displaystyle\dim_{\mathbb{C}}\text{\rm Hom}_{G(O_{F})}(\delta_{\beta,\theta},\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta})\geq 2

which contradicts to the first statement of Proposition 2.3.4.

Now $\displaystyle\pi_{\beta,\theta}$ is a supercuspidal representation of $\displaystyle G(F)$ whose formal degree with respect to the Haar measure $\displaystyle d_{G(F)}(x)$ of $\displaystyle G(F)$ such that $\displaystyle\int_{G(O_{F})}d_{G(F)}(x)=1$ is equal to $\displaystyle\dim\delta_{\beta,\theta}$ . We have completed the proof of Theorem 2.3.1.

3 Kaleta’s $\displaystyle L$ -parameter

3.1 Local Langlands correspondence of elliptic tori

Let $\displaystyle K_{+}/F$ be a finite extension, $\displaystyle K/K_{+}$ a quadratic extension with a non-trivial element $\displaystyle\tau$ of $\displaystyle\text{\rm Gal}(K/K_{+})$ . Let us denote by $\displaystyle L$ an arbitrary Galois extension over $\displaystyle F$ containing $\displaystyle K$ for which let us denote by

\displaystyle\text{\rm Emb}_{F}(K,L)=\left\{\sigma|_{K}\mid\sigma\in\text{\rm Gal}(L/F)\right\}

the set of the embeddings over $\displaystyle F$ of $\displaystyle K$ into $\displaystyle L$ .

Put $\displaystyle O_{K}=O_{K_{+}}\oplus\omega O_{K_{+}}$ with $\displaystyle\omega^{\tau}+\omega=0$ . Then $\displaystyle\text{\rm ord}_{K}(\omega)=e(K/K_{+})-1$ . Let us denote by $\displaystyle\mathbb{V}$ the $\displaystyle\overline{F}$ -algebra of the functions $\displaystyle v$ on $\displaystyle\text{\rm Emb}_{F}(K,\overline{F})$ with values in $\displaystyle\overline{F}$ which is endowed with a symplectic $\displaystyle\overline{F}$ -form

\displaystyle D(u,v)=\frac{1}{2}\sum_{\gamma\in\text{\rm Emb}_{F}(K,\overline{F})}\left(\omega^{-1}\varpi_{K_{+}}^{-d_{+}}\right)^{\gamma}u(\tau\gamma)\cdot v(\gamma)

( $\displaystyle u,v\in\mathbb{V}$ ) where $\displaystyle\mathcal{D}(K_{+}/F)=\mathfrak{p}_{K_{+}}^{d_{+}}$ is the difference of $\displaystyle K_{+}/F$ . The action of $\displaystyle\sigma\in\text{\rm Gal}(\overline{F}/F)$ on $\displaystyle v\in\mathbb{V}$ is defined by $\displaystyle v^{\sigma}(\gamma)=v(\gamma\sigma^{-1})^{\sigma}$ . Then fixed point subspace $\displaystyle{\mathbb{V}}^{\text{\rm Gal}(\overline{F}/L)}=\mathbb{V}(L)$ is the set of the functions on $\displaystyle\text{\rm Emb}_{F}(K,L)$ with values in $\displaystyle L$ , and $\displaystyle{\mathbb{V}}^{\text{\rm Gal}(\overline{F}/F)}=\mathbb{V}(F)$ is identified with $\displaystyle K$ via $\displaystyle v\mapsto v(\text{\bf 1}_{K})$ .

The action of $\displaystyle\sigma\in\text{\rm Gal}(\overline{F}/F)$ on $\displaystyle g\in Sp(\mathbb{V},D)$ is defined by $\displaystyle v\cdot g^{\sigma}=(v^{\sigma^{-1}}\cdot g)^{\sigma}$ . Then the fixed point subgroup $\displaystyle Sp(\mathbb{V},D)^{\text{\rm Gal}(\overline{F}/F)}$ is identified with $\displaystyle Sp(K,D)$ via $\displaystyle g\mapsto g|_{K}$ .

Put $\displaystyle S=\text{\rm Res}_{K/F}\mathbb{G}_{m}$ which is identified with the multiplicative group $\displaystyle\mathbb{V}^{\times}$ . Then $\displaystyle S(F)$ is identified with the multiplicative group $\displaystyle K^{\times}$ .

Let $\displaystyle T$ be a subtorus of $\displaystyle S$ wich is identified with the multiplicative subgroup of $\displaystyle\mathbb{V}^{\times}$ consisting of the functions $\displaystyle s$ on $\displaystyle\text{\rm Emb}_{F}(K,\overline{F})$ to $\displaystyle{\overline{F}}^{\times}$ such that $\displaystyle s(\tau\gamma)=s(\gamma)^{-1}$ for all $\displaystyle\gamma\in\text{\rm Emb}_{F}(K,\overline{F})$ . In other words $\displaystyle T$ is a maximal torus of $\displaystyle Sp(\mathbb{V},D)$ by identifying $\displaystyle s\in T$ with $\displaystyle[v\mapsto v\cdot s]\in Sp(\mathbb{V},D)$ . The fixed point subgroup $\displaystyle T^{\text{\rm Gal}(\overline{F}/F)}=T(F)$ is identified with

\displaystyle U_{K/K_{+}}=\{\varepsilon\in O_{K}^{\times}\mid N_{K/K_{+}}(\varepsilon)=1\}\quad\text{\rm by $\displaystyle s\mapsto s(\text{\bf 1}_{K})$. }

The group $\displaystyle X(S)$ of the characters over $\displaystyle\overline{F}$ of $\displaystyle S$ is a free $\displaystyle\mathbb{Z}$ -module with $\displaystyle\mathbb{Z}$ -basis $\displaystyle\{b_{\delta}\}_{\delta\in\text{\rm Emb}_{F}(K,\overline{F})}$ where $\displaystyle b_{\delta}(s)=s(\delta)$ for $\displaystyle s\in S$ . The dual torus $\displaystyle S^{^}=X(S){\otimes}_{\mathbb{Z}}\mathbb{C}^{\times}$ is identified with the group of the functions $\displaystyle s$ on $\displaystyle\text{\rm Emb}_{F}(K,\overline{F})$ with values in $\displaystyle\mathbb{C}^{\times}$ . The action of $\displaystyle\sigma\in W_{F}\subset\text{\rm Gal}(\overline{F}/F)$ on $\displaystyle S$ induces the action on $\displaystyle X(S)$ such that $\displaystyle b_{\delta}^{\sigma}=b_{\delta\sigma}$ , and hence the action on $\displaystyle s\in S^{^}$ is defined by $\displaystyle s^{\sigma}(\gamma)=s(\gamma\sigma^{-1})$ .

Since we have a bijection $\displaystyle\dot{\rho}\mapsto\rho|_{K}$ of $\displaystyle W_{K}\backslash W_{F}$ onto $\displaystyle\text{\rm Emb}_{F}(K,\overline{F})$ , the $\displaystyle\overline{F}$ -algebra $\displaystyle\mathbb{V}$ (resp. the torus $\displaystyle S$ , $\displaystyle S^{^}$ ) is identified with the set of the left $\displaystyle W_{K}$ -invariant functions on $\displaystyle W_{F}$ with values in $\displaystyle\mathbb{\overline{}}\overline{F}$ (resp. $\displaystyle\overline{F}^{\times}$ , $\displaystyle\mathbb{C}^{\times}$ ). If we denote by $\displaystyle\widetilde{\tau}\in W_{K_{+}}$ a pull back of $\displaystyle\tau\in\text{\rm Gal}(K/K_{+})$ by the restriction mapping $\displaystyle W_{K_{+}}\to\text{\rm Gal}(K/K_{+})$ , the torus $\displaystyle T$ is identified with the set of $\displaystyle s\in S$ such that $\displaystyle s(\widetilde{\tau}\rho)=s(\rho)^{-1}$ for all $\displaystyle\rho\in W_{F}$ . Note that

\displaystyle\widetilde{\tau}^{2}\!\!\pmod{\overline{[W_{K},W_{K}]}}=\delta_{K}(\alpha_{K/K_{+}}(\tau,\tau))

where $\displaystyle[\alpha_{K/K_{+}}]\in H^{2}(\text{\rm Gal}(K/K_{+}),K^{\times})$ is the fundamental class which gives the isomorphism

\displaystyle\text{\rm Gal}(K/K_{+})\,\tilde{\to}\,K_{+}^{\times}/N_{K/K_{+}}(K^{\times})\quad(\sigma\mapsto\alpha_{K/K_{+}}(\sigma,\tau)).

The local Langlands correspondence for the torus $\displaystyle S$ is the isomorphism

H^{1}(W_{F},S^{^})\,\tilde{\to}\,\text{\rm Hom}(W_{K},\mathbb{C}^{\times})

(3.1)

given by $\displaystyle[\alpha]\mapsto[\rho\mapsto\alpha(\rho)(\text{\bf 1}_{K})]$ . The inverse mapping is defined as follows. Let

\displaystyle l:\text{\rm Emb}_{F}(K,\overline{F})\to W_{F}

be a section of the restriction mapping $\displaystyle W_{F}\to\text{\rm Emb}_{F}(K,\overline{F})$ , that is $\displaystyle l(\gamma)|_{K}=\gamma$ for all $\displaystyle\gamma\in\text{\rm Emb}_{F}(K,\overline{F})$ and $\displaystyle l(\text{\bf 1}_{K})=1$ , and put

\displaystyle J(\gamma,\sigma)=l(\gamma)\sigma l(\gamma\sigma)^{-1}\in W_{K}\;\;\text{\rm for $\displaystyle\gamma\in\text{\rm Emb}_{F}(K,\overline{F})$, $\displaystyle\sigma\in W_{F}$.}

Take a $\displaystyle\psi\in\text{\rm Hom}(W_{K},\mathbb{C}^{\times})$ and define $\displaystyle\alpha\in Z^{1}(W_{F},S^{^})$ by

\displaystyle\alpha(\sigma)(\rho)=\alpha(\sigma\rho^{-1})(1)\cdot\alpha(\rho^{-1})(1)\;\;\text{\rm with}\;\;\alpha(\sigma)(1)=\psi\left(J(\text{\bf 1}_{K},\sigma^{-1})^{-1}\right)

for all $\displaystyle\sigma,\rho\in W_{F}$ . Then $\displaystyle\psi\mapsto[\alpha]$ is the inverse mapping of the isomorphism (3.1).

If we restrict the isomorphism (3.1) to continuous group homomorphisms, we have an isomorphism

H^{1}_{\text{\rm conti}}(W_{F},S^{^})\,\tilde{\to}\,\text{\rm Hom}_{\text{\rm conti}}(K^{\times},\mathbb{C}^{\times})

(3.2)

via (3.1) combined with the isomorphism of the local class filed theory

\displaystyle\delta_{K}:K^{\times}\,\tilde{\to}\,W_{K}/\overline{[W_{K},W_{K}]}.

The surjection $\displaystyle x\mapsto x^{1-\tau}$ of $\displaystyle K^{\times}$ onto $\displaystyle U_{K/K_{+}}$ gives a canonical inclusion

\text{\rm Hom}_{\text{\rm conti.}}(U_{K/K_{+}},\mathbb{C}^{\times})\subset\text{\rm Hom}_{\text{\rm conti.}}(K^{\times},\mathbb{C}^{\times}).

(3.3)

The restriction from $\displaystyle S$ to $\displaystyle T$ gives a surjection $\displaystyle X(S)\to X(T)$ whose kernel is the subgroup of $\displaystyle X(S)$ generated by $\displaystyle\{b_{\delta}+b_{\tau\delta}\mid\delta\in\text{\rm Emb}_{F}(K,L)\}$ . Then the dual torus $\displaystyle T^{^}=X(T){\otimes}_{\mathbb{Z}}\mathbb{C}^{\times}$ is identified with the group of the functions $\displaystyle s$ on $\displaystyle\text{\rm Emb}_{F}(K,\overline{F})$ with values in $\displaystyle\mathbb{C}^{\times}$ such that $\displaystyle s(\tau\gamma)=s(\gamma)^{-1}$ for all $\displaystyle\gamma\in\text{\rm Emb}_{F}(K,\overline{F})$ . As above $\displaystyle T^{^}$ is identified with the set of the left $\displaystyle W_{K}$ -invariant functions $\displaystyle s$ of $\displaystyle W_{F}$ with values in $\displaystyle\mathbb{C}^{\times}$ such that $\displaystyle s(\widetilde{\tau}\rho)=s(\rho)^{-1}$ for all $\displaystyle\rho\in W_{F}$ .

Then we have

Proposition 3.1.1

1)

The inclusion $\displaystyle T^{^}\subset S^{^}$ gives a canonical inclusion

$H^{1}_{\text{\rm conti.}}(W_{F},T^{^})\subset H^{1}_{\text{\rm conti.}}(W_{F},S^{^}).$ (3.4)
2)

The restriction of the isomorphism (3.2) to these included subgroups (3.4) and (3.3) gives the isomorphism

$H^{1}_{\text{\rm conti}}(W_{F},T^{^})\,\tilde{\to}\,\text{\rm Hom}_{\text{\rm conti}}(U_{K/K_{+}},\mathbb{C}^{\times}).$ (3.5)

[Proof] See [20] for the arguments with general tori. A direct proof for our specific setting is as follows.

1) Take a $\displaystyle\beta\in Z^{1}(W_{F},T^{^})\subset Z^{1}(W_{F},S^{^})$ such that $\displaystyle\beta\in B^{1}(W_{F},S^{^})$ , that is, there exists a $\displaystyle s\ S^{^}$ such that $\displaystyle\beta(\sigma)=s^{\beta-1}$ for all $\displaystyle\sigma\in W_{F}$ . Chose a $\displaystyle\varepsilon\in\mathbb{C}^{\times}$ such that $\displaystyle\varepsilon^{2}=s(\text{\bf 1}_{K})\cdot s(\tau)$ . The relation $\displaystyle\beta(\sigma)(\tau)=\beta(\sigma)(\text{\bf 1}_{K})^{-1}$ for all $\displaystyle\sigma\in W_{F}$ implies

\displaystyle s(\sigma|)\cdot s(\widetilde{\tau}\sigma)=s(\text{\bf 1}_{K})\cdot s(\tau)=\varepsilon^{2}

for all $\displaystyle\sigma\in W_{F}$ . Then $\displaystyle t=[\sigma\mapsto s(\sigma)\varepsilon^{-1}]$ is an element of $\displaystyle T^{^}$ such that $\displaystyle t^{\sigma-1}=\beta(\sigma)$ for all $\displaystyle\sigma\in W_{F}$ .

2) Put

\displaystyle\text{\rm Emb}_{F}(K,\overline{F})=\{\gamma_{i},\tau\gamma_{i}\mid 1\leq i\leq n\}

and let $\displaystyle l:\text{\rm Emb}_{F}(K,\overline{F})\to W_{F}$ be a section of the restriction mapping $\displaystyle W_{F}\to\text{\rm Emb}_{F}(K,\overline{F})$ such that $\displaystyle l(\tau\gamma_{i})=\widetilde{\tau}l(\gamma_{i})$ ( $\displaystyle 1\leq i\leq n$ ). Take a $\displaystyle\theta\in\text{\rm Hom}_{\text{\rm conti.}}(K^{\times},\mathbb{C}^{\times})$ which corresponds to $\displaystyle\alpha\in Z^{1}(W_{F},S^{^})$ , that is

\displaystyle\alpha(\sigma)(\text{\bf 1}_{K})=\theta(x)

for $\displaystyle\sigma\in W_{F}$ with $\displaystyle x\in K^{\times}$ such that $\displaystyle J(\text{\bf 1}_{K},\sigma^{-1})^{-1}\!\!\pmod{\overline{[W_{K},W_{K}]}}=\delta_{K}(x)$ . For any $\displaystyle\sigma\in W_{F}$ , we have

\displaystyle\alpha(\sigma\widetilde{\tau}^{-1})=\begin{cases}\theta(x^{\tau})&:\sigma^{-1}|_{K}=\gamma_{i},\\ \theta(\alpha_{K/K_{+}}(\tau,\tau)\cdot x^{\tau})&:\sigma^{-1}|_{K}=\tau\gamma_{i}.\end{cases}

Since

\displaystyle\alpha(\sigma)(\rho)=\alpha(\sigma\rho^{-1})(\text{\bf 1}_{K})\cdot\alpha(\rho^{-1})(\text{\bf 1}_{K})^{-1}

for all $\displaystyle\sigma,\rho\in W_{F}$ and $\displaystyle K_{+}^{\times}=N_{K/K_{+}}(K^{\times})\sqcup\alpha_{K/K_{+}}(\tau,\tau)N_{K/K_{+}}(K^{\times})$ , we have $\displaystyle\alpha\in Z^{1}(W_{F},T^{^})$ if and only if $\displaystyle\theta(N_{K/K_{+}}(K^{\times})=1$ , that is, there exists $\displaystyle c\in\text{\rm Hom}_{\text{\rm conti.}}(U_{K/K_{+}},\mathbb{C}^{\times})$ such that $\displaystyle\theta(x)=c(x^{1-\tau})$ ( $\displaystyle x\in K^{\times}$ ). $\displaystyle\blacksquare$

Put ${}^{L}\!T=W_{F}\ltimes T^{^}$ . Then a cohomology class $\displaystyle[\alpha]\in H^{1}_{\text{\rm conti}}(W_{F},T^{^})$ defines a continuous group homomorphism

\widetilde{\alpha}:W_{F}\to\,^{L}T\quad(\sigma\mapsto(\sigma,\alpha(\sigma)))

(3.6)

and $\displaystyle[\alpha]\mapsto\widetilde{\alpha}$ induces a well-defined bijection

\displaystyle H^{1}_{\text{\rm conti}}(W_{F},T^{^})\,\tilde{\to}\,\text{\rm Hom}_{\text{\rm conti}}^{\ast}(W_{F},^{L}T)/\text{\rm``$\displaystyle T^{^}$-conjugate"}

where $\displaystyle\text{\rm Hom}_{\text{\rm conti}}^{\ast}(W_{F},^{L}T)$ denotes the set of the continuous group homomorphisms $\displaystyle\psi$ of $\displaystyle W_{F}$ to ${}^{L}T$ such that $\displaystyle W_{F}\xrightarrow{\psi}\,^{L}T\xrightarrow{\text{\rm proj.}}W_{F}$ is the identity map.

3.2 $\displaystyle\chi$ -datum

In this subsection, let us assume that $\displaystyle K/F$ is a Galois extension and put $\displaystyle\Gamma=\text{\rm Gal}(K/F)$ . For a $\displaystyle\gamma\in\Gamma$ of order two, let us denote by $\displaystyle K_{\gamma}$ the intermediate subfield of $\displaystyle K/F$ such that $\displaystyle\text{\rm Gal}(K/K_{\gamma})=\langle\gamma\rangle$ .

Let us denote by $\displaystyle SO_{2n+1}(\mathbb{C})$ the complex special orthogonal group with respect to the symmetric matrix

\displaystyle S=\begin{bmatrix}S_{1}&0\\ &-2\end{bmatrix}\;\text{\rm with}\;S_{1}=\begin{bmatrix}0&1_{n}\\ 1_{n}&0\end{bmatrix}

and put

\displaystyle\mathbb{T}^{^}=\left\{\begin{bmatrix}t&&\\ &t^{-1}&\\ &&1\end{bmatrix}\biggm{|}t=\begin{bmatrix}t_{1}&&\\ &\ddots&\\ &&t_{n}\end{bmatrix},\;t_{i}\in\mathbb{C}^{\times}\right\}

a maximal torus of $\displaystyle SO_{2n+1}(\mathbb{C})$ . We have an isomorphism $\displaystyle T^{^}\,\tilde{\to}\,\mathbb{T}^{^}$ given by

\displaystyle s\mapsto\text{\rm diag}(s(\gamma_{1}),\cdots,s(\gamma_{n}),s(\gamma_{n+1}),\cdots,s(\gamma_{2n}),1)

where $\displaystyle\text{\rm Emb}_{F}(K,\overline{F})=\{\gamma_{i}\}_{1\leq i\leq 2n}$ where $\displaystyle\gamma_{1}=\text{\bf 1}_{K}$ and $\displaystyle\gamma_{n+i}=\tau\gamma_{i}$ ( $\displaystyle 1\leq i\leq n$ ). The action of $\displaystyle W_{F}$ on $\displaystyle T^{^}$ induces the action on $\displaystyle\mathbb{T}^{^}$ which factors through $\displaystyle\Gamma$ .

The Weyl group $\displaystyle W(\mathbb{T}^{^})=N_{SO_{2n+1}(\mathbb{C})}(\mathbb{T}^{^})/\mathbb{T}^{^}$ on $\displaystyle\mathbb{T}^{^}$ is identified with a subgroup of the permutation group $\displaystyle S_{2n}$ generated by

\displaystyle\begin{pmatrix}1&\cdots&n&n+1&\cdots&2n\\ \sigma(1)&\cdots&\sigma(n)&n+\sigma(1)&\cdots&n+\sigma(n)\end{pmatrix}\;\text{\rm with $\displaystyle\sigma\in S_{n}$ and}

\displaystyle\begin{pmatrix}1&2&\cdots&n&n+1&n+2&\cdots&2n\\ n+1&2&\cdots&n&1&n+2&\cdots&2n\end{pmatrix}.

Then any $\displaystyle w\in W(\mathbb{T}^{^})$ is represented by

\displaystyle\widetilde{w}=\begin{bmatrix}[w]&0\\ 0&\det[w]\end{bmatrix}\in N_{SO_{2n+1}(\mathbb{C})}(\mathbb{T}^{^}),

where $\displaystyle[w]\in GL_{2n}(\mathbb{Z})$ is the permutation matrix corresponding to $\displaystyle w\in W(\mathbb{T}^{^})\subset S_{2n}$ .

For any $\displaystyle\gamma\in\text{\rm Emb}_{F}(K,\overline{F})=\Gamma$ , let us denote by $\displaystyle a_{\gamma}$ an element of $\displaystyle X(T^{^})$ such that $\displaystyle a_{\gamma}(s)=s(\gamma)$ for all $\displaystyle s\in T^{^}$ . Then

\displaystyle\Phi(T^{^})=\left\{a_{\gamma}\cdot a_{\gamma^{\prime}},a_{\gamma}\mid\gamma,\gamma^{\prime}\in\Gamma,\;\gamma\neq\gamma^{\prime}\right\}.

is the set of the roots of $\displaystyle SO_{2n+1}(\mathbb{C})$ with respect to $\displaystyle T^{^}=\mathbb{T}^{^}$ with the simple roots

\displaystyle\Delta=\{\alpha_{i}=a_{\gamma_{i}}\cdot a_{\tau\gamma_{i+1}},\alpha_{n}=a_{\gamma_{n}}\mid 1\leq i<n\}.

Let $\displaystyle\{X_{\alpha},X_{-\alpha},H_{\alpha}\}$ be the standard triple associate with a simple root $\displaystyle\alpha\in\Delta$ . Then $\displaystyle s_{\alpha}\in W(\mathbb{T}^{^})$ is represented by

\displaystyle n(s_{\alpha})=\exp(X_{\alpha})\cdot\exp(-X_{-\alpha})\cdot\exp(X_{\alpha})\in N_{SO_{2n+1}(\mathbb{C})}(\mathbb{T}^{^})

and $\displaystyle W(\mathbb{T}^{^})$ is generated by $\displaystyle S=\{s_{\alpha}\}_{\alpha\in\Delta}$ . For any $\displaystyle w\in W(\mathbb{T}^{^})$ , let $\displaystyle w=s_{1}s_{2}\cdots s_{r}$ ( $\displaystyle s_{i}\in S$ ) be a reduced presentation and put

\displaystyle n(w)=n(s_{1})n(s_{2})\cdots n(s_{r})\in N_{SO_{2n+1}(\mathbb{C})}(\mathbb{T}^{^}).

Then $\displaystyle r(w)=\widetilde{w}^{-1}n(w)\in\mathbb{T}^{^}$ .

The action of $\displaystyle\sigma\in W_{F}$ on $\displaystyle X(T^{^})$ induced from the action on $\displaystyle T^{^}$ is such that $\displaystyle a_{\gamma}^{\sigma}=a_{\gamma\sigma}$ for all $\displaystyle\gamma\in\text{\rm Emb}_{F}(K,\overline{F})$ , and it determines an element $\displaystyle w(\sigma)\in W(\mathbb{T}^{^})$ . Then [11] shows that the $\displaystyle 2$ -cocycle $\displaystyle t\in Z^{2}(W_{F},\mathbb{T}^{^})$ defined by

\displaystyle t(\sigma,\sigma^{\prime})=n(w(\sigma\sigma^{\prime}))^{-1}n(w(\sigma))\cdot n(w(\sigma^{\prime}))\quad(\sigma,\sigma^{\prime}\in W_{F})

is split by $\displaystyle r_{p}:W_{F}\to\mathbb{T}^{^}$ defined by $\displaystyle\chi$ -data as follows.

For any $\displaystyle\lambda\in\Phi(T^{^})$ , put

\displaystyle\Gamma_{\lambda}=\{\sigma\in\Gamma\mid\lambda^{\sigma}=\lambda\},\quad\Gamma_{\pm\lambda}=\{\sigma\in\Gamma\mid\lambda^{\sigma}=\pm\lambda\}

and put $\displaystyle F_{\lambda}=L^{\Gamma_{\lambda}}$ , $\displaystyle F_{\pm\lambda}=L^{\Gamma_{\pm\lambda}}$ . Then $\displaystyle(F_{\lambda}:F_{\pm\lambda})=\text{\rm$\displaystyle 1$ or $\displaystyle 2$}$ , and $\displaystyle\lambda$ is called symmetric if $\displaystyle(F_{\lambda}:F_{\pm\lambda})=2$ .

The Galois group $\displaystyle\Gamma$ acts on $\displaystyle\Phi(T^{^})$ and

\displaystyle\Phi(T^{^})/\Gamma=\{a_{\text{\bf 1}_{K}}a_{\gamma},a_{\text{\bf 1}_{K}}\mid 1\neq\gamma\in\Gamma\}.

If $\displaystyle\lambda=a_{\text{\bf 1}_{K}}a_{\gamma}$ , then $\displaystyle\lambda$ is symmetric if and only if $\displaystyle\gamma\neq\tau$ . If further $\displaystyle\gamma^{2}\neq 1$ , then $\displaystyle F_{\lambda}=K$ and $\displaystyle F_{\pm\lambda}=K_{+}$ and choose a continuous character $\displaystyle\chi_{\lambda}:F_{\lambda}^{\times}=K^{\times}\to\mathbb{C}^{\times}$ such that $\displaystyle\chi_{\lambda}|_{F_{\pm\lambda}^{\times}}:K_{+}^{\times}\to\{\pm 1\}$ is the character of the quadratic extension $\displaystyle K/K_{+}$ . We may assume that $\displaystyle\chi_{a_{\text{\bf 1}_{K}}a_{\gamma^{-1}}}=\chi_{a_{\text{\bf 1}_{K}}a_{\gamma}}^{-1}$ .

If $\displaystyle\gamma^{2}=1$ , then $\displaystyle F_{\lambda}=K_{\gamma}$ and $\displaystyle F_{\pm\lambda}=E=K_{\gamma}\cap K_{+}$ and choose a continuous character $\displaystyle\chi_{\lambda}:F_{\lambda}^{\times}=K_{\gamma}^{\times}\to\mathbb{C}^{\times}$ such that $\displaystyle\chi_{\lambda}|_{F_{\pm\lambda}^{\times}}:E^{\times}\to\{\pm 1\}$ is the character of the quadratic extension $\displaystyle K_{\gamma}/E$ .

If $\displaystyle\lambda=a_{\text{\bf 1}_{K}}$ then $\displaystyle F_{\lambda}=K$ and $\displaystyle F_{\pm\lambda}=K_{+}$ and choose a continuous character $\displaystyle\chi_{\lambda}:F_{\lambda}^{\times}=K^{\times}\to\mathbb{C}^{\times}$ such that $\displaystyle\chi_{\lambda}|_{F_{\pm\lambda}^{\times}}:K_{+}^{\times}\to\{\pm 1\}$ is the character of the quadratic extension $\displaystyle K/K_{+}$ .

These characters are parts of a system of $\displaystyle\chi$ -data $\displaystyle\chi_{\lambda}:F_{\lambda}\to\mathbb{C}^{\times}$ ( $\displaystyle\lambda\in\Phi(\mathbb{T}^{^})$ ) such that

1)

$\displaystyle\chi_{-\lambda}=\chi_{\lambda}^{-1}$ and $\displaystyle\chi_{\lambda^{\sigma}}=\chi_{\lambda}(x^{\sigma^{-1}})$ for all $\displaystyle\sigma\in\Gamma$ , and
2)

$\displaystyle\chi_{\lambda}=1$ if $\displaystyle\lambda$ is not symmetric.

With this $\displaystyle\chi$ -data and the gauge

\displaystyle p:\Phi(\mathbb{T}^{^})\to\{\pm 1\}\;\text{\rm s.t.}\;p(\lambda)=\begin{cases}1&:\lambda>0,\\ -1&:\lambda<0,\end{cases}

the mechanism of [11] gives a $\displaystyle r_{p}:W_{F}\to\mathbb{T}^{^}$ such that

\displaystyle t(\sigma,\sigma^{\prime})=r_{p}(\sigma)^{\sigma^{\prime}}r_{p}(\sigma\sigma^{\prime})^{-1}r_{p}(\sigma^{\prime})\;\;\text{\rm for all $\displaystyle\sigma,\sigma^{\prime}\in W_{F}$}

and

	$\displaystyle\displaystyle r_{p}(\sigma)=$	$\displaystyle\displaystyle\prod_{\gamma\in\Gamma,\gamma^{2}\neq 1}\prod_{0<\lambda\in\{a_{\text{\bf 1}_{K}}a_{\gamma}\}_{\Gamma}}\chi_{\lambda}(x)^{\check{\lambda}}\times\prod_{\stackrel{{\scriptstyle\scriptstyle\gamma\in\Gamma,\gamma^{2}=1}}{{\gamma\neq 1,\tau}}}\prod_{0<\lambda\in\{a_{\text{\bf 1}_{K}}a_{\gamma}\}_{\Gamma}}\chi_{\lambda}(N_{K/F_{\lambda}}(x))^{\check{\lambda}}$
		$\displaystyle\displaystyle\times\prod_{0<\lambda\in\{a_{\text{\bf 1}_{K}}\}_{\Gamma}}\chi_{\lambda}(x)^{\check{\lambda}}$

if $\displaystyle\dot{\sigma}=(1,x)\in W_{K/F}=\Gamma\ltimes_{\alpha_{K/F}}K^{\times}$ , where $\displaystyle\{\alpha\}_{\Gamma}$ is the $\displaystyle\Gamma$ -orbit of $\displaystyle\alpha\in\Phi(\mathbb{T}^{^})$ and $\displaystyle\check{\lambda}$ is the co-root of $\displaystyle\lambda$ . Then we have a group homomorphism

{}^{L}T=W_{F}\ltimes\mathbb{T}^{^}\to SO_{2n+1}(\mathbb{C})\quad((\sigma,s)\mapsto n(w(\sigma))r_{p}(\sigma)^{-1}s).

(3.7)

If we put $\displaystyle r(\sigma)=r(w(\sigma))$ for $\displaystyle\sigma\in W_{F}$ , we have

\displaystyle t(\sigma,\sigma^{\prime})=r(\sigma)^{\sigma^{\prime}}r(\sigma\sigma^{\prime})^{-1}r(\sigma^{\prime})\qquad(\sigma,\sigma^{\prime}\in W_{F}).

Now $\displaystyle\chi_{p}(\sigma)=r(\sigma)\cdot r_{p}(\sigma)^{-1}$ ( $\displaystyle\sigma\in W_{F}$ ) define an element of $\displaystyle Z^{1}(W_{F},\mathbb{T}^{^})$ and the group homomorphism (3.7) is

{}^{L}T=W_{F}\ltimes\mathbb{T}^{^}\to SO_{2n+1}(\mathbb{C})\quad((\sigma,s)\mapsto\widetilde{w}(\sigma)\chi_{p}(\sigma)\cdot s).

(3.8)

Let $\displaystyle c\in\text{\rm Hom}_{\text{\rm conti}}(U_{K/K_{+}},\mathbb{C}^{\times})$ be the character corresponding to the cohomology class $\displaystyle[\chi_{p}]\in H_{\text{\rm conti.}}^{1}(W_{F},\mathbb{T}^{^})$ by the local Langlands correspondence of torus (3.5).

3.3 Explicit value of $\displaystyle c(-1)$

From now on, we will assume that $\displaystyle K/F$ is a tamely ramified Galois extension and put $\displaystyle\Gamma=\text{\rm Gal}(K/F)$ .

The structure of the Galois group $\displaystyle\text{\rm Gal}(K/F)$ is well understood:

\text{\rm Gal}(K/F)=\langle\delta,\rho\rangle

(3.9)

where $\displaystyle\text{\rm Gal}(K/K_{0})=\langle\delta\rangle$ with the maximal unramified subextension $\displaystyle K_{0}/F$ of $\displaystyle K/F$ and $\displaystyle\rho|_{K_{0}}\in\text{\rm Gal}(K_{0}/F)$ is the inverse of the Frobenius automorphism. There exists a prime element $\displaystyle\varpi_{K}$ of $\displaystyle K$ such that $\displaystyle\varpi_{K}^{e}\in K_{0}$ . Then $\displaystyle\sigma\mapsto\varpi_{K}^{1-\sigma}\!\!\pmod{\mathfrak{p}_{K}}$ is an injective group homomorphism of $\displaystyle\text{\rm Gal}(K/K_{0})$ into $\displaystyle\mathbb{K}^{\times}$ , and hence $\displaystyle e|q^{f}-1$ . Put $\displaystyle\rho^{f}=\delta^{m}$ with $\displaystyle 0\leq m<e$ . We have a relation $\displaystyle\rho^{-1}\delta\rho=\delta^{q}$ due to Iwasawa [9] and hence

\displaystyle\delta^{m}=\rho^{-1}\delta^{m}\rho=\delta^{qm}

that is $\displaystyle m(q-1)\equiv 0\!\!\pmod{e}$ . So we have

\rho^{f(q-1)}=1

(3.10)

Since $\displaystyle f$ divides $\displaystyle\text{\rm ord}(\rho)$ , we have

\displaystyle\text{\rm ord}(\rho)=f\cdot\frac{e}{\text{\rm GCD}\{e,m\}}.

The structure of the elements of order two in $\displaystyle\text{\rm Gal}(K/F)$ plays an important role in our arguments, and we have

Proposition 3.3.1

$\displaystyle H=\{\gamma\in\text{\rm Gal}(K/F)\mid\gamma^{2}=1\}\subset Z(\text{\rm Gal}(K/F))$ and

\displaystyle H=\begin{cases}\{1,\delta^{\frac{e}{2}}\}&:f=\text{\rm odd}\;\text{\rm or}\;\left\{\begin{array}[]{l}e=\text{\rm even},\\ m=\text{\rm odd}\end{array}\right.\\ \{1,\rho^{\frac{f}{2}}\delta^{-\frac{m}{2}}\}&:e=\text{\rm odd}\,,m=\text{\rm even}\\ \{1,\rho^{\frac{f}{2}}\delta^{\frac{e-m}{2}}\}&:e=\text{\rm odd}\,,m=\text{\rm odd}\\ \{1,\delta^{\frac{e}{2}},\rho^{\frac{f}{2}}\delta^{-\frac{m}{2}},\rho^{\frac{f}{2}}\delta^{\frac{e-m}{2}}\}&:f=\text{\rm even}\,,e=\text{\rm even}\,,m=\text{\rm even}.\end{cases}

For $\displaystyle\gamma\in\text{\rm Gal}(K/F)$ of order two, the quadratic extension $\displaystyle K/K_{\gamma}$ is ramified if and only if $\displaystyle\gamma\in\text{\rm Gal}(K/K_{0})$ .

[Proof] Take a $\displaystyle 1\neq\gamma\in\text{\rm Gal}(K/F)$ such that $\displaystyle\gamma^{2}=1$ .

If $\displaystyle\gamma\in\text{\rm Gal}(K/K_{0})$ , then $\displaystyle e$ is even and $\displaystyle\gamma=\delta^{\frac{e}{2}}$ is the unique element of order $\displaystyle 2$ of the normal subgroup $\displaystyle\text{\rm Gal}(K/K_{0})$ . So $\displaystyle\gamma\in Z(\text{\rm Gal}(K/F))$ . In this case $\displaystyle K_{0}\subset K_{\gamma}$ and $\displaystyle K/K_{\gamma}$ is ramified extension.

Assume that $\displaystyle\gamma\not\in\text{\rm Gal}(K/K_{0})$ . Then $\displaystyle\gamma|_{K_{0}}\in\text{\rm Gal}(K_{0}/F)$ is of order two (hence $\displaystyle f=2f^{\prime}$ is even), and $\displaystyle\gamma=\rho^{f^{\prime}}\delta^{a}$ with $\displaystyle 0\leq a<e$ . Then $\displaystyle K/K_{\gamma}$ is unramified extension, because if it was not the case we have $\displaystyle f(K_{\gamma}/F)=f(K/F)$ and hence $\displaystyle K_{0}\subset K_{\gamma}$ which means

\displaystyle\gamma\in\text{\rm Gal}(K/K_{\gamma})\subset\text{\rm Gal}(K/K_{0}),

contradicting to the assumption $\displaystyle\gamma\not\in\text{\rm Gal}(K/K_{0})$ . Then $\displaystyle f(K_{\gamma}/F)=f^{\prime}$ and $\displaystyle e(K_{\gamma}/F)=e$ , and hence $\displaystyle e|q^{f^{\prime}}-1$ . So we have

\displaystyle 1=\gamma^{2}=\rho^{f}\rho^{-f^{\prime}}\delta^{a}\rho^{f^{\prime}}\delta^{a}=\delta^{m+aq^{f^{\prime}}+a}=\delta^{2a+m},

hence $\displaystyle 2a\equiv-m\!\!\pmod{e}$ . Then $\displaystyle a\equiv-\frac{m}{2}\;\text{\rm or}\;\frac{e-m}{2}\!\!\pmod{e}$ if $\displaystyle e$ is even (hence $\displaystyle m$ is even), and

\displaystyle a\!\!\pmod{e}=\begin{cases}-\frac{m}{2}&:\text{\rm if $\displaystyle m$ is even},\\ \frac{e-m}{2}&:\text{\rm if $\displaystyle m$ is odd}\end{cases}

if $\displaystyle e$ is odd. We have $\displaystyle e|q^{f^{\prime}}-1$ hence

\displaystyle\delta\gamma=\rho^{f^{\prime}}\delta^{q^{f^{\prime}}+a}=\rho^{f^{\prime}}\delta^{1+a}=\gamma\delta.

Now we have

\rho^{f^{\prime}(q-1)}=1.

(3.11)

In fact $\displaystyle\text{\rm Gal}(K_{\gamma}/F)=\langle\delta^{\prime},\rho^{\prime}\rangle$ with $\displaystyle\delta^{\prime}=\delta|_{K_{\gamma}},\rho^{\prime}=\rho|_{K_{\gamma}}$ . Then $\displaystyle(\rho^{\prime})^{f^{\prime}(q-1)}=1$ , that is

\displaystyle\rho^{f^{\prime}(q-1)}\in\text{\rm Gal}(K/K_{\gamma})=\langle\gamma\rangle.

If $\displaystyle\rho^{f^{\prime}(q-1)}\neq 1$ , then $\displaystyle\rho^{f^{\prime}(q-1)}=\gamma=\rho^{f^{\prime}}\delta^{a}$ , therefore

\displaystyle\rho^{f^{\prime}q}=\rho^{f}\delta^{a}=\delta^{m+a}\in\text{\rm Gal}(K/K_{0})

and hence $\displaystyle f$ divides $\displaystyle f^{\prime}q$ , contradicting to the assumption that $\displaystyle q$ is odd. Now we have

\displaystyle\gamma\rho=\rho^{f^{\prime}+1}\delta^{qa}=\rho\gamma\cdot\delta^{a(q-1)}.

For $\displaystyle a=-\frac{m}{2}$ or $\displaystyle a=\frac{e-m}{2}$ , we have $\displaystyle a(q-1)\equiv 0\!\!\pmod{e}$ if and only if

\displaystyle\frac{q-1}{2}\equiv 0\!\!\pmod{\frac{e}{\text{\rm GCD}\{e,m\}}}

which is equivalent to $\displaystyle\rho^{f\cdot\frac{q-1}{2}}=\rho^{f^{\prime}(q-1)}=1$ . Then (3.11) implies $\displaystyle\gamma\rho=\rho\gamma$ . Then we have $\displaystyle\gamma$ is an element of the center of $\displaystyle\text{\rm Gal}(K/F)$ . $\displaystyle\blacksquare$

Put $\displaystyle\widetilde{c}(x)=c(x^{1-\tau})$ for $\displaystyle x\in K^{\times}$ . Then we have

	$\displaystyle\displaystyle\widetilde{c}(x)$	$\displaystyle\displaystyle=\chi_{p}(1,x)(\text{\bf 1}_{K})$
		$\displaystyle\displaystyle=\prod_{\gamma\in\Gamma,\gamma^{2}\neq 1}\chi_{a_{\text{\bf 1}_{K}}a_{\gamma}}(x)\times\prod_{\stackrel{{\scriptstyle\scriptstyle\gamma\in\Gamma,\gamma^{2}=1}}{{\gamma\neq 1,\tau}}}\chi_{a_{\text{\bf 1}_{K}}a_{\gamma}}(N_{K/K_{\gamma}}(x))\times\chi_{a_{\text{\bf 1}_{K}}}(x)^{2}.$

Since $\displaystyle\chi_{a_{\text{\bf 1}_{K}}a_{\gamma^{-1}}}=\chi_{a_{\text{\bf 1}_{K}}a_{\gamma}}^{-1}$ for $\displaystyle\gamma\in\Gamma$ , we have

\displaystyle\widetilde{c}(x)=\chi_{a_{\text{\bf 1}_{K}}}(x^{1-\tau})\quad(x\in K^{\times})

if $\displaystyle H=\{1,\tau\}$ , and

\displaystyle\widetilde{c}(x)=\chi_{a_{\text{\bf 1}_{K}}a_{\delta^{\prime}}}(N_{K/K_{\delta^{\prime}}}(x))\cdot\chi_{a_{\text{\bf 1}_{K}}a_{\tau\delta^{\prime}}}(N_{K/K_{\tau\delta^{\prime}}}(x))\cdot\chi_{a_{\text{\bf 1}_{K}}}(x^{1-\tau})\quad(x\in K^{\times})

if $\displaystyle H=\{1,\tau,\delta^{\prime}=\delta^{\frac{e}{2}},\tau\delta^{\prime}\}$ . In this case, since $\displaystyle f$ is even, $\displaystyle K/K_{+}$ is unramified so that $\displaystyle\tau\not\in\text{\rm Gal}(K/K_{0})=\langle\delta\rangle$ . We have

Proposition 3.3.2

If $\displaystyle|H|=2$ , then

\displaystyle c(-1)=\begin{cases}(-1)^{\frac{q-1}{2}}&:\text{\rm if $\displaystyle K/K_{+}$ is ramified},\\ 1&:\text{\rm if $\displaystyle K/K_{+}$ is unramified,}.\end{cases}

If $\displaystyle|H|=4$ , then

\displaystyle c(-1)=-(-1)^{\frac{q^{f_{+}}-1}{2}}.

Note that $\displaystyle K/F$ is totally ramified if $\displaystyle K/K_{+}$ is ramified.

[Proof] If $\displaystyle|H|=2$ , we have $\displaystyle c(x)=\chi_{a_{\text{\bf 1}_{K}}}(x)$ for $\displaystyle x\in U_{K/K_{+}}$ so that

\displaystyle c(-1)=\left(-1,K/K_{+}\right)=\begin{cases}1&:\text{\rm if $\displaystyle K/K_{+}$ is unramified,}\\ (-1)^{\frac{q-1}{2}}&:\text{\rm if $\displaystyle K/K_{+}$ is ramified}.\end{cases}

From now on, we will consider the case of $\displaystyle|H|=4$ . Put $\displaystyle H=\{1,\tau,\delta^{\prime},\tau\delta^{\prime}\}$ and let $\displaystyle E=K^{H}$ be the fixed subfield of $\displaystyle K/F$ . Then we have

\displaystyle\text{\rm Gal}(K_{\delta^{\prime}}/E)=\langle\tau|_{K_{\delta^{\prime}}}\rangle,\quad\text{\rm Gal}(K_{\tau\delta^{\prime}}/E)=\langle\tau|_{K_{\tau\delta^{\prime}}}\rangle.

Put $\displaystyle K_{\delta^{\prime}}=E(\eta)$ with $\displaystyle\eta^{2}\in E$ , or equivalently $\displaystyle\eta^{\tau}=-\eta$ . Then we have

	$\displaystyle\displaystyle c(-1)$	$\displaystyle\displaystyle=\widetilde{c}(\eta)=\chi_{a_{\text{\bf 1}_{K}}a_{\delta^{\prime}}}(N_{K/K_{\delta^{\prime}}}(\eta))\cdot\chi_{a_{\text{\bf 1}_{K}}a_{\tau\delta^{\prime}}}(N_{K/K_{\tau\delta^{\prime}}}(\eta))\cdot\chi_{a_{\text{\bf 1}_{K}}}(\eta^{1-\tau})$
		$\displaystyle\displaystyle=\left(\eta^{2},K_{\delta^{\prime}}/E\right)\cdot\left(-\eta^{2},K_{\tau\delta^{\prime}}/E\right)\cdot\left(-1,K/K_{+}\right).$

Since $\displaystyle K/K_{+}$ is unramified, we have $\displaystyle\left(-1,K/K_{+}\right)=1$ . Since $\displaystyle N_{K_{\delta^{\prime}}/E}(\eta)=-\eta^{2}$ , we have $\displaystyle\left(-\eta^{2},K_{\delta^{\prime}}/E\right)=1$ . Since $\displaystyle K_{\delta^{\prime}}/E$ is unramified, we have $\displaystyle\left(-1,K_{\delta^{\prime}}/E\right)=1$ . Hence $\displaystyle\left(\eta^{2},K_{\delta^{\prime}}/E\right)=1$ . By the standard formula of the norm residue symbol, we have

\left(\eta,K/K_{\delta^{\prime}}\right)=\left(-\eta^{2},K/E\right)\in\text{\rm Gal}(K/K_{\delta^{\prime}})\subset\text{\rm Gal}(K/E)

(3.12)

since $\displaystyle N_{K_{\delta^{\prime}}/E}(\eta)=-\eta^{2}$ , and

\displaystyle\left(-\eta^{2},K/E\right)=(\left(-\eta^{2},K_{\delta^{\prime}}/E\right),\left(-\eta^{2},K_{\tau\delta^{\prime}}/E\right))

in $\displaystyle\text{\rm Gal}(K/E)=\text{\rm Gal}(K_{\delta^{\prime}}/E)\times\text{\rm Gal}(K_{\tau\delta^{\prime}}/E)$ by $\displaystyle\sigma=(\sigma|_{K_{\delta^{\prime}}},\sigma|_{K_{\tau\delta^{\prime}}})$ . Note that we have $\displaystyle\left(-\eta^{2},K_{\delta^{\prime}}/E\right)=1$ . So if $\displaystyle\left(\eta,K/K_{\delta^{\prime}}\right)=1$ , then $\displaystyle\left(-\eta^{2},K_{\tau\delta^{\prime}}/E\right)=1$ . If $\displaystyle\left(\eta,K/K_{\delta^{\prime}}\right)\neq 1$ , then $\displaystyle\left(\eta,K/K_{\delta^{\prime}}\right)=\delta^{\prime}$ , hence

\displaystyle\left(-\eta^{2},K_{\tau\delta^{\prime}}/E\right)=\delta^{\prime}|_{K_{\tau\delta^{\prime}}}\neq 1.

So we have $\displaystyle\left(-\eta^{2},K_{\tau\delta^{\prime}}/E\right)=\left(\eta,K/K_{\delta^{\prime}}\right)$ . The restriction mapping $\displaystyle\text{\rm Gal}(K/E)\to\text{\rm Gal}(K_{+}/E)$ sends $\displaystyle\left(-\eta^{2},K/E\right)$ to $\displaystyle\left(-\eta^{2},K_{+}/E\right)$ . Since the restriction mapping gives the isomorphism

\displaystyle\text{\rm Gal}(K/K_{\delta^{\prime}})\,\tilde{\to}\,\text{\rm Gal}(K_{+}/E).

Hence (3.12) shows $\displaystyle\left(\eta,K/K_{\delta^{\prime}}\right)=\left(-\eta^{2},K_{+}/E\right)$ . Since $\displaystyle K_{+}/E$ is a ramified quadratic extension and $\displaystyle\eta^{2}\in E$ is not square in $\displaystyle E$ , we have

\displaystyle\left(-\eta^{2},K_{+}/E\right)=\left(-1,K_{+}/E\right)\cdot\left(\eta^{2},K_{+}/E\right)=(-1)^{\frac{q^{f_{+}}-1}{2}}\cdot(-1).

$\displaystyle\blacksquare$

The following proposition will be used in the next two sections.

Proposition 3.3.3

We can choose the $\displaystyle\chi$ -data $\displaystyle\{\chi_{\lambda}\}_{\lambda\in\Phi(\mathbb{T}^{^})}$ so that $\displaystyle c(x)=1$ for all $\displaystyle x\in U_{K/K_{+}}\cap(1+\mathfrak{p}_{K}^{2})$ .

[Proof] If $\displaystyle K/K_{+}$ is ramified, then $\displaystyle K/F$ is totally ramified and $\displaystyle c(x)=\chi_{a_{\text{\bf 1}_{K}}}(x)$ for $\displaystyle x\in U_{K/K_{+}}$ . Since

\displaystyle\left(1+\mathfrak{p}_{K_{+}},K/K_{+}\right)=1\;\;\text{\rm and}\;\;\left(1+\mathfrak{p}_{K}^{2}\right)\cap K_{+}^{\times}=1+\mathfrak{p}_{K_{+}},

we can assume that $\displaystyle\chi_{a_{\text{\bf 1}_{K}}}$ is trivial on $\displaystyle 1+\mathfrak{p}_{K}^{2}$ . Then $\displaystyle c(x)=1$ for all $\displaystyle x\in U_{K/K_{+}}\cap(1+\mathfrak{p}_{K}^{2})$ .

Assume that $\displaystyle K/K_{+}$ is unramified. Since

\displaystyle\left(1+\mathfrak{p}_{K_{+}},K/K_{+}\right)=1\;\;\text{\rm and}\;\;\left(1+\mathfrak{p}_{K}\right)\cap K_{+}^{\times}=1+\mathfrak{p}_{K_{+}},

we can choose $\displaystyle\chi_{a_{\text{\bf 1}_{K}}}$ so that $\displaystyle\chi_{a_{\text{\bf 1}_{K}}}(1+\mathfrak{p}_{K})=1$ . If further $\displaystyle|H|=4$ , then $\displaystyle K_{\delta^{\prime}}/E$ is unramified, and $\displaystyle K_{\tau\delta^{\prime}}/E$ is ramified. Since

\displaystyle(1+\mathfrak{p}_{K_{\delta^{\prime}}})\cap E^{\times}=(1+\mathfrak{p}_{K_{\tau\delta^{\prime}}}^{2})\cap E=1+\mathfrak{p}_{E}

and $\displaystyle(x,K_{\delta^{\prime}}/E)=(x,K_{\tau\delta^{\prime}}/E)=1$ for all $\displaystyle x\in 1+\mathfrak{p}_{E}$ , we can assume that

\displaystyle\chi_{a_{\text{\bf 1}_{K}}a_{\delta^{\prime}}}(1+\mathfrak{p}_{K_{\delta^{\prime}}})=1,\quad\chi_{a_{\text{\bf 1}_{K}}a_{\tau\delta^{\prime}}}(1+\mathfrak{p}_{K_{\tau\delta^{\prime}}})=1.

Since $\displaystyle K/K_{\delta^{\prime}}$ is ramified and $\displaystyle K/K_{\tau\delta^{\prime}}$ is unramified, we have

\displaystyle N_{K/K_{\delta^{\prime}}}(1+\mathfrak{p}_{K}^{2})=1+\mathfrak{p}_{K_{\delta^{\prime}}},\quad N_{K/K_{\tau\delta^{\prime}}}(1+\mathfrak{p}_{K}^{2})=1+\mathfrak{p}_{K_{\tau\delta^{\prime}}}^{2}

Hence $\displaystyle\widetilde{c}(x)=1$ for all $\displaystyle x\in 1+\mathfrak{p}_{K}^{2}$ . Because $\displaystyle K/K_{+}$ is unramified, we can prove by induction on $\displaystyle k$ that $\displaystyle x\mapsto x^{1-\tau}$ is surjection of $\displaystyle 1+\mathfrak{p}_{K}^{k}$ onto $\displaystyle U_{K/K_{+}}\cap(1+\mathfrak{p}_{K}^{k})$ . Then $\displaystyle c(x)=1$ for all $\displaystyle x\in U_{K/K_{+}}\cap(1+\mathfrak{p}_{K}^{2})$ . $\displaystyle\blacksquare$

3.4 $\displaystyle L$ -parameters associated with characters of tame elliptic tori

By local Langlands correspondence of tori described in Proposition 3.1.1, the continuous character $\displaystyle\theta$ of $\displaystyle U_{K/K_{+}}$ which parametrizes the irreducible representation $\displaystyle\delta_{\beta,\theta}$ of $\displaystyle Sp_{2n}(O_{F})$ determines the cohomology class $\displaystyle[\alpha]\in H^{1}_{\text{\rm conti}}(W_{F},T^{^})$ . Then we have a group homomorphism

\varphi:W_{F}\xrightarrow{\widetilde{\alpha}}{{}^{L}T}\xrightarrow{\text{\rm\eqref{eq:modified-langlands-shelstad-homomorphism-of-torus-to-dual-group}}}SO_{2n+1}(\mathbb{C}).

(3.13)

The construction of $\displaystyle\varphi$ shows that $\displaystyle\varphi(\sigma)\in SO_{2n+1}(\mathbb{C})$ is of the form

\varphi(\sigma)=\begin{bmatrix}\varphi_{1}(\sigma)&0\\ 0&\det\varphi_{1}(\sigma)\end{bmatrix}\;\text{\rm with}\;\varphi_{1}(\sigma)\in O(S_{1},\mathbb{C})\quad(S_{1}=\begin{bmatrix}0&1_{n}\\ 1_{n}&0\end{bmatrix})

(3.14)

for $\displaystyle\sigma\in W_{F}$ . The definition of (3.8) shows that

	$\displaystyle\displaystyle\text{\rm tr}\varphi_{1}(\sigma)$	$\displaystyle\displaystyle=\sum_{\gamma\in\text{\rm Emb}_{F}(K,\overline{F}),\gamma\sigma=\gamma}\chi_{p}(\sigma)(\gamma)\cdot\alpha(\sigma)(\gamma)$
		$\displaystyle\displaystyle=\sum_{\dot{\gamma}\in W_{K}\backslash W_{F},\gamma\sigma\gamma^{-1}\in W_{K}}\chi_{p}(\sigma)(\gamma)\cdot\alpha(\sigma(\gamma)$
		$\displaystyle\displaystyle=\sum_{\dot{\gamma}\in W_{K}\backslash W_{F},\gamma\sigma\gamma^{-1}\in W_{K}}\psi_{c}\cdot\psi_{\theta}(\gamma\sigma\gamma^{-1})$

for $\displaystyle\sigma\in W_{F}$ . Here $\displaystyle\psi_{c}$ (resp. $\displaystyle\psi_{\theta}$ ) is the element of $\displaystyle\text{\rm Hom}_{\text{\rm conti}}(W_{K},\mathbb{C})$ corresponding to $\displaystyle c$ (resp. $\displaystyle\theta$ ) by

\displaystyle\text{\rm Hom}_{\text{\rm conti}}(U_{K/K_{+}},\mathbb{C}^{\times})\xrightarrow{\eqref{eq:canonical-inclusion-of-hom-group-of-elliptic-tori}}\text{\rm Hom}_{\text{\rm conti}}(K^{\times},\mathbb{C}^{\times})\xrightarrow{\delta_{K}}\text{\rm Hom}_{\text{\rm conti.}}(W_{K},\mathbb{C}^{\times}).

This shows that $\displaystyle\varphi_{1}$ is the induced representation of $\displaystyle W_{F}$ from the character $\displaystyle\psi_{c}\cdot\psi_{\theta}$ of $\displaystyle W_{K}$ . So $\displaystyle\varphi_{1}$ factors through the canonical surjection

\displaystyle W_{F}\to W_{K/F}=W_{F}/\overline{[W_{K},W_{K}]}

and, if we put $\displaystyle\vartheta=c\cdot\theta$ and $\displaystyle\widetilde{\vartheta}(x)=\vartheta(x^{1-\tau})$ ( $\displaystyle x\in K^{\times}$ ), we have

\text{\rm tr}\varphi_{1}(\sigma,x)=\begin{cases}0&:\sigma\neq 1,\\ \sum_{\gamma\in\text{\rm Gal}(K/F)}\widetilde{\vartheta}(x^{\gamma})&:\sigma=1\end{cases}

(3.15)

for $\displaystyle(\sigma,x)\in W_{K/F}=\text{\rm Gal}(K/F)\ltimes_{\alpha_{K/F}}K^{\times}$ with the fundamental class $\displaystyle[\alpha_{K/F}]\in H^{2}(\text{\rm Gal}(K/F),K^{\times})$ .

The representation space $\displaystyle V_{\vartheta}$ of the induced representation $\displaystyle\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\vartheta}$ is the complex vector space of the $\displaystyle\mathbb{C}$ -valued function $\displaystyle v$ on $\displaystyle\text{\rm Gal}(K/F)$ with the action of $\displaystyle(\sigma,x)\in W_{K/F}$

\displaystyle(x\cdot v)(\gamma)=\widetilde{\vartheta}(x^{\gamma})\cdot v(\gamma),\quad(\sigma\cdot v)(\gamma)=\widetilde{\vartheta}(\alpha_{K/F}(\sigma,\sigma^{-1}\gamma))\cdot v(\sigma^{-1}\gamma).

A $\displaystyle\mathbb{C}$ -basis $\displaystyle\{v_{\rho}\}_{\rho\in\text{\rm Gal}(K/F)}$ of $\displaystyle V_{\vartheta}$ is defined by

\displaystyle v_{\rho}(\gamma)=\begin{cases}1&:\gamma=\rho,\\ 0&:\gamma\neq\rho.\end{cases}

Then

\displaystyle x\cdot v_{\rho}=\widetilde{\vartheta}(x^{\rho})\cdot v_{\rho},\quad\sigma\cdot v_{\rho}=\widetilde{\vartheta}(\alpha_{K/F}(\sigma,\rho))\cdot v_{\sigma\rho}

for $\displaystyle(\sigma,x)\in W_{K/F}$ . The following proposition will be used to analyze $\displaystyle\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\vartheta}$ in detail.

Proposition 3.4.1

Assume $\displaystyle l\geq 2$ , then

\displaystyle\text{\rm Min}\left\{2\leq k\in\mathbb{Z}\biggm{|}\begin{array}[]{l}\widetilde{\vartheta}(\alpha)=1\\ \forall\alpha\in 1+\mathfrak{p}_{K}^{k}\end{array}\right\}=\begin{cases}e(r-1)+1&:\text{\rm$\displaystyle K/K_{+}$ is unramified,}\\ e(r-1)&:\text{\rm$\displaystyle K/K_{+}$ is ramified.}\end{cases}

For an integer $\displaystyle k\geq 2$

\displaystyle\left\{\sigma\in\text{\rm Gal}(K/F)\biggm{|}\begin{array}[]{l}\widetilde{\vartheta}(x^{\sigma})=\widetilde{\vartheta}(x)\\ \text{\rm for $\displaystyle\forall x\in 1+\mathfrak{p}_{K}^{k}$}\end{array}\right\}=\begin{cases}\text{\rm Gal}(K/F)&:k>e(r-1),\\ \text{\rm Gal}(K/K_{0})&:k=e(r-1),\\ \{1\}&:k<e(r-1).\end{cases}

[Proof] Note that $\displaystyle\vartheta(x)=\theta(x)$ for all $\displaystyle x\in U_{K/K_{+}}\cap(1+\mathfrak{p}_{K}^{2})$ (by Proposition 3.3.3) and $\displaystyle\theta(x)=1$ for all $\displaystyle x\in U_{K/K_{+}}\cap(1+\mathfrak{p}_{K}^{er})$ . Take an integer $\displaystyle k$ such that $\displaystyle 0\leq k\leq el^{\prime}$ , and hence $\displaystyle 2\leq el\leq er-k$ . Then, for any $\displaystyle x\in O_{K}$ , we have

\displaystyle(1+\varpi_{F}^{r}\varpi_{K}^{-k}x)^{1-\tau}\equiv 1+\varpi_{F}^{r}(\varpi_{K}^{-k}x-\varpi_{K}^{-k\tau}x^{\tau})\!\!\pmod{\mathfrak{p}_{K}^{er}}

since $\displaystyle 2(er-k)\geq er$ . Hence, for $\displaystyle\alpha=1+\varpi_{F}\varpi_{K}^{-k}x\in 1+\mathfrak{p}_{K}^{er-k}$ ( $\displaystyle x\in O_{K}$ ), we have

\widetilde{\vartheta}(\alpha)=\psi\left(T_{K/F}\left((\varpi_{K}^{-k}x-\varpi_{K}^{-k\tau}x^{\tau})\beta\right)\right)=\psi\left(2T_{K/F}(\varpi_{K}^{-k}x\beta)\right).

(3.16)

1) The statement $\displaystyle\widetilde{\vartheta}(\alpha)=1$ for all $\displaystyle\alpha\in 1+\mathfrak{p}_{K}^{er-k}$ is equivalent to the statement $\displaystyle T_{K/F}\left(\varpi_{K}^{-k}x\beta\right)\in O_{F}$ for all $\displaystyle x\in O_{K}$ , or to the statement $\displaystyle\varpi_{K}^{-k}(\beta)\in\mathcal{D}(K/F)^{-1}=\mathfrak{p}_{K}^{1-e}$ , and hence $\displaystyle\text{\rm ord}_{K}(\beta)\geq k-e-1$ . Since

\displaystyle\text{\rm ord}_{K}(\beta)=\begin{cases}0&:\text{\rm$\displaystyle K/K_{+}$ is unramified,}\\ 1&:\text{\rm$\displaystyle K/K_{+}$ is ramified}\end{cases}

the proof is completed.

2) Because $\displaystyle K/F$ is tamely ramified, we have

	$\displaystyle\displaystyle V_{t}(K/F)$	$\displaystyle\displaystyle=\{\sigma\in\text{\rm Gal}(K/F)\mid\text{\rm ord}_{K}(x^{\sigma}-x)\geq t+1\;\forall x\in O_{K}\}$
		$\displaystyle\displaystyle=\begin{cases}\text{\rm Gal}(K/F)&:t<0,\\ \text{\rm Gal}(K/K_{0})&:0\leq t<1,\\ \{1\}&:1\leq t.\end{cases}$		(3.17)

Take a $\displaystyle\sigma\in\text{\rm Gal}(K/F)$ . Then, by (3.16), we have

\displaystyle\widetilde{\vartheta}(\alpha^{\sigma})=\psi\left(2T_{K/F}(\varpi_{K}^{-k\sigma}x^{\sigma}\beta)\right)=\psi\left(2T_{K/F}(\varpi_{K}^{-k}x\beta^{\sigma})\right).

So the statement $\displaystyle\widetilde{\vartheta}(\alpha^{\sigma})=\widetilde{\vartheta}(\alpha)$ for all $\displaystyle\alpha\in 1+\mathfrak{p}_{K}^{er-k}$ is equivalent to the statement $\displaystyle\varpi_{K}^{-k}(\beta^{\sigma}-\beta)\in\mathcal{D}(K/F)^{-1}=\mathfrak{p}_{K}^{1-e}$ , or to the statement

\displaystyle\text{\rm ord}_{K}(x^{\sigma}-x)\geq k-e+1\;\;\text{\rm for all $\displaystyle x\in O_{K}$}

since $\displaystyle O_{K}=O_{F}[\beta]$ , which is equivalent to $\displaystyle\sigma\in V_{k-e}$ . Then (3.17) completes the proof. $\displaystyle\blacksquare$

Proposition 3.4.2

The induced representation $\displaystyle\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\vartheta}$ is irreducible.

[Proof] Take a $\displaystyle 0\neq T\in\text{\rm End}_{W_{K/F}}(V_{\vartheta})$ . Since

\displaystyle Tv_{\rho}=T(\rho\cdot v_{1})=\rho\cdot Tv_{1}

for all $\displaystyle\rho\in\text{\rm Gal}(K/F)$ , we have $\displaystyle Tv_{1}\neq 0$ . If $\displaystyle(Tv_{1})(\gamma)\neq 0$ for a $\displaystyle\gamma\in\text{\rm Gal}(K/F)$ , then we have

	$\displaystyle\displaystyle\widetilde{\vartheta}(x^{\gamma})\cdot(Tv_{1})(\gamma)$	$\displaystyle\displaystyle=(x\cdot Tv_{1})(\gamma)=T(x\cdot v_{1})(\gamma)$
		$\displaystyle\displaystyle=(T(\widetilde{\vartheta}(x)\cdot v_{1}))(\gamma)=\widetilde{\vartheta}(x)\cdot(Tv_{1})(\gamma),$

and hence $\displaystyle\widetilde{\vartheta}(x^{\gamma})=\widetilde{\vartheta}(x)$ for all $\displaystyle x\in K^{\times}$ . Then $\displaystyle\gamma=1$ by Proposition 3.4.1. This means $\displaystyle Tv_{1}=c\cdot v_{1}$ with a $\displaystyle c\in\mathbb{C}^{\times}$ . Then

\displaystyle Tv_{\rho}=\rho\cdot(Tv_{1})=c\cdot v_{\rho}

for all $\displaystyle\rho\in\text{\rm Gal}(K/F)$ , and hence $\displaystyle T$ is a homothety. $\displaystyle\blacksquare$

Remark 3.4.3

The proof of Proposition 3.4.2 shows that the induced representation $\displaystyle\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\vartheta}$ is irreducible if $\displaystyle\widetilde{\vartheta}$ is a character of $\displaystyle K^{\times}$ such that $\displaystyle\widetilde{\vartheta}(x^{\sigma})=\widetilde{\vartheta}(x)$ for all $\displaystyle x\in K^{\times}$ with $\displaystyle\sigma\in\text{\rm Gal}(K/F)$ implies $\displaystyle\sigma=1$ .

4 Formal degree conjecture

In this section, we will assume that $\displaystyle K/F$ is a tamely ramified Galois extension of degree $\displaystyle 2n$ and put $\displaystyle\Gamma=\text{\rm Gal}(K/F)$ , We will keep the notations of the preceding sections.

4.1 $\displaystyle\gamma$ -factor of adjoint representation

The admissible representation of the Weil-Deligne group $\displaystyle W_{F}\times SL_{2}(\mathbb{C})$ to $\displaystyle SO_{2n+1}(\mathbb{C})$ corresponding to the triple $\displaystyle(\varphi,SO_{2n+1}(\mathbb{C}),0)$ as explained in the appendix A.6 is

W_{F}\times SL_{2}(\mathbb{C})\xrightarrow{\text{\rm projection}}W_{F}\xrightarrow{\varphi}SO_{2n+1}(\mathbb{C})

(4.1)

whcih is also denoted by $\displaystyle\varphi$ . The purpose of this subsection is to determine the $\displaystyle\gamma$ -factor $\displaystyle\gamma(\varphi,\text{\rm Ad},\psi,d(x),s)$ whose definition and the basic properties are presented in the appendix A.6. Our result is

Theorem 4.1.1

\displaystyle\gamma(\varphi,\text{\rm Ad},\psi,d(x),0)=w(\text{\rm Ad}\circ\varphi)\cdot q^{n^{2}r}\times\begin{cases}1&:\text{\rm$\displaystyle K/K_{+}$ is ramified},\\ \frac{2}{1+q^{-f_{+}}}&:\text{\rm$\displaystyle K/K_{+}$ is unramified}\end{cases}

where $\displaystyle\psi$ is a continuous unitary additive character of $\displaystyle F$ such that

\displaystyle\{x\in F\mid\psi(xO_{F})=1\}=O_{F}

and $\displaystyle d(x)$ is the Haar measure on $\displaystyle F$ such that $\displaystyle\int_{O_{F}}d(x)=1$ .

The rest of this subsection is devoted to the proof of the theorem.

Let us use the notation of (3.9)

\displaystyle\Gamma=\text{\rm Gal}(K/F)=\langle\delta,\rho\rangle,

that is, $\displaystyle\text{\rm Gal}(K/K_{0})=\langle\delta\rangle$ with the maximal unramified subextension $\displaystyle K_{0}/F$ of $\displaystyle K/F$ and $\displaystyle\rho|_{K_{0}}\in\text{\rm Gal}(K_{0}/F)$ is the inverse of the Frobenius automorphism. Put

\displaystyle\rho\delta\rho^{-1}=\delta^{l},\quad\rho^{f}=\delta^{m}\quad(0\leq i,m<e,\;ql\equiv 1\!\!\pmod{e}).

By the canonical surjection

\displaystyle W_{F}\to W_{F}/\overline{[W_{K},W_{K}]}=W_{K/F}=\text{\rm Gal}(K/F){\ltimes}_{\alpha_{K/F}}K^{\times}\subset\text{\rm Gal}(K^{\text{\rm ab}}/F),

$\displaystyle I_{F}=\text{\rm Gal}(F^{\text{\rm alg}}/F^{\text{\rm ur}})\subset W_{F}$ is mapped onto

\displaystyle\text{\rm Gal}(K/K_{0}){\ltimes}_{\alpha_{K/F}}O_{K}^{\times}=\text{\rm Gal}(K^{\text{\rm ab}}/F^{\text{\rm ur}}).

The representation space $\displaystyle V_{\vartheta}$ of $\displaystyle\varphi_{1}=\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\vartheta}$ has a $\displaystyle W_{K/F}$ -invariant non-degenerate symmetric form

\displaystyle S_{1}(u,v)=\sum_{\gamma\in\Gamma}\widetilde{\vartheta}\left(\alpha_{K/F}(\gamma,\tau)\right)^{-1}\cdot u(\gamma)v(\gamma\tau)\quad(u,v\in V_{\vartheta})

which is unique up to constant multiple, by Proposition B.0.1. Put

\displaystyle u_{\sigma}=\widetilde{\vartheta}\left(\alpha_{K/F}(\sigma,\tau)\right)\cdot v_{\sigma}\in V_{\vartheta}

for $\displaystyle\sigma\in\Gamma$ . Then we have

\alpha\cdot v_{\beta}=\widetilde{\vartheta}\left(\alpha_{K/F}(\alpha,\beta)\right)\cdot v_{\alpha\beta},\quad\alpha\cdot u_{\beta}=\widetilde{\vartheta}\left(\alpha_{K/F}(\alpha,\beta)\right)^{-1}\cdot u_{\alpha\beta}

(4.2)

and

\displaystyle S_{1}(v_{\alpha},v_{\beta})=S_{1}(u_{\alpha},u_{\beta})=0,\quad S_{1}(v_{\alpha},u_{\alpha})=\begin{cases}1&:\alpha=\beta,\\ 0&:\alpha\neq\beta\end{cases}

for $\displaystyle\alpha,\beta\in\Gamma$ . Fixing a representatives $\displaystyle\mathcal{S}$ of $\displaystyle\Gamma/\langle\tau\rangle$ , we will identify the orthogonal group $\displaystyle O(V,S_{1})$ of the symmetric form $\displaystyle S_{1}$ with the matrix group $\displaystyle O(S_{1},\mathbb{C})$ of (3.14) by means of the $\displaystyle\mathbb{C}$ -basis $\displaystyle\{v_{\sigma},u_{\sigma}\}_{\sigma\in\mathcal{S}}$ of $\displaystyle V_{\vartheta}$ which we will call the canonical basis associated with $\displaystyle\mathcal{S}$ . Then we have

\displaystyle\varphi(x)=\begin{bmatrix}[x]&&\\ &[x]^{-1}&\\ &&1\end{bmatrix}\in SO_{2n+1}(\mathbb{C})\;\;\text{\rm with}\;\;[x]=\text{\rm diag}(\widetilde{\vartheta}(x^{\sigma}))_{\sigma\in\mathcal{S}}

for $\displaystyle x\in K^{\times}\subset W_{K/F}$ so that the centralizer of $\displaystyle\varphi(O_{K}^{\times})$ in $\displaystyle SO_{2n+1}(\mathbb{C})$ is

Z_{SO_{2n+1}(\mathbb{C})}(\varphi(O_{K}^{\times}))=\left\{\begin{bmatrix}a&&\\ &a^{-1}&\\ &&1\end{bmatrix}\biggm{|}\text{\rm$\displaystyle a$=diagonal$\displaystyle\in GL_{n}(\mathbb{C})$}\right\}

(4.3)

and the space $\displaystyle\widehat{\mathfrak{g}}^{O_{K}^{\times}}$ of the $\displaystyle\text{\rm Ad}\circ\varphi(O_{K}^{\times})$ -fixed vectors in

\displaystyle\widehat{\mathfrak{g}}=\mathfrak{so}_{2n+1}(\mathbb{C})=\{X\in\mathfrak{gl}_{2n+1}(\mathbb{C})\mid XS+S\,^{t}X=0\}

\widehat{\mathfrak{g}}^{O_{K}^{\times}}=\left\{\begin{bmatrix}A&&\\ &-A&\\ &&0\end{bmatrix}\biggm{|}\text{\rm$\displaystyle A$=diagonal$\displaystyle\in\mathfrak{gl}_{n}(\mathbb{C})$}\right\}

(4.4)

by Proposition 3.4.1.

Let us denote by $\displaystyle\mathcal{A}_{\varphi}$ the centralizer of $\displaystyle\text{\rm Im}(\varphi)$ in $\displaystyle SO_{2n+1}(\mathbb{C})$ . We have

Proposition 4.1.2

\displaystyle L(\varphi,\text{\rm Ad},s)=\begin{cases}1&:\text{\rm$\displaystyle K/K_{+}$ is ramified},\\ \frac{1}{1+q^{-f_{+}s}}&:\text{\rm$\displaystyle K/K_{+}$ is unramified}\end{cases}

and

\displaystyle\mathcal{A}_{\varphi}=\left\{\begin{bmatrix}\pm 1_{2n}&0\\ 0&1\end{bmatrix}\right\}.

[Proof] Assume that $\displaystyle K/K_{+}$ is ramified. Then $\displaystyle K/F$ is totally ramified and $\displaystyle\text{\rm Gal}(K/F)=\langle\delta\rangle$ a cyclic group of order $\displaystyle 2n$ with $\displaystyle\tau=\delta^{n}$ . Put $\displaystyle\mathcal{S}=\{\delta^{i}\}_{0\leq i<n}$ which is a representatives of $\displaystyle\Gamma/\langle\tau\rangle$ . Since $\displaystyle K/F$ is a cyclic extension, we have $\displaystyle\alpha_{K/F}(\alpha,\beta)\in F^{\times}$ for all $\displaystyle\alpha,\beta\in\Gamma$ , the canonical basis associated with $\displaystyle\mathcal{S}$ is

\displaystyle v_{i}=v_{\delta^{i-1}},\quad u_{i}=v_{\delta^{n+i-1}}\quad(1\leq i\leq n).

Then (4.2) shows

\varphi(\delta)=\begin{bmatrix}0&1&0\\ 1_{2n-1}&0&0\\ 0&0&-1\end{bmatrix}\in SO_{2n+1}(\mathbb{C}).

(4.5)

Then the centralizer $\displaystyle\mathcal{A}_{\varphi}$ of $\displaystyle\text{\rm Im}(\varphi)$ in $\displaystyle SO_{2n+1}(\mathbb{C})$ is

\displaystyle\mathcal{A}_{\varphi}=\left\{\begin{bmatrix}\pm 1_{2n}&0\\ 0&1\end{bmatrix}\right\},

and the space $\displaystyle\widehat{\mathfrak{g}}^{I_{F}}$ of the $\displaystyle\text{\rm Ad}\circ\varphi(I_{F})$ -fixed vectors in $\displaystyle\widehat{\mathfrak{g}}$ is $\displaystyle\{0\}$ so that we have $\displaystyle L(\varphi,\text{\rm Ad},s)=1$ .

Now assume that $\displaystyle K/K_{+}$ is unramified. Then $\displaystyle\tau=\delta^{a}\rho^{f_{+}}$ with $\displaystyle 0\leq a<e$ by Proposition 3.3.1. Put $\displaystyle\mathcal{S}=\{\delta^{i}\rho^{j}\}_{0\leq i<e,0\leq j<f_{+}}$ which is a representatives of $\displaystyle\Gamma/\langle\tau\rangle$ . The associated basis

\displaystyle v_{ij}=v_{\delta^{i-1}\rho^{j-1}},\quad u_{ij}=u_{\delta^{i-1}\rho^{j-1}}\quad(1\leq i\leq e,1\leq j\leq f_{+})

is ordered lexicographically. Then

\displaystyle\varphi(\delta)=\begin{bmatrix}\Delta&&\\ &{}^{t}\Delta^{-1}&\\ &&1\end{bmatrix}\in SO_{2n+1}(\mathbb{C})\;\text{\rm with}\;\Delta=\begin{bmatrix}\Delta_{1}&&\\ &\ddots&\\ &&\Delta_{f_{+}}\end{bmatrix},

where

\displaystyle\Delta_{j}=\begin{bmatrix}0&0&0&\cdots&0&\alpha_{e,j}\\ \alpha_{1,j}&0&0&\cdots&0&0\\ &\alpha_{2,j}&0&\cdots&0&0\\ &&\ddots&\ddots&\vdots&\vdots\\ &&&\ddots&0&0\\ &&&&\alpha_{e-1,j}&0\end{bmatrix}

with $\displaystyle\alpha_{i,j}=\widetilde{\vartheta}\left(\alpha_{K/F}(\delta,\delta^{i-1}\rho^{j-1})\right)$ . The action of $\displaystyle\varphi(\delta)$ on (4.4) shows that the space $\displaystyle\widehat{\mathfrak{g}}^{I_{F}}$ of the $\displaystyle\text{\rm Ad}\circ\varphi(I_{F})$ -fixed vectors in $\displaystyle\widehat{\mathfrak{g}}$ is

\displaystyle\widehat{\mathfrak{g}}^{I_{F}}=\left\{\begin{bmatrix}A&&\\ &-A&\\ &&0\end{bmatrix}\biggm{|}A=\begin{bmatrix}a_{1}1_{e}&&&\\ &a_{2}1_{e}&&\\ &&\ddots&\\ &&&a_{f_{+}}1_{e}\end{bmatrix}\right\}.

Since

\displaystyle\rho\cdot\delta^{i-1}\rho^{j-1}=\begin{cases}\delta^{l(i-1)}\rho^{j}=\delta^{i^{\prime}-1}\rho-j&:1\leq j<f_{+},\\ \delta^{l(i-1)-a}\tau=\delta^{i^{\prime\prime}-1}\tau&:j=f_{+}\end{cases}\qquad(1\leq i^{\prime},i^{\prime\prime}\leq e)

for $\displaystyle 1\leq i\leq e$ , let $\displaystyle[l]$ and $\displaystyle[l,a]$ be the permutation matrices of the permutations

\displaystyle\begin{pmatrix}1&2&\cdots&e\\ 1^{\prime}&2^{\prime}&\cdots&e^{\prime}\end{pmatrix},\qquad\begin{pmatrix}1&2&\cdots&e\\ 1^{\prime\prime}&2^{\prime\prime}&\cdots&e^{\prime\prime}\end{pmatrix}

respectively. Then we have

\displaystyle\varphi(\rho)=\begin{bmatrix}A&B&0\\ C&D&0\\ 0&0&(-1)^{e}\end{bmatrix}\in SO_{2n+1}(\mathbb{C})

with

\displaystyle A=\begin{bmatrix}0&0&0&\cdots&0&0\\ P_{1}&0&0&\cdots&0&0\\ &P_{2}&0&\cdots&0&0\\ &&\ddots&\ddots&\vdots&\vdots\\ &&&\ddots&0&0\\ &&&&P_{f_{+}-1}&0\end{bmatrix},\quad B=\begin{bmatrix}&&&&&&&Q_{f_{+}}\\ &&&&&&0&\\ &&&&\mathinner{\mkern 1.0mu\raise 1.0pt\hbox{.}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise 7.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 1.0mu}&&&\\ &&0&&&&&\\ &0&&&&&&\\ 0&&&&&&&\end{bmatrix},

\displaystyle C=\begin{bmatrix}&&&&&&&P_{f_{+}}\\ &&&&&&0&\\ &&&&\mathinner{\mkern 1.0mu\raise 1.0pt\hbox{.}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise 7.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 1.0mu}&&&\\ &&0&&&&&\\ &0&&&&&&\\ 0&&&&&&&\end{bmatrix},\quad D=\begin{bmatrix}0&0&0&\cdots&0&0\\ Q_{1}&0&0&\cdots&0&0\\ &Q_{2}&0&\cdots&0&0\\ &&\ddots&\ddots&\vdots&\vdots\\ &&&\ddots&0&0\\ &&&&Q_{f_{+}-1}&0\end{bmatrix}

where

	$\displaystyle\displaystyle P_{j}$	$\displaystyle\displaystyle=\begin{cases}\left[\begin{pmatrix}1&2&\cdots&e\\ 1^{\prime}&2^{\prime}&\cdots&e^{\prime}\end{pmatrix}\right]\begin{bmatrix}\alpha_{1j}&&&\\ &\alpha_{2j}&&\\ &&\ddots&\\ &&&\alpha_{ej}\end{bmatrix}&:1\leq j<f_{+},\\ \left[\begin{pmatrix}1&2&\cdots&e\\ 1^{\prime\prime}&2^{\prime\prime}&\cdots&e^{\prime\prime}\end{pmatrix}\right]\begin{bmatrix}\beta_{1}&&&\\ &\beta_{2}&&\\ &&\ddots&\\ &&&\beta_{e}\end{bmatrix}&:j=f_{+},\end{cases}$
	$\displaystyle\displaystyle Q_{j}$	$\displaystyle\displaystyle=\begin{cases}\left[\begin{pmatrix}1&2&\cdots&e\\ 1^{\prime}&2^{\prime}&\cdots&e^{\prime}\end{pmatrix}\right]\begin{bmatrix}\alpha_{1j}^{-1}&&&\\ &\alpha_{2j}^{-1}&&\\ &&\ddots&\\ &&&\alpha_{ej}^{-1}\end{bmatrix}&:1\leq j<f_{+},\\ \left[\begin{pmatrix}1&2&\cdots&e\\ 1^{\prime\prime}&2^{\prime\prime}&\cdots&e^{\prime\prime}\end{pmatrix}\right]\begin{bmatrix}\beta_{1}^{-1}&&&\\ &\beta_{2}^{-1}&&\\ &&\ddots&\\ &&&\beta_{e}^{-1}\end{bmatrix}&:j=f_{+}\end{cases}$

with

	$\displaystyle\displaystyle\alpha_{ij}$	$\displaystyle\displaystyle=\widetilde{\vartheta}\left(\alpha_{K/F}(\rho,\delta^{i-1}\rho^{j-1})\right),$
	$\displaystyle\displaystyle\beta_{i}$	$\displaystyle\displaystyle=\widetilde{\vartheta}\left(\alpha_{K/F}(\rho,\delta^{i-1}\rho^{f_{+}-1})\cdot\alpha_{K/F}(\delta^{i^{\prime\prime}-1},\tau)^{-1}\right).$

Then the adjoint action of $\displaystyle\varphi(\rho)$ on $\displaystyle\widehat{\mathfrak{g}}^{I_{F}}$ gives

\displaystyle\det\left(1_{f_{+}}-t\cdot\text{\rm Ad}\circ\varphi(\rho)|_{\widehat{\mathfrak{g}}^{I_{F}}}\right)=1+t^{-f_{+}}

so that we have $\displaystyle L(\varphi,\text{\rm Ad},s)=\left(1+q^{-f_{+}s}\right)^{-1}$ . Finally the centralizer of $\displaystyle\text{\rm Im}(\varphi)$ in $\displaystyle SO_{2n+1}(\mathbb{C})$ is

\displaystyle\mathcal{A}_{\varphi}=\left\{\begin{bmatrix}\pm 1_{2n}&0\\ 0&1\end{bmatrix}\right\}.

$\displaystyle\blacksquare$

Next we will calculate the Artin conductor of $\displaystyle\text{\rm Ad}\circ\varphi$ . Note that the complex vector space $\displaystyle\widehat{\mathfrak{g}}$ is isomorphic to the space of alternating matrices

\displaystyle\text{\rm Alt}_{2n+1}(\mathbb{C})=\{X\in M_{2n+1}(\mathbb{C})\mid X+\,^{t}X=0\}

and $\displaystyle\text{\rm Ad}\circ\varphi$ on $\displaystyle\widehat{\mathfrak{g}}$ is isomorphic to $\displaystyle{\bigwedge}^{2}\varphi$ on $\displaystyle\text{\rm Alt}_{2n+1}(\mathbb{C})$ . Since $\displaystyle\varphi=\varphi_{1}\oplus\det\varphi_{1}$ and $\displaystyle\varphi_{1}=\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\vartheta}$ with $\displaystyle(\det\varphi_{1})|_{K^{\times}}=1$ , we have

\displaystyle{\bigwedge}^{2}\varphi=({\bigwedge}^{2}\varphi_{1})\oplus(\varphi_{1}\otimes\det\varphi_{1})=({\bigwedge}^{2}\varphi_{1})\oplus\varphi_{1}.

Then $\displaystyle\chi_{\text{\rm Ad}\circ\varphi}=\chi_{{\wedge}^{2}\varphi_{1}}+\chi_{\varphi_{1}}$ and the character formula

\displaystyle\chi_{\varphi_{1}}(g)=\begin{cases}0&:\sigma\neq 1,\\ \sum_{\gamma\in\Gamma}\widetilde{\vartheta}(x^{\gamma})&:\sigma=1\end{cases}

for $\displaystyle g=(\sigma,x)\in W_{K/F}=\Gamma{\ltimes}_{\alpha_{K}/F}K^{\times}$ gives

	$\displaystyle\displaystyle\chi_{{\wedge}^{2}\varphi_{1}}(g)$	$\displaystyle\displaystyle=\frac{1}{2}\left\{\chi_{\varphi_{1}}(g)^{2}-\chi_{\varphi_{1}}(g^{2})\right\}$
		$\displaystyle\displaystyle=\begin{cases}0&:\sigma^{2}\neq 1,\\ -\frac{1}{2}\sum_{\gamma\in\Gamma}\widetilde{\vartheta}\left(\alpha_{K/F}(\sigma,\sigma)^{\gamma}\cdot x^{(1+\sigma)\gamma}\right)&:\sigma^{2}=1,\sigma\neq 1,\\ \frac{1}{2}\sum_{\stackrel{{\scriptstyle\scriptstyle\alpha,\gamma\in\Gamma}}{{\gamma\neq 1}}}\widetilde{\vartheta}\left(x^{\alpha(1+\gamma)}\right)&:\sigma=1.\end{cases}$		(4.6)

Now we have

Proposition 4.1.3

The Artin conductor of $\displaystyle\text{\rm Ad}\circ\varphi$ is

\displaystyle a(\text{\rm Ad}\circ\varphi)=2n^{2}r.

[Proof] Let us denote by $\displaystyle K^{(k)}=K_{\varpi_{K},k}$ ( $\displaystyle k=1,2,\cdots$ ) the field of $\displaystyle\varpi_{K}^{k}$ -th division points of Lubin-Tate theory over $\displaystyle K$ . Then we have an isomorphism

\displaystyle\delta_{K}:1+\mathfrak{p}_{K}^{k}\,\tilde{\to}\,\text{\rm Gal}(K^{\text{\rm ab}}/K^{(k)}K^{\text{\rm ur}}).

Because the character $\displaystyle\widetilde{\vartheta}:K^{\times}\to\mathbb{C}^{\times}$ comes from a character of

\displaystyle G_{\beta}(O_{F}/\mathfrak{p}^{r})\subset\left(O_{K}/\mathfrak{p}_{K}^{er}\right)^{\times},

$\displaystyle\varphi$ is trivial on $\displaystyle\text{\rm Gal}(K^{\text{\rm ab}}/K^{(er)}K^{\text{\rm ur}})$ . Note that $\displaystyle K^{(er)}K^{\text{\rm ur}}=K^{(er)}F^{\text{\rm ur}}$ is a finite extension of $\displaystyle F^{\text{\rm ur}}$ . If us use the upper numbering

\displaystyle V^{s}=V_{t}(K^{(er)}F^{\text{\rm ur}}/F^{\text{\rm ur}})

of the higher ramification group, where $\displaystyle t\mapsto s$ is the inverse of Hasse function whose graph is

then $\displaystyle\delta_{K}$ induces the isomorphism

\displaystyle(1+\mathfrak{p}_{K}^{k})/(1+\mathfrak{p}_{K}^{er})\,\tilde{\to}\,\text{\rm Gal}(K^{(er)}K^{\text{\rm ur}}/K^{(k)}K^{\text{\rm ur}})=V^{s}

for $\displaystyle k-1<s\leq k$ ( $\displaystyle k=1,2,\cdots$ ), and hence, for $\displaystyle V_{t}=V_{t}(K^{(er)}F^{\text{\rm ur}}/F^{\text{\rm ur}})$ , we have

\displaystyle|V_{t}|=\begin{cases}e\cdot q^{nr}(1-q^{-f})&:t=0,\\ q^{nr-fk}&:q^{f(k-1)}-1<t\leq q^{fk}-1.\end{cases}

By the definition

\displaystyle a(\text{\rm Ad}\circ\varphi)=\sum_{t=0}^{\infty}\left(\dim_{\mathbb{C}}\widehat{\mathfrak{g}}-\dim_{\mathbb{C}}\widehat{\mathfrak{g}}^{V_{t}}\right)\cdot(V_{0}:V_{t})^{-1}.

We have

\displaystyle\dim_{\mathbb{C}}\widehat{\mathfrak{g}}^{V_{0}}=\dim_{\mathbb{C}}\widehat{\mathfrak{g}}^{I_{F}}=\begin{cases}0&:\text{\rm$\displaystyle K/K_{+}$ is ramified},\\ f_{+}&:\text{\rm$\displaystyle K/K_{+}$ is unramified}\end{cases}

as shown in the proof of Proposition 4.1.2. For $\displaystyle q^{f(k-1)}-1<t\leq q^{fk}-1$ with $\displaystyle k>0$ , we have

	$\displaystyle\displaystyle\|V_{t}\|\cdot\dim_{\mathbb{C}}\widehat{\mathfrak{g}}^{V_{t}}$	$\displaystyle\displaystyle=\sum_{g\in V_{t}}\chi_{\text{\rm Ad}\circ\varphi}(g)=\frac{1}{2}\sum_{\dot{x}\in V_{t}}\sum_{\stackrel{{\scriptstyle\scriptstyle\alpha,\gamma\in\Gamma}}{{\gamma\neq 1}}}\widetilde{\vartheta}(x^{\alpha(1+\gamma)})+\sum_{\dot{x}\in V_{t}}\widetilde{\vartheta}(x^{\gamma})$
		$\displaystyle\displaystyle=n\cdot\sum_{\dot{x}\in V_{t}}\sum_{\tau\neq\gamma\in\Gamma}\widetilde{\vartheta}(x^{1-\gamma})+2n\cdot\sum_{\dot{x}\in V_{t}}\widetilde{\vartheta}(x)$

where $\displaystyle V_{t}$ is identified with $\displaystyle(1+\mathfrak{p}_{K}^{k})/(1+\mathfrak{p}_{K}^{er})$ .

If $\displaystyle K/K_{+}$ is unramified, then $\displaystyle\tau\not\in\text{\rm Gal}(K/K_{0})$ , and Proposition 3.4.1 gives

\displaystyle\sum_{\dot{x}\in V_{t}}\sum_{\tau\neq\gamma\in\Gamma}\widetilde{\vartheta}(x^{1-\gamma})=|V_{t}|\times\begin{cases}2n-1&:k>e(r-1),\\ e&:k=e(r-1),\\ 1&:k<e(r-1)\end{cases}

and

\displaystyle\sum_{\dot{x}\in V_{t}}\widetilde{\vartheta}(x)=\begin{cases}|V_{t}|&:k>e(r-1),\\ 0&:k\leq e(r-1).\end{cases}

So we have

\displaystyle\dim_{\mathbb{C}}\widehat{\mathfrak{g}}^{V_{t}}=\begin{cases}n(2n+1)&:k>e(r-1),\\ ne&:k=e(r-1),\\ n&:k<e(r-1).\end{cases}

Then we have

	$\displaystyle\displaystyle a(\text{\rm Ad}\circ\varphi)$	$\displaystyle\displaystyle=n(2n+1)-f_{+}+\{n(2n+1)-n\}\cdot e^{-1}\cdot\{e(r-1)-1\}$
		$\displaystyle\displaystyle\hphantom{=n(2n+1)-\;f_{+}}+\{n(2n+1)-ne\}\cdot e^{-1}$
		$\displaystyle\displaystyle=2n^{2}r.$

If $\displaystyle K/K_{+}$ is ramified, then $\displaystyle K/F$ is totally ramified and we have

\displaystyle\sum_{\dot{x}\in V_{t}}\sum_{\tau\neq\gamma\in\Gamma}\widetilde{\vartheta}(x^{1-\gamma})=|V_{t}|\times\begin{cases}2n-1&:k\geq e(r-1),\\ 1&:k<e(r-1)\end{cases}

and

\displaystyle\sum_{\dot{x}\in V_{t}}\widetilde{\vartheta}(x)=\begin{cases}|V_{t}|&:k\geq e(r-1),\\ 0&:k<e(r-1)\end{cases}

by Proposition 3.4.1. The we have

	$\displaystyle\displaystyle a(\text{\rm Ad}\circ\varphi)$	$\displaystyle\displaystyle=n(2n+1)+\{n(2n+1)-n\}\cdot(2n)^{-1}\cdot\{2n(r-1)-1\}$
		$\displaystyle\displaystyle=2n^{2}r.$

$\displaystyle\blacksquare$

Since

\displaystyle\gamma(\varphi,\text{\rm Ad},\psi,d(x),0)=\varepsilon(\varphi,\text{\rm Ad},\psi,d(x))\cdot\frac{L(\varphi,\text{\rm Ad},1)}{L(\varphi,\text{\rm Ad},0)}

and

\displaystyle\varepsilon(\varphi,\text{\rm Ad},\psi,d(x))=w(\text{\rm Ad}\circ\varphi)\cdot q^{a(\text{\rm Ad}\circ\varphi)/2},

Proposition 4.1.2 and Proposition 4.1.3 give the proof of Theorem 4.1.1.

4.2 $\displaystyle\gamma$ -factor of principal parameter

Let $\displaystyle\text{\rm Sym}_{2n}$ be the symmetric tensor representation of $\displaystyle SL_{2}(\mathbb{C})$ on the space $\displaystyle\mathcal{P}_{2n}$ of the complex coefficient homogeneous polynomials of $\displaystyle X,Y$ of degree $\displaystyle 2n$ . Then

\displaystyle\langle f,g\rangle=\left.f\left(-\frac{\partial}{\partial Y},\frac{\partial}{\partial X}\right)g(X,Y)\right|_{(X,Y)=(0,0)}\quad(f,g\in\mathcal{P}_{2n})

defines a $\displaystyle SL_{2}(\mathbb{C})$ -invariant non-degenerate symmetric complex bilinear form on the complex vector space $\displaystyle\mathcal{P}_{2n}$ . For the $\displaystyle\mathbb{C}$ -basis $\displaystyle\left\{v_{k}=\frac{1}{(k-1)!}X^{2n+1-k}Y^{k-1}\right\}_{k=1,2,\cdots,2n+1}$ of $\displaystyle\mathcal{P}_{2n}$ , we have

\displaystyle\langle v_{k},v_{l}\rangle=\begin{cases}0&:k+l=2n+2,\\ (-1)^{k-1}=(-1)^{l-1}&:k+l=2n+2\end{cases}

and the identification

\displaystyle SO(\mathcal{P}_{2n},\langle\,,\rangle)=SO_{2n+1}(\mathbb{C})=\{g\in SL_{2n+1}(\mathbb{C})\mid gJ_{2n+1}\,^{t}g=J_{2n+1}\}

where

\displaystyle J_{2n+1}=\begin{bmatrix}&&&&1\\ &&&-1&\\ &&\mathinner{\mkern 1.0mu\raise 1.0pt\hbox{.}\mkern 2.0mu\raise 4.0pt\hbox{.}\mkern 2.0mu\raise 7.0pt\vbox{\kern 7.0pt\hbox{.}}\mkern 1.0mu}&&\\ &-1&&&\\ 1&&&&\end{bmatrix}.

The Lie algebra of $\displaystyle SO_{2n+1}(\mathbb{C})$ is

\displaystyle\mathfrak{so}_{2n+1}(\mathbb{C})=\{X\in\mathfrak{gl}_{2n+1}(\mathbb{C})\mid XJ_{2n+1}+J_{2n+1}\,^{t}X=0\}

and

\displaystyle d\,\text{\rm Sym}_{2n}\begin{bmatrix}0&1\\ 0&0\end{bmatrix}=N_{0}=\begin{bmatrix}0&1&&&\\ &0&1&&\\ &&\ddots&\ddots&\\ &&&0&1\\ &&&&0\end{bmatrix}\in\widehat{\mathfrak{g}}

is the nilpotent element in $\displaystyle\mathfrak{so}_{2n+1}(\mathbb{C})$ associated with the standard épinglage of the standard root system of $\displaystyle\mathfrak{so}_{2n+1}(\mathbb{C})$ . Then

\varphi_{0}:W_{F}\times SL_{2}(\mathbb{C})\xrightarrow{\text{\rm proj.}}SL_{2}(\mathbb{C})\xrightarrow{\text{\rm Sym}_{2n}}SO_{2n+1}(\mathbb{C})

(4.7)

is a representation of Weil-Deligne group with the associated triplet $\displaystyle(\rho_{0},SO_{2n+1}(\mathbb{C}),N_{0})$ such that $\displaystyle\rho_{0}|_{I_{F}}$ is trivial and

\displaystyle\rho_{0}(\widetilde{\text{\rm Fr}})=\begin{bmatrix}q^{-n}&&&&\\ &q^{-(n-1)}&&&\\ &&\ddots&&\\ &&&q^{(n-1)}&\\ &&&&q^{n}\end{bmatrix}\in SO_{2n+1}(\mathbb{C}).

Now

\{N_{0}^{2k-1}\mid k=1,2,\cdots,n\}

(4.8)

is a $\displaystyle\mathbb{C}$ -basis of

\displaystyle\widehat{\mathfrak{g}}_{N_{0}}=\{X\in\widehat{\mathfrak{g}}\mid[X,N_{0}]=0\}.

The representation matrix of $\displaystyle\text{\rm Ad}\circ\rho_{0}(\widetilde{\text{\rm Fr}})\in GL_{\mathbb{C}}(\widehat{\mathfrak{g}})$ is

\displaystyle\begin{bmatrix}q^{-1}&&&\\ &q^{-3}&&\\ &&\ddots&\\ &&&q^{-(2n-1)}\end{bmatrix}

so that we have

	$\displaystyle\displaystyle L(\varphi_{0},\text{\rm Ad},s)$	$\displaystyle\displaystyle=\det\left(1-q^{-s}\cdot\text{\rm Ad}\circ\rho_{0}(\widetilde{\text{\rm Fr}})\|_{\widehat{\mathfrak{g}}_{N_{0}}}\right)^{-1}$
		$\displaystyle\displaystyle=\prod_{k=1}^{n}\left(1-q^{-(s+2k-1)}\right)^{-1}.$

On the other hand [7, p.448] shows

\displaystyle\varepsilon(\varphi_{0},\text{\rm Ad},\psi,d(x))=q^{n^{2}}.

Since the symmetric tensor representation $\displaystyle\text{\rm Sym}_{2n}$ is self-dual, we have

	$\displaystyle\displaystyle\gamma(\varphi_{0},\text{\rm Ad},\psi,d(x),0)$	$\displaystyle\displaystyle=\varepsilon(\varphi_{0},\text{\rm Ad},\psi,d(x))\cdot\frac{L(\varphi_{0},\text{\rm Ad},1)}{L(\varphi_{0},\text{\rm Ad},0)}$
		$\displaystyle\displaystyle=q^{n^{2}}\cdot\prod_{k=1}^{n}\frac{1-q^{-(2k-1)}}{1-q^{-2k}}.$		(4.9)

4.3 Verification of formal degree conjecture

Let $\displaystyle d_{G(F)}$ be the Haar measure on $\displaystyle G(F)$ such that $\displaystyle\int_{G(O_{F})}d_{G(F)}(x)=1$ . Then the Euler-Poincaré measure $\displaystyle\mu_{G(F)}$ on $\displaystyle G(F)=Sp_{2n}(F)$ is (see [12, p.150, Th.7])

\displaystyle d\mu_{G(F)}(x)=(-1)^{n}q^{n^{2}}\prod_{k=1}^{n}\left(1-q^{-(2k-1)}\right)\cdot d_{G(F)}(x).

Then Theorem 2.3.1 implies that the formal degree of the supercuspidal representation $\displaystyle\pi_{\beta,\theta}=\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$ with respect to the absolute value of the Euler-Poincaré measure on $\displaystyle G(F)$ is

q^{n^{2}(r-1)}\cdot\prod_{k=1}^{n}\frac{1-q^{-2k}}{1-q^{-(2k-1)}}\times\begin{cases}\frac{1}{2}&:\text{\rm$\displaystyle K/K_{+}$ is ramified},\\ \frac{1}{1+q^{-f_{+}}}&:\text{\rm$\displaystyle K/K_{+}$ is unramified}.\end{cases}

(4.10)

Since the order of the centralizer $\displaystyle\mathcal{A}_{\varphi}$ of $\displaystyle\text{\rm Im}(\varphi)$ in $\displaystyle SO_{2n+1}(\mathbb{C})$ is two (Proposition 4.1.2), Theorem 4.1.1 and (4.9) gives the following

Theorem 4.3.1

The formal degree of the supercuspidal representation $\displaystyle\pi_{\beta,\theta}=\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}$ with respect to the absolute value of the Euker-Poincaré measure on $\displaystyle G(F)$ is

\displaystyle\frac{1}{|\mathcal{A}_{\varphi}|}\cdot\left|\frac{\gamma(\varphi,\text{\rm Ad},\psi,d(x),0)}{\gamma(\varphi_{0},\text{\rm Ad},\psi,d(x),0)}\right|.

Since $\displaystyle\mathcal{A}_{\varphi}$ is a finite abelian group, all the irreducible representation of $\displaystyle\mathcal{A}_{\varphi}$ is one-dimensional. So Theorem 4.3.1 says that the formal degree conjecture is valid if we consider (4.1) as the Arthur-Langlands parameter of the supercuspidal representation $\displaystyle\pi_{\beta,\theta}$ and (4.7) as the principal parameter of $\displaystyle G(F)=Sp_{2n}(F)$ .

5 Root number conjecture

5.1 Structure of adjoint representation

We will identify the representations of $\displaystyle W_{K/F}$ with the representations of $\displaystyle W_{F}$ which factor through the canonical surjection

\displaystyle W_{F}\to W_{F}/\overline{[W_{K}.W_{K}]}=W_{K/F}.

We will also regard a representation of $\displaystyle\Gamma$ as the representation of $\displaystyle W_{K/F}$ via the projection $\displaystyle W_{K/F}\to\Gamma$ .

As we have seen in the subsection 4.1

\text{\rm$\displaystyle\text{\rm Ad}\circ\varphi$ on $\displaystyle\widehat{\mathfrak{g}}$}={\bigwedge}^{2}\varphi={\bigwedge}^{2}\varphi_{1}\oplus\varphi_{1}

(5.1)

with $\displaystyle\varphi_{1}=\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\vartheta}$ . Now we have

Theorem 5.1.1

\displaystyle{\bigwedge}^{2}\varphi_{1}=\bigoplus_{\pi(\tau)\neq 1}\pi^{\dim\pi}\oplus\bigoplus_{\{\gamma\neq\gamma^{-1}\}\subset\Gamma}\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\vartheta}_{\gamma}\oplus\bigoplus_{\stackrel{{\scriptstyle\scriptstyle 1,\tau\neq\gamma\in\Gamma}}{{\gamma^{2}=1}}}\text{\rm Ind}_{W_{K/K_{\gamma}}}^{W_{K/F}}\chi_{\gamma}.

Here $\displaystyle\bigoplus_{\pi(\tau)\neq 1}$ denotes the direct sum over the equivalence classes $\displaystyle\pi$ of the irreducible representations of $\displaystyle\Gamma$ such that $\displaystyle\pi(\tau)\neq 1$ . The direct sum $\displaystyle\bigoplus_{\{\gamma\neq\gamma^{-1}\}\subset\Gamma}$ is over the subsets $\displaystyle\{\gamma,\gamma^{-1}\}\subset\Gamma$ such that $\displaystyle\gamma^{2}\neq 1$ , and $\displaystyle\widetilde{\vartheta}_{\gamma}(x)=\widetilde{\vartheta}(x^{1+\gamma})$ ( $\displaystyle x\in K^{\times}$ ). For a $\displaystyle\gamma\in\Gamma$ of order two, the unitary character $\displaystyle\chi_{\gamma}$ of $\displaystyle W_{K/K_{\gamma}}$ is defined by

	$\displaystyle\displaystyle\chi_{\gamma}:W_{K/K_{\gamma}}=W_{K_{\gamma}}/\overline{[W_{K},W_{K}]}$	$\displaystyle\displaystyle\xrightarrow[\;\text{\rm can.}\;\;]{}W_{K_{\gamma}}/\overline{[W_{K_{\gamma}},W_{K_{\gamma}}]}$
		$\displaystyle\displaystyle\xrightarrow[\delta_{K_{\gamma}}^{-1}]{\sim}K_{\gamma}^{\times}\xrightarrow[(\ast,K/K_{\gamma})\cdot\widetilde{\vartheta}]{}\mathbb{C}^{\times}$

with the subfield $\displaystyle F\subset K_{\gamma}\subset K$ such that $\displaystyle\text{\rm Gal}(K/K_{\gamma})=\langle\gamma\rangle$ and

\displaystyle(x,K/K_{\gamma})=\begin{cases}1&:x\in N_{K/K_{\gamma}}(K^{\times}),\\ -1&:x\not\in N_{K/K_{\gamma}}(K^{\times}).\end{cases}

The rest of this subsection is devoted to the proof of the theorem.

5.1.1

Take a $\displaystyle\gamma\in\Gamma$ of order two. Note that the group homomorphism $\displaystyle x\mapsto\left(x,K/K_{\gamma}\right)$ induces the inverse of the isomorphism

\displaystyle\text{\rm Gal}(K/K_{\gamma})\,\tilde{\to}\,K_{\gamma}^{\times}/N_{K/K_{\gamma}}(K^{\times})\quad(\sigma\mapsto\alpha_{K/F}(\sigma,\gamma))

if we identify $\displaystyle\text{\rm Gal}(K/K_{\gamma})$ with $\displaystyle\{\pm 1\}$ . Then the commutative diagram

implies that we have

\displaystyle\chi_{\gamma}(\sigma,x)=\text{\rm sign}(\sigma)\cdot\widetilde{\vartheta}\left(\alpha_{K/F}(\sigma,\gamma)\cdot x^{1+\gamma}\right)

for $\displaystyle(\gamma,x)\in W_{K/K_{\gamma}}=\text{\rm Gal}(K/K_{\gamma}){\ltimes}_{\alpha_{K/F}}K^{\times}\subset W_{K/F}$ where

\displaystyle\text{\rm sign}(\sigma)=\begin{cases}1&:\sigma=1,\\ -1&:\sigma=\gamma.\end{cases}

By means of the cocycle relation of $\displaystyle\alpha_{K/F}$ , we have

\displaystyle\chi_{\gamma}(\alpha^{-1}(\sigma,x)\alpha)=\begin{cases}\widetilde{\vartheta}(x^{\alpha(1+\gamma)})&:\sigma=1,\\ -\widetilde{\vartheta}\left(\alpha_{K/F}(\gamma,\gamma)^{\alpha}x^{\alpha(1+\gamma)}\right)&:\sigma=\gamma\end{cases}

for any $\displaystyle(\sigma,x)\in W_{K/K_{\gamma}}$ and $\displaystyle\alpha\in\sigma$ . Note that the elements of $\displaystyle\Gamma$ of order two are central as shown by Proposition 3.3.1. The the character of the induced representation $\displaystyle\pi_{\gamma}=\text{\rm Ind}_{W_{K/K_{\gamma}}}^{W_{K/F}}\chi_{\gamma}$ is

	$\displaystyle\displaystyle\chi_{\pi_{\gamma}}(\sigma,x)$	$\displaystyle\displaystyle=\begin{cases}0&:\sigma\not\in\text{\rm Gal}(K/K_{\gamma}),\\ \sum_{\dot{\alpha}\in\Gamma/\langle\gamma\rangle}\chi_{\gamma}\left(\alpha^{-1}(\sigma,x)\alpha\right)&:\sigma\in\text{\rm Gal}(K/K_{\gamma})\\ \end{cases}$
		$\displaystyle\displaystyle=\begin{cases}0&:\sigma\neq 1,\gamma,\\ \sum_{\dot{\alpha}\in\Gamma/\langle\gamma\rangle}\widetilde{\vartheta}\left(x^{\alpha(1+\gamma)}\right)&:\sigma=1,\\ -\sum_{\dot{\alpha}\in\Gamma/\langle\gamma\rangle}\widetilde{\vartheta}\left(\alpha_{K/F}(\gamma,\gamma)^{\alpha}x^{\alpha(1+\gamma)}\right)&:\sigma=\gamma.\end{cases}$		(5.2)

5.1.2

Since $\displaystyle\tau\in\Gamma$ is a central element, the character of the induced representation $\displaystyle R_{\tau}=\text{\rm Ind}_{\langle\tau\rangle}^{\Gamma}\text{\bf 1}_{\langle\tau\rangle}$ is

\displaystyle\chi_{R_{\tau}}(\sigma)=\begin{cases}0&:\sigma\neq 1,\tau,\\ (\Gamma:\langle\tau\rangle)=n&:\sigma=1,\tau.\end{cases}

For an irreducible representation $\displaystyle\pi$ of $\displaystyle\Gamma$ , we have $\displaystyle\pi(\tau)=\pm\text{\bf 1}$ , and we have

\displaystyle\langle\pi,R_{\tau}\rangle=|\Gamma|^{-1}(n\cdot\chi_{\pi}(1)+n\cdot\chi_{\pi}(\tau))=\begin{cases}\dim\pi&:\pi(\tau)=\text{\bf 1},\\ 0&:\pi(\tau)=-\text{\bf 1}.\end{cases}

Hence $\displaystyle R_{\tau}\oplus\bigoplus_{\pi(\tau)\neq 1}\pi^{\dim\pi}$ is the regular representation $\displaystyle R_{\Gamma}=\text{\rm Ind}_{\{1\}}^{\Gamma}\text{\bf 1}$ , and we have

\left(\chi_{R_{\Gamma}}-\chi_{R_{\tau}}\right)(\sigma)=\begin{cases}n&:\sigma=1,\\ -n&:\sigma=\tau,\\ 0&:\sigma\neq 1,\tau.\end{cases}

(5.3)

5.1.3

Recall the character formula (4.6). Since $\displaystyle\widetilde{\vartheta}(x)=\vartheta(x^{1-\tau})$ ( $\displaystyle x\in K^{\times}$ ) and the elements of $\displaystyle\Gamma$ of order two are central (Proposition 3.3.1), we have

\displaystyle\frac{1}{2}\sum_{\alpha\in\Gamma}\widetilde{\vartheta}(x^{\alpha(1+\tau)})=\frac{1}{2}|\Gamma|=n.

Since

\displaystyle\sum_{\alpha\in\Gamma}\widetilde{\vartheta}\left(x^{\alpha(1+\gamma^{-1})}\right)=\sum_{\alpha\in\Gamma}\widetilde{\vartheta}(x^{\alpha(1+\gamma)})

for any $\displaystyle\gamma$ , we have

	$\displaystyle\displaystyle\chi_{{\wedge}^{2}\varphi_{1}}(1,x)$	$\displaystyle\displaystyle=\frac{1}{2}\sum_{\stackrel{{\scriptstyle\scriptstyle\alpha,\gamma\in\Gamma}}{{\gamma\neq 1}}}\widetilde{\vartheta}(x^{\alpha(1+\gamma)})$
		$\displaystyle\displaystyle=n+\sum_{\{\gamma\neq\gamma^{-1}\}\subset\Gamma}\sum_{\alpha\in\Gamma}\widetilde{\vartheta}(x^{\alpha(1+\gamma)})+\sum_{\stackrel{{\scriptstyle\scriptstyle 1,\tau\neq\gamma\in\Gamma}}{{\gamma^{2}=1}}}\sum_{\dot{\alpha}\in\Gamma/\langle\gamma\rangle}\widetilde{\vartheta}(x^{\alpha(1+\gamma)}).$

Take a $\displaystyle\sigma\in\Gamma$ of order two. Then the cocycle relation of $\displaystyle\alpha_{K/F}$ gives $\displaystyle\alpha_{K/F}(\sigma,\sigma)^{\sigma}=\alpha_{K/F}(\sigma,\sigma)$ . Since $\displaystyle\sigma$ is a central element of $\displaystyle\Gamma$ , we have

\displaystyle\alpha_{K/F}(\sigma,\sigma)^{\sigma\alpha}=\alpha_{K/F}(\sigma,\sigma)^{\alpha}

for any $\displaystyle\alpha\in\Gamma$ . If $\displaystyle\sigma=\tau$ , we have

\displaystyle\chi_{{\wedge}^{2}\varphi_{1}}(\tau,x)=-\frac{1}{2}\sum_{\alpha\in\Gamma}\widetilde{\vartheta}(x^{\alpha(1+\tau)})=-\frac{1}{2}|\Gamma|=-n.

If $\displaystyle\sigma\neq\tau$ , then we have

\displaystyle\chi_{{\wedge}^{2}\varphi_{1}}(\sigma,x)=-\sum_{\dot{\alpha}\in\Gamma/\langle\sigma\rangle}\widetilde{\vartheta}\left(\alpha_{K/F}(\sigma,\sigma)^{\alpha}\cdot x^{\alpha(1+\sigma)}\right).

Since the character of the induced representation $\displaystyle\rho_{\gamma}=\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\vartheta}_{\gamma}$ for $\displaystyle\gamma\in\Gamma$ such that $\displaystyle\gamma^{2}\neq 1$ is

\displaystyle\chi_{\rho_{\gamma}}(\sigma,x)=\begin{cases}0&:\sigma\neq 1,\\ \sum_{\alpha\in\Gamma}\widetilde{\vartheta}(x^{\alpha(1+\gamma)})&:\sigma=1,\end{cases}

the formulae (5.2) and (5.3) gives

\displaystyle\chi_{{\wedge}^{2}\varphi_{1}}=\chi_{R_{\Gamma}}-\chi_{R_{\tau}}+\sum_{\{\gamma\neq\gamma^{-1}\}\subset\Gamma}\chi_{\sigma_{\gamma}}+\sum_{\stackrel{{\scriptstyle\scriptstyle 1,\tau\neq\gamma\in\Gamma}}{{\gamma^{2}=1}}}\chi_{\pi_{\gamma}}

which complete the proof of Theorem 5.1.1.

5.2 Root number of adjoint representation

By the decomposition 5.1 and Theorem 5.1.1, the adjoint representation $\displaystyle\text{\rm Ad}\circ\varphi$ of the Weil group $\displaystyle W_{F}$ on $\displaystyle\widehat{\mathfrak{g}}$ is written as a direct sum of representations induced from abelian characters. Using this decomposition, we can calculate the $\displaystyle\varepsilon$ -factor of the adjoint representation. The result is

Theorem 5.2.1

With respect to a additive character $\displaystyle\psi$ of $\displaystyle F$ such that

\displaystyle\{x\in F\mid\psi(xO_{F})=1\}=O_{F}

and the Haar measure $\displaystyle d(x)$ on $\displaystyle F$ such that $\displaystyle\int_{O_{F}}d(x)=1$ , we have

\displaystyle\varepsilon(\varphi,\text{\rm Ad},\psi,d(x))=w(\text{\rm Ad}\circ\varphi)\cdot q^{n^{2}r}

with the root number

\displaystyle w(\text{\rm Ad}\circ\varphi)=\vartheta(-1)\times\begin{cases}(-1)^{\frac{q-1}{2n}\cdot\frac{n(n+1)}{2}}&:\text{\rm$\displaystyle K/K_{+}$ is ramified},\\ 1&:\text{\rm$\displaystyle K/K_{+}$ is unramified and $\displaystyle|H|=2$},\\ -(-1)^{\frac{q^{f_{+}}-1}{2}}&:\text{\rm$\displaystyle K/K_{+}$ is unramified and $\displaystyle|H|=4$}.\end{cases}

Here $\displaystyle H=\{\gamma\in\Gamma\mid\gamma^{2}=1\}$ whose structure is given in Proposition 3.3.1.

Note that if $\displaystyle K/K_{+}$ is ramified, then $\displaystyle K/F$ is totally ramified and hence $\displaystyle 2n=(K:F)$ divides $\displaystyle q-1$ .

The rest of this devoted to the proof of the theorem.

5.2.1

To begin with

\displaystyle\varepsilon(\varphi,\text{\rm Ad},\psi,d(x))=\varepsilon(\text{\rm Ad}\circ\varphi,\psi,d(x))

by the definition. Define the additive character $\displaystyle\psi_{F}$ of $\displaystyle F$ by

\displaystyle\psi_{F}:F\xrightarrow{T_{F/\mathbb{Q}_{p}}}\mathbb{Q}_{p}\xrightarrow{\text{\rm canonical}}\mathbb{Q}_{p}/\mathbb{Z}_{p}\,\tilde{\to}\,\mathbb{Q}/\mathbb{Z}\xrightarrow{\exp(2\pi\sqrt{-1}\ast)}\mathbb{C}^{\times}.

Then

\displaystyle\{x\in F\mid\psi_{F}(xO_{F})=1\}=\mathcal{D}(F/\mathbb{Q}_{p})^{-1}=\mathfrak{p}_{F}^{-d(F)}

and $\displaystyle\psi_{K}=\psi_{F}\circ T_{K/F}$ . Let $\displaystyle d_{F}(x)$ be the Haar measure on $\displaystyle F$ such that

\displaystyle\int_{O_{F}}d_{F}(x)=q^{-d(F)}.

Then

\displaystyle\varepsilon(\text{\rm Ad}\circ\varphi,\psi,d(x))=q^{-n(2n+1)\cdot d(F)/2}\varepsilon(\text{\rm Ad}\circ\varphi,\psi_{F},d_{F}(x)).

Put

\displaystyle\varepsilon(\ast,\psi_{F})=\varepsilon(\ast,\psi_{F},d_{F}(X)),\quad\lambda(K/F,\psi_{F})=\lambda(K/F,\psi_{F},d_{F}(x),d_{K}(x))

for the sake of simplicity. By (5.1) and Theorem 5.1.1, we have

\displaystyle\text{\rm Ad}\circ\varphi=\Pi_{1}\oplus\Pi_{2}\oplus\Pi_{3}

with $\displaystyle\Pi_{1}=\bigoplus_{\pi(\tau)\neq 1}\pi^{\dim\pi}$ and

\displaystyle\Pi_{2}=\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\vartheta}\oplus\bigoplus_{\{\gamma\neq\gamma^{-1}\}\subset\Gamma}\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\vartheta}_{\gamma},\qquad\Pi_{3}=\bigoplus_{\stackrel{{\scriptstyle\scriptstyle 1,\tau\neq\gamma\in\Gamma}}{{\gamma^{2}=1}}}\text{\rm Ind}_{W_{K/K_{\gamma}}}^{W_{K/F}}\chi_{\gamma},

were $\displaystyle\Pi_{3}$ appears only if $\displaystyle|H|=4$ . Note also that we can change in $\displaystyle\Pi_{2}$ , the definition of $\displaystyle\widetilde{\vartheta}_{\gamma}$ to $\displaystyle\widetilde{\vartheta}_{\gamma}(x)=\widetilde{\vartheta}(x^{1-\gamma})$ for $\displaystyle\gamma\in\Gamma$ such that $\displaystyle\gamma\neq\gamma^{-1}$ by replacing $\displaystyle\gamma$ with $\displaystyle\tau\gamma$ . Now we have

\displaystyle\varepsilon(\text{\rm Ad}\circ\varphi,\psi_{F})=\varepsilon(\Pi_{1},\psi_{F})\cdot\varepsilon(\Pi_{2},\psi_{F})\cdot\varepsilon(\Pi_{3},\psi_{F}).

Since $\displaystyle\Pi_{1}=\text{\rm Ind}_{W_{K}}^{W_{F}}\text{\bf 1}_{W_{K}}-\text{\rm Ind}_{W_{K_{+}}}^{W_{F}}\text{\bf 1}_{W_{K_{+}}}$ , we have

	$\displaystyle\displaystyle\varepsilon(\Pi_{1},\psi_{F})=\varepsilon\left(\text{\rm Ind}_{W_{K}}^{W_{F}}\text{\bf 1}_{W_{K}},\psi_{F}\right)\cdot\varepsilon\left(\text{\rm Ind}_{W_{K_{+}}}^{W_{F}}\text{\bf 1}_{W_{K_{+}}}\psi_{F}\right)^{-1}$
	$\displaystyle\displaystyle=\lambda(K/F,\psi_{F})\varepsilon(\text{\bf 1}_{W_{K}},\psi_{K})\cdot\lambda(K_{+}/F,\psi_{F})^{-1}\varepsilon(\text{\bf 1}_{W_{K_{+}}},\psi_{K_{+}})^{-1}$

and

\displaystyle\varepsilon(\text{\bf 1}_{W_{K}},\psi_{K})=q_{K}^{d(K)/2}=q^{f(e\cdot d(F)+e-1)/2}

where $\displaystyle q_{K}=|O_{K}/\mathfrak{p}_{K}|=q^{f}$ . Since $\displaystyle K/F$ is tamely ramified, we have $\displaystyle d(K)=e\cdot d(F)+e-1$ . Similarly we have

\displaystyle\varepsilon(\text{\bf 1}_{W_{K_{+}}},\psi_{K_{+}})=q^{f_{+}(e_{+}\cdot d(F)+e_{+}-1)/2}.

On the other hand, we have

\displaystyle\varepsilon(\Pi_{2},\psi_{F})=\varepsilon(\widetilde{\vartheta},\psi_{K})\prod_{\{\gamma\neq\gamma^{-1}\}\subset\Gamma}\varepsilon(\widetilde{\vartheta}_{\gamma},\psi_{K})\times\begin{cases}\lambda(K/F,\psi_{F})^{n}&:|H|=2,\\ \lambda(K/F,\psi_{F})^{n-1}&:|H|=4.\end{cases}

Now we have

\displaystyle\varepsilon(\widetilde{\vartheta},\psi_{K})=G_{\psi_{K}}\left(\widetilde{\vartheta}^{-1},\varpi_{K}^{-(d(K)+f(\widetilde{\vartheta}))}\right)\cdot\widetilde{\vartheta}(\varpi_{K})^{d(K)+f(\widetilde{\vartheta})}\cdot q_{K}^{(d(K)+f(\widetilde{\vartheta}))/2}.

Since $\displaystyle\widetilde{\vartheta}|_{K_{+}^{\times}}=1$ and $\displaystyle K=K_{+}(\beta)$ , Theorem 3 of [6] says that

\displaystyle G_{\psi_{K}}\left(\widetilde{\vartheta}^{-1},\varpi_{K}^{-(d(K)+f(\widetilde{\vartheta}))}\right)\cdot\widetilde{\vartheta}(\varpi_{K})^{d(K)+f(\widetilde{\vartheta})}=\widetilde{\vartheta}(\beta)=\vartheta(-1).

So we have

\displaystyle\varepsilon(\widetilde{\vartheta},\psi_{K})=\vartheta(-1)\cdot q^{f(e\cdot d(F)+e-1+f(\widetilde{\vartheta}))/2}.

Similarly we have

\displaystyle\varepsilon(\widetilde{\vartheta}_{\gamma},\psi_{K})=q^{f(e\cdot d(F)+e-1+f(\widetilde{\vartheta}_{\gamma}))/2}

for $\displaystyle\gamma\in\Gamma$ such that $\displaystyle\gamma^{2}\neq 1$ , since $\displaystyle\widetilde{\vartheta}_{\gamma}(\beta)=\vartheta((-1)^{1-\gamma})=1$ .

5.2.2

Assume that $\displaystyle K/K_{+}$ is ramified. Then $\displaystyle K/F$ is totally ramified and $\displaystyle H=\{1,\tau=\delta^{n}\}$ . Since $\displaystyle e=2n$ is even, we have

\displaystyle\lambda(K/F,\psi_{F})=(-1)^{\frac{q-1}{2n}\cdot\frac{n(n+1)}{2}}G_{\psi_{F}}(\left(\frac{\ast}{F}\right),\varpi_{F}^{-(d(F)+1)})

by Proposition A.3.5. Similarly we have

\displaystyle\lambda(K_{+}/F,\psi_{F})=\begin{cases}(-1)^{\frac{q-1}{n}\cdot\frac{n(n+2)}{8}}G_{\psi_{F}}(\left(\frac{\ast}{F}\right),\varpi_{F}^{-(d(F)+1)})&:\text{\rm$\displaystyle n$ is even},\\ 1&:\text{\rm$\displaystyle n$ is odd}.\end{cases}

So we have

\displaystyle\varepsilon(\Pi_{1},\psi_{F})=q^{(n\cdot d(F)+n)/2}\times\begin{cases}(-1)^{\frac{q-1}{4}}&:\text{\rm$\displaystyle n$ is even},\\ (-1)^{\frac{q-1}{2}\cdot\frac{n+1}{2}}G_{\psi_{F}}(\left(\frac{\ast}{F}\right),\varpi_{F}^{-(d(F)+1)})&:\text{\rm$\displaystyle n$ is odd}.\end{cases}

By Proposition 3.4.1, we have

\displaystyle f(\widetilde{\vartheta})=\text{\rm Min}\{0<k\in\mathbb{Z}\mid\widetilde{\vartheta}(1+\mathfrak{p}_{K}^{k})=1\}=2n(r-1)

and

\displaystyle f(\widetilde{\vartheta}_{\gamma})=\text{\rm Min}\{0<k\in\mathbb{Z}\mid\widetilde{\vartheta}_{\gamma}(1+\mathfrak{p}_{K}^{k})=1\}=2n(r-1)

for $\displaystyle\gamma\in\Gamma$ such that $\displaystyle\gamma^{2}\neq 1$ . Then we have

\displaystyle\varepsilon(\Pi_{2},\psi_{F})=\vartheta(-1)\cdot q^{n^{2}d(F)+n^{2}r-n/2}\lambda(K/F,\psi_{F})^{n}

and

\displaystyle\lambda(K/F,\psi_{F})^{n}=(-1)^{\frac{q-1}{2}\cdot\frac{n(n+1)}{2}}G_{\psi_{F}}(\left(\frac{\ast}{F}\right),\varpi_{F}^{-(d(F)+1)})^{n}.

Since

\displaystyle G_{\psi_{F}}(\left(\frac{\ast}{F}\right),\varpi_{F}^{-(d(F)+1)})^{2}=\left(\frac{-1}{F}\right)=(-1)^{\frac{q-1}{2}},

we have

\displaystyle\lambda(K/F,\psi_{F})^{n}=(-1)^{\frac{q-1}{s}\cdot\frac{n}{2}}\cdot(-1)^{\frac{q-1}{s}\cdot\frac{n}{2}}=1

if $\displaystyle n$ is even, and

	$\displaystyle\displaystyle\lambda(K/F,\psi_{F})^{n}$	$\displaystyle\displaystyle=(-1)^{\frac{q-1}{2}\cdot\frac{n+1}{2}}\cdot(-1)^{\frac{q-1}{2}\cdot\frac{n-1}{2}}G_{\psi_{F}}(\left(\frac{\ast}{F}\right),\varpi_{F}^{-(d(F)+1)})$
		$\displaystyle\displaystyle=G_{\psi_{F}}(\left(\frac{\ast}{F}\right),-\varpi_{F}^{-(d(F)+1)})$

if $\displaystyle n$ is odd. So we finally get

	$\displaystyle\displaystyle\varepsilon(\text{\rm Ad}\circ\varphi,\psi_{F})$	$\displaystyle\displaystyle=\varepsilon(\Pi_{1},\psi_{F})\cdot\varepsilon(\Pi_{2},\psi_{F})$
		$\displaystyle\displaystyle=\vartheta(-1)\cdot q^{n(2n+1)d(F)/2+n^{2}r}\times\begin{cases}(-1)^{\frac{q-1}{4}}&:\text{\rm$\displaystyle n$ is even},\\ (-1)^{\frac{q-1}{2}\cdot\frac{n+1}{2}}&:\text{\rm$\displaystyle n$ is odd}.\end{cases}$

5.2.3

Assume that $\displaystyle K/K_{+}$ is unramified and $\displaystyle|H|=2$ . In this case, Proposition 3.3.1 shows that $\displaystyle e=e_{+}$ is odd, and $\displaystyle H=\{1,\tau\}$ . Then $\displaystyle f=2f_{+}$ is even, since $\displaystyle ef=2n$ . By Proposition A.3.5, we have

\displaystyle\lambda(K/F,\psi_{F})=(-1)^{(f-1)d(F)}=(-1)^{d(F)},\quad\lambda(K_{+}/F,\psi_{F})=(-1)^{(f_{+}-1)d(F)}.

So we have

\displaystyle\varepsilon(\Pi_{1},\psi_{F},d_{F}(x))=(-1)^{f_{+}\cdot d(F)}q^{(n\cdot d(F)+n-f_{+})/2}.

By Proposition 3.4.1, we have

\displaystyle f(\widetilde{\vartheta})=\text{\rm Min}\{0<k\in\mathbb{Z}\mid\widetilde{\vartheta}(1+\mathfrak{p}_{K}^{k})=1\}=e(r-1)+1,

and

	$\displaystyle\displaystyle f(\widetilde{\vartheta}_{\gamma})$	$\displaystyle\displaystyle=\text{\rm Min}\{0<k\in\mathbb{Z}\mid\widetilde{\vartheta}_{\gamma}(1+\mathfrak{p}_{K}^{k})=1\}$
		$\displaystyle\displaystyle=\begin{cases}e(r-1)&:\gamma\in\{\delta^{\pm 1},\delta^{\pm 2},\cdots,\delta^{\pm\frac{e-1}{2}}\},\\ e(r-1)+1&:\text{\rm otherwise}\end{cases}$

for a $\displaystyle\gamma\in\Gamma$ such that $\displaystyle\gamma\neq\gamma^{-1}$ . So we have

\displaystyle\varepsilon(\Pi_{2},\psi_{F},d_{F}(x))=\vartheta(-1)\cdot(-1)^{n\cdot d(F)}q^{n^{2}(d(F)+r)-)n-f_{+})/2}.

Then finally we have

	$\displaystyle\displaystyle\varepsilon(\text{\rm Ad}\circ\varphi,\psi_{F})$	$\displaystyle\displaystyle=\varepsilon(\Pi_{1},\psi_{F})\cdot\varepsilon(\Pi_{2},\psi_{F})$
		$\displaystyle\displaystyle=\vartheta(-1)\cdot q^{n(2n+1)d(F)/2+n^{2}r}.$

5.2.4

Assume that $\displaystyle K/K_{+}$ is unramified and $\displaystyle|H|=4$ . In this case, Proposition 3.3.1 shows that $\displaystyle e=e_{+},f=2f_{+}$ and $\displaystyle m$ are all even, and

\displaystyle H=\{1,\tau,\delta^{\prime}=\delta^{\frac{e}{2}},\tau^{\prime}=\delta^{\prime}\tau\}.

Put

\displaystyle E=K_{+}\cap K_{\tau^{\prime}}=K_{\tau^{\prime}}\cap K_{\delta^{\prime}}=K_{\delta^{\prime}}\cap K_{+}.

Then $\displaystyle K/K_{+},K/K_{\tau^{\prime}}$ and $\displaystyle K_{\delta^{\prime}}/E$ are unramified quadratic extension, on the other hand $\displaystyle K_{+}/E,K_{\tau^{\prime}}/E$ and $\displaystyle K/K_{\delta^{\prime}}$ are ramified quadratic extension. $\displaystyle K_{0}\subset K_{\delta^{\prime}}\subset K$ and $\displaystyle E_{0}=E\cap K_{0}$ is the maximal unramified subextension of $\displaystyle E/F$ .

By Proposition A.3.6, we have

\displaystyle\lambda(K/F,\psi_{F})=-(-1)^{\frac{q^{f_{+}}-1}{2}}.

By Proposition 3.4.1, we have

\displaystyle f(\widetilde{\vartheta})=\text{\rm Min}\{0<k\in\mathbb{Z}\mid\widetilde{\vartheta}(1+\mathfrak{p}_{K}^{k})=1\}=e(r-1)+1,

and

	$\displaystyle\displaystyle f(\widetilde{\vartheta}_{\gamma})$	$\displaystyle\displaystyle=\text{\rm Min}\{0<k\in\mathbb{Z}\mid\widetilde{\vartheta}_{\gamma}(1+\mathfrak{p}_{K}^{k})=1\}$
		$\displaystyle\displaystyle=\begin{cases}e(r-1)&:\gamma\in\{\delta^{\pm 1},\delta^{\pm 2},\cdots,\delta^{\pm\frac{e-1}{2}}\},\\ e(r-1)+1&:\text{\rm otherwise}\end{cases}$

for a $\displaystyle\gamma\in\Gamma$ such that $\displaystyle\gamma\neq\gamma^{-1}$ . So we have

\varepsilon(\Pi_{1},\psi_{F})\cdot\varepsilon(\Pi_{2},\psi_{F})\\ =\vartheta(-1)\cdot q^{n(2n-1)d(F)/2+n(n-1)r+f_{+}/2}\cdot\lambda(K_{+}/F,\psi_{F})^{-1}.

(5.4)

Proposition 5.2.2

	$\displaystyle\displaystyle\varepsilon(\Pi_{3},\psi_{F})=q^{n\cdot d(F)+nr-f_{+}/2}$	$\displaystyle\displaystyle\cdot\lambda(K_{\tau^{\prime}}/F,\psi_{F})$
		$\displaystyle\displaystyle\times\begin{cases}-1&:\text{\rm$\displaystyle e/2$ is even},\\ (-1)^{d(F)+\frac{q^{f_{+}}-1}{2}}&:\text{\rm$\displaystyle e/2$ is odd}.\end{cases}$

[Proof] We have

\displaystyle\varepsilon(\Pi_{3},\psi_{F})=\prod_{\gamma\in\{\delta^{\prime},\tau^{\prime}\}}\lambda(K_{\gamma}/F,\psi_{F})\cdot\varepsilon(\widetilde{\chi}_{\gamma},\psi_{K_{\gamma}})

where $\displaystyle\widetilde{\chi}_{\gamma}(x)=(x,K/K_{\gamma})\cdot\widetilde{\vartheta}(x)$ ( $\displaystyle x\in K_{\gamma}^{\times}$ ).

1) The case $\displaystyle\gamma=\tau^{\prime}$ . Since $\displaystyle K/K_{\gamma}$ is unramified, we have

\displaystyle(x,K/K_{\gamma})=(-1)^{\text{\rm ord}_{K_{\gamma}}(x)}\quad(x\in K_{\gamma}^{\times})

and $\displaystyle N_{K/K_{\gamma}}(1+\mathfrak{p}_{K}^{k})=1+\mathfrak{p}_{K_{\gamma}}^{k}$ ( $\displaystyle 0<k\in\mathbb{Z}$ ). Then we have

\displaystyle f(\chi_{\gamma})=\text{\rm Min}\{0<k\in\mathbb{Z}\mid\chi_{\gamma}(1+\mathfrak{p}_{K_{\gamma}}^{k})=1\}=e(r-1)

because $\displaystyle\chi_{\gamma}(1+\mathfrak{p}_{K_{\gamma}}^{k})=1$ if and only if $\displaystyle\widetilde{\vartheta}(x^{1-\delta^{\prime}})=1$ for all $\displaystyle x\in 1+\mathfrak{p}_{K}^{k}$ which is equivalent to $\displaystyle k\geq e(r-1)$ by Proposition 3.4.1. Since $\displaystyle K_{\gamma}/E$ is ramified quadratic extension, we have $\displaystyle K_{\gamma}=E(\sqrt{\varpi_{E}})$ where $\displaystyle\varpi_{E}$ is a prime element of $\displaystyle E$ . Then $\displaystyle\widetilde{\chi}_{\gamma}|_{E^{\times}}=1$ and

\displaystyle\widetilde{\chi}_{\gamma}(\sqrt{\varpi_{E}})=(\sqrt{\varpi_{E}},K/K_{\gamma})\cdot\widetilde{\vartheta}(\sqrt{\varpi_{E}})=-\vartheta(-1),

hence we have

\displaystyle G_{\psi_{K_{\gamma}}}(\widetilde{\chi}_{\gamma}^{-1},-\varpi_{K_{\gamma}}^{-(d(K_{\gamma})+f(\widetilde{\chi}_{\gamma}))})\cdot\widetilde{\chi}_{\gamma}(\varpi_{K_{\gamma}})^{d(K_{\gamma})+f(\widetilde{\chi}_{\gamma})}=-\vartheta(-1)

by Theorem 3 of [6]. Then we have

\displaystyle\varepsilon(\widetilde{\chi}_{\gamma},\psi_{K_{\gamma}})=-\vartheta(-1)\cdot q^{(n\cdot d(F)+nr-f_{+})/2}.

2) The case $\displaystyle\gamma=\delta^{\prime}$ . In this case $\displaystyle K/K_{\gamma}$ is ramified quadratic extension. Then we have

\displaystyle f(\widetilde{\chi}_{\gamma})=\text{\rm Min}\{0<k\in\mathbb{Z}\mid\widetilde{\chi}_{\gamma}(1+\mathfrak{p}_{K_{\gamma}}^{k})=1\}=\frac{e}{2}\cdot(r-1)+1

because $\displaystyle\widetilde{\chi}_{\gamma}(1+\mathfrak{p}_{K_{\gamma}}^{k})=1$ if and only if $\displaystyle\widetilde{\vartheta}(x^{1-\tau^{\prime}})=1$ for all $\displaystyle x\in 1+\mathfrak{p}_{K}^{2k}$ which is equivalent to $\displaystyle k\geq\frac{e}{2}\cdot(r-1)+1$ . There exists a prime element $\displaystyle\varpi_{K_{\gamma}}$ of $\displaystyle K_{\gamma}$ such that $\displaystyle K=K_{\gamma}(\sqrt{\varpi_{K_{\gamma}}})$ , and we have

\displaystyle(\varpi_{K_{\gamma}},K/K_{\gamma})=(-1,K/K_{\gamma})=(-1)^{\frac{q^{f}-1}{2}}=1

since $\displaystyle f=2f_{+}$ is even. On the other hand $\displaystyle K_{\gamma}/E$ is unramified quadratic extension, and we have

\displaystyle(\varepsilon,K/K_{\gamma})=(\varepsilon,K_{\gamma}/E)=1

for all $\displaystyle\varepsilon\in O_{E}^{\times}$ . Hence $\displaystyle\widetilde{\chi}_{\gamma}|_{E^{\times}}=1$ . If we put $\displaystyle K_{\gamma}=E(\sqrt{\varepsilon})$ with $\displaystyle\varepsilon\in O_{E}^{\times}$ , then we have

\displaystyle G_{\psi_{K_{\gamma}}}(\widetilde{\chi}_{\gamma}^{-1},-\varpi_{K_{\gamma}}^{-(d(K_{\gamma})+f(\widetilde{\chi}_{\gamma}))})=\widetilde{\chi}_{\gamma}(\sqrt{\varepsilon})=(\sqrt{\varepsilon},K/K_{\gamma})\cdot\vartheta(-1)

by Theorem 3 of [6]. It is shown in the proof of Proposition 3.3.2 that

\displaystyle(\sqrt{\varepsilon},K/K_{\gamma})=-(-1)^{\frac{q^{f_{+}}-1}{2}}.

So we have

\displaystyle\varepsilon(\widetilde{\chi}_{\gamma},\psi_{K_{\gamma}})=-(-1)^{\frac{q^{f_{+}}-1}{2}}\vartheta(-1)\cdot q^{(n\cdot d(F)+nr)/2}.

Since $\displaystyle K_{\gamma}/E$ is unramified quadratic extension, we have

\displaystyle\lambda(K_{\gamma}/F,\psi_{F})=\begin{cases}-(-1)^{\frac{q^{f_{+}}-1}{2}}&:\text{\rm$\displaystyle e/2$ is even},\\ (-1)^{d(F)}&:\text{\rm$\displaystyle e/2$ is odd}.\end{cases}

by Proposition A.3.6. $\displaystyle\blacksquare$

Proposition 5.2.3

\displaystyle\lambda(K_{\tau^{\prime}}/F,\psi_{F})\cdot\lambda(K_{+}/F,\psi_{F})^{-1}=\begin{cases}1&:\text{\rm$\displaystyle e/2$ is even},\\ (-1)^{d(F)+1}&:\text{\rm$\displaystyle e/2$ is odd}.\end{cases}

[Proof] Since $\displaystyle K_{\delta^{\prime}}/E$ is an unramified quadratic extension, put $\displaystyle K_{\delta^{\prime}}=E(\sqrt{\varepsilon})$ with $\displaystyle\varepsilon\in O_{E}^{\times}$ . Then $\displaystyle{\sqrt{\varepsilon}}^{\tau^{\prime}}=-\sqrt{\varepsilon}$ . Since $\displaystyle K_{+}/E$ is a ramified quadratic extension, we have $\displaystyle K_{+}=E(\varpi_{K_{+}})$ with a prime element $\displaystyle\varpi_{K_{+}}$ of $\displaystyle K_{+}$ such that $\displaystyle\varpi_{K_{+}}^{2}\in E$ . Then $\displaystyle{\varpi_{K_{+}}}^{\tau^{\prime}}=-\varpi_{K_{+}}$ , and hence $\displaystyle\varpi_{K_{\tau^{\prime}}}=\sqrt{\varepsilon}\cdot\varpi_{K_{+}}$ is a prime element of $\displaystyle K_{\tau^{\prime}}$ such that $\displaystyle K_{\tau^{\prime}}=E(\varpi_{K_{\tau^{\prime}}})$ and $\displaystyle{\varpi_{K_{\tau^{\prime}}}}^{2}\in E$ . Then

\displaystyle\varpi_{+}=N_{K_{+}/E_{0}}(\varpi_{K_{+}})\;\;\text{\rm and}\;\;\varpi_{\tau^{\prime}}=N_{K_{\tau^{\prime}}/E_{0}}(\varpi_{K_{\tau^{\prime}}})

are prime elements of $\displaystyle E_{0}$ , since $\displaystyle K_{+}/E_{0}$ and $\displaystyle K_{\tau^{\prime}}/E_{0}$ are totally ramified extension. On the other hand, we have

\displaystyle N_{K_{\tau^{\prime}}/E}(\varpi_{K_{\tau^{\prime}}})=-{\varpi_{K_{\tau^{\prime}}}}^{2}=-\varepsilon\cdot{\varpi_{K_{+}}}^{2}=\varepsilon\cdot n_{K_{+}/E}(\varpi_{K_{+}}),

and hence $\displaystyle\varpi_{\tau^{\prime}}=N_{E/E_{0}}(\varepsilon)\cdot\varpi_{+}$ . Now we have

		$\displaystyle\displaystyle\lambda(K_{\tau^{\prime}}/F,\psi_{F})\cdot\lambda(K_{+}/F,\psi_{F})^{-1}$
	$\displaystyle\displaystyle=$	$\displaystyle\displaystyle G_{\psi_{E_{0}}}(\left(\frac{\ast}{E_{0}}\right),-\varpi_{\tau^{\prime}}^{-(d(E_{0})+1)})\cdot G_{\psi_{E_{0}}}(\left(\frac{\ast}{E_{0}}\right),-\varpi_{+}^{-(d(E_{0})+1)})^{-1}$
	$\displaystyle\displaystyle=$	$\displaystyle\displaystyle\left(\frac{N_{E/E_{0}}(\varepsilon)}{E_{0}}\right)^{d(F)+1}$

by Proposition A.3.5. Since $\displaystyle K_{+}/E_{0}$ is a tamely totally ramified extension, and hence a cyclic extension, let $\displaystyle E_{0}\subset M\subset K_{+}$ be the intermediate field such that $\displaystyle(M:E_{0})=2$ . Then

\displaystyle\left(\frac{N_{E/E_{0}}(\varepsilon)}{E_{0}}\right)=(N_{E/E_{0}}(\varepsilon),M/E_{0}).

If $\displaystyle e/2$ is even, then $\displaystyle M\subset E$ because $\displaystyle(E:E_{0})=e/2$ , and hence

\displaystyle\left(\frac{N_{E/E_{0}}(\varepsilon)}{E_{0}}\right)=(N_{M/E_{0}}\left(N_{E/M}(\varepsilon)\right),M/E_{0})=1.

Assume that $\displaystyle e/2$ is odd. Since $\displaystyle K_{+}/E$ is a ramified quadratic extension and $\displaystyle\varepsilon\in O_{E}^{\times}$ is not square, we have

\displaystyle(\varepsilon,K_{+}/E)=\left(\frac{\varepsilon}{E}\right)=-1.

On the other hand, we have

\displaystyle(\varepsilon,K_{+}/E)=\left(N_{E/E_{0}}(\varepsilon),K_{+}/E_{0}\right)\in\text{\rm Gal}(K_{+}/E)\subset\text{\rm Gal}(K_{+}/E_{0})

and $\displaystyle\left(N_{E/E_{0}}(\varepsilon),K_{+}/E_{0}\right)$ is mapped to $\displaystyle\left(N_{E/E_{0}}(\varepsilon),M/E_{0}\right)$ by the restriction mapping

\displaystyle\text{\rm Gal}(K_{+}/E_{0})\to\text{\rm Gal}(M/E_{0}).

$\displaystyle M\not\subset E$ Since $\displaystyle(E:E_{0})=e/2$ is odd, hence $\displaystyle K_{+}=ME$ and $\displaystyle M\cap E=E_{0}$ . Then the restriction mapping gives the isomorphism

\displaystyle\text{\rm Gal}(K_{+}/E)\,\tilde{\to}\,\text{\rm Gal}(M/E_{0}),

hence we have

\displaystyle\left(\frac{N_{E/E_{0}}(\varepsilon)}{E_{0}}\right)=\left(N_{E/E_{0}},M/E_{0}\right)=\left(\varepsilon,K_{+}/E\right)=-1.

$\displaystyle\blacksquare$

(5.4) and Proposition 5.2.2 combined with Proposition 5.2.3 gives

	$\displaystyle\displaystyle\varepsilon(\text{\rm Ad}\circ\varphi,\psi_{F})$	$\displaystyle\displaystyle=\varepsilon(\Pi_{1},\psi_{F})\cdot\varepsilon(\Pi_{2},\psi_{F})\cdot\varepsilon(\Pi_{3},\psi_{F})$
		$\displaystyle\displaystyle=\vartheta(-1)\cdot w^{n(2n+1)\cdot d(F)/2+n^{2}r}\times\begin{cases}-1&:\text{\rm$\displaystyle e/2$ is even},\\ -(-1)^{\frac{q^{f_{+}}-1}{2}}&:\text{\rm$\displaystyle e/2$ is odd}.\end{cases}$

Since $\displaystyle K_{+}/F$ is a tamely ramified extension such that $\displaystyle e(K_{+}/F)=e$ and $\displaystyle f(K_{+}/F)=f_{+}$ , and $\displaystyle e$ is even, $\displaystyle e/2$ divides $\displaystyle(q^{f_{+}}-1)/2$ . Hence $\displaystyle(-1)^{\frac{q^{f_{+}}-1}{2}}=1$ if $\displaystyle e/2$ is even. The proof of the formula of Theorem 5.2.1 is completed.

5.3 Verification of root number conjecture

Let $\displaystyle D$ be the maximal torus of $\displaystyle Sp_{2n}$ consisting of the diagonal matrices. The group $\displaystyle X^{\vee}(D)$ of the one-parameter subgroups of $\displaystyle D$ is identified with $\displaystyle\mathbb{Z}^{n}$ by $\displaystyle m\mapsto u_{m}$ where

\displaystyle u_{m}(t)=\begin{bmatrix}t^{m}&0\\ 0&t^{-m}\end{bmatrix}\;\;\text{\rm with}\;\;t^{m}=\begin{bmatrix}t^{m_{1}}&&\\ &\ddots&\\ &&t^{m_{n}}\end{bmatrix}\in GL_{n},

or we will denote by $\displaystyle u_{m}=\sum_{i=1}^{n}m_{i}\cdot u_{i}$ . Then the set of the co-roots of $\displaystyle SP_{2n}$ with respect to $\displaystyle D$ is

\displaystyle\Phi^{\vee}(D)=\{\pm(u_{i}\pm u_{j}),\pm u_{k}\mid 1\leq i<j\leq n,1\leq k\leq n\}.

Now we have

	$\displaystyle\displaystyle 2\cdot\rho$	$\displaystyle\displaystyle=\sum_{1\leq i<j\leq n}(u_{i}-u_{j})+\sum_{1\leq i<j\leq n}(u_{i}+u_{j})+\sum_{k=1}^{n}2u_{k}$
		$\displaystyle\displaystyle=2\sum_{i=1}^{n}\{2(n-i)+1\}\cdot u_{i}.$

So the special central element is

\displaystyle\epsilon=2\cdot\rho(-1)=-1_{2n}\in Sp_{2n}(F).

If we recall

\displaystyle\pi_{\beta,\theta}=\text{\rm ind}_{G(O_{F})}^{G(F)}\delta_{\beta,\theta}\;\;\text{\rm with}\;\;\delta_{\beta,\theta}=\text{\rm Ind}_{G(O_{F}/\mathfrak{p}_{F}^{r};\beta)}^{G(O_{F}/\mathfrak{p}_{F}^{r})}\sigma_{\beta,\theta}

and the construction of $\displaystyle\sigma_{\beta,\theta}$ , we have

\displaystyle\pi_{\beta,\theta}(\epsilon)=\delta_{\beta,\theta}(\epsilon)=\sigma_{\beta,\theta}(\epsilon)=\theta(-1).

Since $\displaystyle\vartheta=c\cdot\theta$ , Theorem 5.2.1 and Proposition 3.3.2 show that

\displaystyle w(\text{\rm Ad}\circ\varphi)=\theta(-1)\times\begin{cases}1&:\text{\rm$\displaystyle K/K_{+}$ is unramified},\\ (-1)^{\frac{q-1}{2n}\cdot\frac{n(n-1)}{2}}&:\text{\rm$\displaystyle K/K_{+}$ is ramified}.\end{cases}

So we have proved the following theorem.

Theorem 5.3.1

If $\displaystyle K/F$ is not totally ramified or $\displaystyle K/F$ is totally ramified and

\displaystyle\frac{q-1}{2}\cdot(n-1)\equiv 0\!\!\pmod{4},

then we have $\displaystyle w(\text{\rm Ad}\circ\varphi)=\pi_{\beta,\theta}(\epsilon)$ .

This theorem says that the root number conjecture is valid if we consider $\displaystyle\varphi$ as the Langlands parameter of the supercuspidal representation $\displaystyle\pi_{\beta,\theta}$ under the required conditions.

6 The case of $\displaystyle Sp_{4}(F)$

In this section, let us assume that $\displaystyle K/F$ is a quintic Galois extension, and consider a candidate of the Langlands parameter of the supercuspidal representation $\displaystyle\pi_{\beta,\theta}$ of $\displaystyle Sp_{4}(F)$ different from the parameter considered in the subsection 3.4. Note that $\displaystyle\Gamma=\text{\rm Gal}(K/F)$ is a cyclic group if and only if $\displaystyle K/F$ is unramified or totally ramified.

The proofs are omitted because they are quite similar to those of the preceding sections.

6.1 Another candidate for the Langlands parameter

The character $\displaystyle\theta$ of $\displaystyle U_{K/K_{+}}$ which parametrizes the supercuspidal representation $\displaystyle\pi_{\beta,\theta}$ defines the character $\displaystyle\widetilde{\theta}$ of $\displaystyle K^{\times}$ by $\displaystyle\widetilde{\theta}(x)=\theta(x^{1-\tau})$ . Then the representation space $\displaystyle V_{\theta}$ of the induced representation $\displaystyle\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\theta}$ has $\displaystyle W_{K/F}$ -quasi invariant anti-symmetric form

\displaystyle D_{\nu}(\varphi,\psi)=\sum_{\gamma\in\text{\rm Gal(K/F)}}\nu(\gamma)\cdot\widetilde{\theta}\left(\alpha_{K/F}(\gamma,\tau)\right)^{-1}\varphi(\gamma)\psi(\gamma\tau)

where $\displaystyle\nu$ is a character of $\displaystyle G\text{\rm Gal}(K/F)$ such that $\displaystyle\nu(\tau)=-1$ (c.f. appendix B). Let us identify $\displaystyle GSp(V_{\theta},D_{\nu})$ with $\displaystyle GSp_{4}(\mathbb{C})$ by means of the symplectic basis $\displaystyle\{u_{\rho},v_{\rho}\}_{\dot{\rho}\in\Gamma/\langle\tau\rangle}$ . Then we have a group homomorphism

\varphi:W_{F}\xrightarrow{\text{\rm can.}}W_{K/F}\xrightarrow{\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\theta}}GSp_{4}(\mathbb{C})\xrightarrow{(\ast)}SO_{5}(\mathbb{C})

(6.1)

where $\displaystyle(\ast)$ is the accidental surjection. The admissible representation of the Weil-Deligne group $\displaystyle W_{F}\times SL_{2}(\mathbb{C})$ to $\displaystyle SO_{5}(\mathbb{C})$ corresponding to the triple $\displaystyle(\varphi,SO_{5}(\mathbb{C}),0)$ as explained in appendix A.6 is

W_{F}\times SL_{2}(\mathbb{C})\xrightarrow{\text{\rm proj.}}W_{F}\xrightarrow{\varphi}SO_{5}(\mathbb{C})

(6.2)

which is also denoted by $\displaystyle\varphi$ .

6.2 Formal degree conjecture

By writing down the parameter (6.2) explicitly as in the subsection 4.1, we have

Z_{SO_{5}(\mathbb{C})}(\text{\rm Im}\varphi)\simeq=\begin{cases}\mathbb{Z}/2\mathbb{Z}&:\text{\rm$\displaystyle K/F$ is unramified or totally ramified},\\ \mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}&:\text{\rm otherwise}\end{cases}

(6.3)

and

L(\varphi,\text{\rm Ad},s)=\begin{cases}1&:\text{\rm$\displaystyle K/K_{+}$ is ramified,}\\ \frac{1}{1+q^{-f_{+}s}}&:\text{\rm$\displaystyle K/K_{+}$ is unramified.}\end{cases}

(6.4)

The Artin conductor of $\displaystyle\text{\rm Ad}\circ\varphi$ is

a(\text{\rm Ad}\circ\varphi)=8r.

(6.5)

Then

\frac{1}{\left|Z_{SO_{5}(\mathbb{C})}(\text{\rm Im}\varphi)\right|}\cdot\left|\frac{\gamma(\varphi,\text{\rm Ad},0)}{\gamma(\varphi_{0},\text{\rm Ad},0)}\right|

(6.6)

gives the formal degree of the supercuspidal representation $\displaystyle\pi_{\beta,\theta}$ given by (4.10) if $\displaystyle K/F$ is unramified or totally ramified. If $\displaystyle K/F$ is ramified not totally ramified, this is not the case, that is, the order of the centralizer $\displaystyle Z_{SO_{5}(\mathbb{C})}(\text{\rm Im}\varphi)$ is twice as big as required, in other words, the image $\displaystyle\text{\rm Im}(\varphi)$ of the parameter is too small.

6.3 The root number conjecture

Since the parameter (6.2) failed the formal degree conjecture if $\displaystyle K/F$ is ramified not totally ramified, we will consider in this subsection, the root number conjecture in the case of $\displaystyle K/F$ being unramified or totally ramified.

In this case $\displaystyle K/F$ is a cyclic extension. So we put $\displaystyle\text{\rm Gal}(K/F)=\langle\rho\rangle$ so that $\displaystyle\tau=\rho^{2}$ . Then the representation (6.1) has a decomposition $\displaystyle\varphi=\varphi_{1}\oplus\det\varphi_{1}$ with

\displaystyle\varphi_{1}:W_{F}\xrightarrow{\text{\rm can.}}W_{K/F}\xrightarrow{\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\theta}_{\rho}}O_{4}(\mathbb{C})

where $\displaystyle\widetilde{\theta}_{\rho}(x)=\widetilde{\theta}(x^{1-\rho})$ ( $\displaystyle x\in K^{\times}$ ). Then we have

\displaystyle\text{\rm Ad}\circ\varphi=\bigoplus_{\stackrel{{\scriptstyle\chi\in\widehat{\Gamma}}}{{\chi(\tau)\neq 1}}}\chi\oplus\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\theta}^{2}\oplus\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\theta}_{\rho}.

The epsilon factor with respect to the additive character and the Haar measure normalized as in Theorem 5.2.1 is

\displaystyle\varepsilon(\varphi,\text{\rm Ad},\psi,d(x))=w(\text{\rm Ad}\circ\varphi)\cdot q^{4r}

with the root number

\displaystyle w(\text{\rm Ad}\circ\varphi)=\begin{cases}1&:\text{\rm$\displaystyle K/F$ is unramified,}\\ (-1)^{\frac{q-1}{4}}&:\text{\rm$\displaystyle K/F$ is totally ramified.}\end{cases}

This means that the root number conjecture is valid if and only if

\displaystyle\theta(-1)=\begin{cases}1&:\text{\rm$\displaystyle K/F$ is unramified,}\\ (-1)^{\frac{q-1}{4}}&:\text{\rm$\displaystyle K/F$ is totally ramified.}\end{cases}

In other words, the parameter (6.2) is note the Langlands parameter of the supercuspidal representation $\displaystyle\pi_{\beta,\theta}$ in general.

Appendix A Local factors

Fix an algebraic closure $\displaystyle F^{\text{\rm alg}}$ of $\displaystyle F$ in which we will take every algebraic extensions of $\displaystyle F$ . Put

\displaystyle\nu_{F}(x)=(F(x):F)^{-1}\text{\rm ord}_{F}(N_{F(x)/F}(x))\;\text{\rm for $\displaystyle\forall x\in F^{\text{\rm alg}}$}

and

\displaystyle O_{K}=\{x\in F^{\text{\rm alg}}\mid\nu_{F}(x)\geq 0\},\quad\mathfrak{p}_{K}=\{x\in F^{\text{\rm alg}}\mid\nu(F)(x)>0\}.

Then $\displaystyle\mathbb{K}=O_{K}/\mathfrak{p}_{K}$ is an algebraic extension of $\displaystyle\mathbb{F}=O_{F}/\mathfrak{p}_{F}$ . If $\displaystyle K/F$ is a finite extension, fix a generator $\displaystyle\varpi_{K}\in O_{K}$ of $\displaystyle\mathfrak{p}_{K}$ .

A.1 Weil group

Let $\displaystyle F^{\text{\rm ur}}$ be the maximal unramified extension of $\displaystyle F$ and $\displaystyle\text{\rm Fr}\in\text{\rm Gal}(F^{\text{\rm ur}}/F)$ the inverse of the Frobenius automorphism of $\displaystyle F^{\text{\rm ur}}$ over $\displaystyle F$ . The the Weil group $\displaystyle W_{F}$ of $\displaystyle F$ is

\displaystyle W_{F}=\left\{\sigma\in\text{\rm Gal}(F^{\text{\rm alg}}/F)\mid\sigma|_{F^{\text{\rm ur}}}\in\langle\text{\rm Fr}\rangle\right\}

The group $\displaystyle W_{F}$ is a locally compact group with respect to the topology such that $\displaystyle I_{F}=\text{\rm Gal}(F^{\text{\rm alg}}/F^{\text{\rm ur}})$ is an open compact subgroup of $\displaystyle W_{F}$ .

Let $\displaystyle F^{\text{\rm ab}}$ be the maximal abelian extension of $\displaystyle F$ in $\displaystyle F^{\text{\rm alg}}$ . Then

\displaystyle\overline{[W_{F},W_{F}]}=\text{\rm Gal}(F^{\text{\rm alg}}/F^{\text{\rm ab}})

and

\displaystyle W_{F}/\overline{[W_{F},W_{F}]}\xrightarrow[\text{\rm res.}]{\sim}\{\sigma\in\text{\rm Gal}(F^{\text{\rm ab}}/F)\mid\sigma|_{F^{\text{\rm ur}}}\in\langle\text{\rm Fr}\rangle\}.

So, by the local class field theory, there exists a topological group isomorphism

\displaystyle\delta_{F}:F^{\times}\,\tilde{\to}\,W_{F}/\overline{[W_{F},W_{F}]}

such that $\displaystyle\delta_{F}(\varpi)|_{F^{\text{\rm ur}}}=\text{\rm Fr}$ . Fix a $\displaystyle\widetilde{\text{\rm Fr}}\in\text{\rm Gal}(F^{\text{\rm alg}}/F)$ such that $\displaystyle\widetilde{\text{\rm Fr}}|_{F^{\text{\rm ab}}}=\delta_{F}(\varpi)$ . Then

\displaystyle W_{F}=\langle\widetilde{\text{\rm Fr}}\rangle\ltimes\text{\rm Gal}(F^{\text{\rm alg}}/F^{\text{\rm ur}}).

Let $\displaystyle K/F$ be a finite extension in $\displaystyle F^{\text{\rm alg}}$ . Then $\displaystyle K^{\text{\rm ur}}=K\cdot F^{\text{\rm ur}}$ and

\displaystyle W_{K}=\{\sigma\in\text{\rm Gal}(F^{\text{\rm alg}}/K)\mid\sigma|_{F^{\text{\rm ur}}}\in\langle\text{\rm Fr}^{f}\rangle\}=\{\sigma\in W_{F}\mid\sigma|_{K}=1\},

where $\displaystyle f=(\mathbb{K}:\mathbb{F})$ , is a closed subgroup of $\displaystyle W_{F}$ . If further $\displaystyle K/F$ is a Galois extension, then $\displaystyle\overline{[W_{K},W_{K}]}\triangleleft W_{F}$ and

\displaystyle W_{K/F}=W_{F}/\overline{[W_{K},W_{K}]}=\{\sigma\in\text{\rm Gal}(K^{\text{\rm ab}}/F)\mid\sigma|_{F^{\text{\rm ur}}}\in\langle\text{\rm Fr}\rangle\}

is called the relative Weil group of $\displaystyle K/F$ . Then we have a exact sequence

\displaystyle 1\to K{\times}\xrightarrow{\delta_{K}}W_{K/F}\xrightarrow{\text{\rm res.}}\text{\rm Gal}(K/F)\to 1

which is the group extension associated with the fundamental calss

\displaystyle[\alpha_{K/F}]\in H^{2}(\text{\rm Gal}(K/F),K^{\times}),

that is, we can identify $\displaystyle W_{K/F}=\text{\rm Gal}(K/F)\times K^{\times}$ with the group operation

\displaystyle(\sigma,x)\cdot(\tau,y)=(\sigma\tau,\alpha_{K/F}(\sigma,\tau)\cdot xy).

Let $\displaystyle K_{0}=K\cap F^{\text{\rm ur}}$ be the maximal unramified subextension of $\displaystyle K/F$ . Then the fundamental calss can be chosen so that $\displaystyle\alpha_{K/F}(\sigma,\tau)\in O_{K}^{\times}$ for all $\displaystyle\sigma,\tau\in\text{\rm Gal}(K/K_{0})$ , and the image $\displaystyle I_{K/F}$ of $\displaystyle I_{F}=\text{\rm Gal}(F^{\text{\rm alg}}/F^{\text{\rm ur}})\subset W_{F}$ under the canonical surjection $\displaystyle W_{F}\to W_{K/F}$ is identified with $\displaystyle\text{\rm Gal}(K/K_{0})\times O_{K}^{\times}$ .

A.2 Artin conductor of representations of Weil group

Let $\displaystyle(\Phi,V)$ be a finite dimensional continuous complex representation of the Weil group $\displaystyle W_{F}$ . Since $\displaystyle I_{F}\cap\text{\rm Ker}(\Phi)$ is an open subgroup of $\displaystyle I_{F}=\text{\rm Gal}(F^{\text{\rm alg}}/F^{\text{\rm ur}})$ , there exists a finite Galois extension $\displaystyle K/F^{\text{\rm ur}}$ such that $\displaystyle text{\rm Gal}(F^{\text{\rm alg}}/K)\subset\text{\rm Ker}(\Phi)$ . Let

	$\displaystyle\displaystyle V_{k}=$	$\displaystyle\displaystyle V_{k}(K/F^{\text{\rm ur}})$
	$\displaystyle\displaystyle=$	$\displaystyle\displaystyle\left\{\sigma\in\text{\rm Gal}(K/F^{\text{\rm ur}})\mid x^{\sigma}\equiv x\!\!\pmod{\mathfrak{p}_{K}^{k+1}}\,\text{\rm for}\,\forall x\in O_{K}\right\}$

be the $\displaystyle k$ -th ramification group of $\displaystyle K/F^{\text{\rm ur}}$ put

\displaystyle\widetilde{V}_{k}=\left[\text{\rm Gal}(F^{\text{\rm alg}}/F^{\text{\rm ur}})\xrightarrow{\text{\rm res.}}\text{\rm Gal}(K/F^{\text{\rm ur}})\right]^{-1}V_{k}

for $\displaystyle k=0,1,2,3,\cdots$ . So $\displaystyle\widetilde{V}_{0}=I_{F}$ . The Artin conductor $\displaystyle a(\Phi)=a(V)$ is defined by

\displaystyle a(\Phi)=a(V)=\sum_{k=0}^{\infty}\dim_{\mathbb{C}}(V/V^{\Phi(\widetilde{V}_{k})})\cdot|V_{0}/V_{k}|^{-1}

where

\displaystyle V^{\Phi(\widetilde{V}_{k})}=\{v\in V\mid\Phi(\widetilde{V}_{k})v=v\}\quad(k=0,1,2,3,\cdots).

A.3 $\displaystyle\varepsilon$ -factor of representations of Weil group

Fix a continuous unitary character $\displaystyle\psi:F\to\mathbb{C}^{\times}$ of the additive group $\displaystyle F$ and a Haar measure $\displaystyle d(x)$ of $\displaystyle F$ .

Langlands and Deligne [1] show that, for every finite dimensional continuous complex representation $\displaystyle(\Phi,V)$ of $\displaystyle W_{F}$ , there exists a complex constant

\displaystyle\varepsilon(\Phi,\psi,d(x))=\varepsilon(V,\psi,d(x))

which satisfies the following relations:

1)

an exact sequence

$\displaystyle 1\to V^{\prime}\to V\to V^{\prime\prime}\to 1$

implies

$\displaystyle\varepsilon(V,\psi,d(x))=\varepsilon(V^{\prime},\psi,d(x))\cdot\varepsilon(V^{\prime\prime},\psi,d(x)),$
2)

for a positive real number $\displaystyle r$

$\displaystyle\varepsilon(\Phi,\psi,r\cdot d(x))=r^{\dim\Phi}\cdot\varepsilon(\Phi,\psi,d(x)),$

for any finite extension $\displaystyle K/F$ and a finite dimensional continuous complex representation $\displaystyle\phi$ of $\displaystyle W_{K}$ , we have

\displaystyle\varepsilon\left(\text{\rm Ind}_{W_{K}}^{W_{F}}\phi,\psi,d(x)\right)=\varepsilon\left(\phi,\psi\circ T_{K/F},d^{(K)}(x)\right)\cdot\lambda(K/F,\psi)^{\dim\phi}

where $\displaystyle d^{(K)}(x)$ is a Haar measure of $\displaystyle K$ and

\displaystyle\lambda(K/F,\psi)=\lambda(K/F,\psi,d(x),d^{(K)}(x))=\frac{\varepsilon\left(\text{\rm Ind}_{W_{K}}^{W_{F}}\text{\bf 1}_{K},\psi,d(x)\right)}{\varepsilon\left(\text{\bf 1}_{K},\psi\circ T_{K/F},d^{(K)}(x)\right)},

4)

if $\displaystyle\dim\Phi=1$ , then $\displaystyle\Phi$ factors through $\displaystyle W_{F}/\overline{[W_{F},W_{F}]}$ and put

$\displaystyle\chi:F^{\times}\xrightarrow{\delta_{F}}W_{F}/\overline{[W_{F},W_{F}]}\xrightarrow{\Phi}\mathbb{C}~{}{\times}.$

Then we have

$\displaystyle\varepsilon(\Phi,\psi,d(x))=\varepsilon(\chi,\psi,d(x))$

where the right hand side is the $\displaystyle\varepsilon$ -factor of Tate [18].

By the definition of $\displaystyle\lambda(K/F,\psi)$ , we have the following chain rule for the finite extensions:

Proposition A.3.1

For finite extensions $\displaystyle F\subset K\subset L$ , we have

\displaystyle\lambda(L/F,\psi)=\lambda(L/K,\psi\circ T_{K/F})\cdot\lambda(K/F,\psi)^{(L:K)}.

If the Haar measure $\displaystyle d(x)$ of $\displaystyle F$ is normalized so that the Fourier transform

\displaystyle\widehat{\varphi}(y)=\int_{F}\varphi(x)\cdot\psi(-xy)d(x)

has inverse transform

\displaystyle\varphi(x)=\int_{F}\widehat{\varphi}(y)\cdot\psi(xy)d(y),

in other words

\displaystyle\int_{O_{F}}d(x)=q^{-n(\psi)/2}\;\;\text{\rm with}\;\;\{x\in F\mid\psi(xO_{F})=1\}=\mathfrak{p}_{F}^{-n(\psi)},

then the explicit value of the $\displaystyle\varepsilon$ -factor $\displaystyle\varepsilon(\chi,\psi,d(x))$ is

1)

if $\displaystyle\chi|_{O_{F}^{\times}}=1$ , then

$\varepsilon(\chi,\psi,d(x))=\chi(\varpi)^{n(\psi)}\cdot q^{n(\psi)/2},$ (A.1)

if $\displaystyle\chi|_{O_{F}^{\times}}\neq 1$ , then

\varepsilon(\chi,\psi,d(x))=G_{\psi}(\chi^{-1},-\varpi^{-(n(\psi)+f(\chi))})\cdot\chi(\varpi)^{n(\psi)+f(\chi)}\cdot q^{-(n(\psi)+f(\chi))/2}

(A.2)

where $\displaystyle f(\chi)=\text{\rm Min}\{0<n\in\mathbb{Z}\mid\chi(1+\mathfrak{p}_{F}^{n})=1\}$ and

\displaystyle G_{\psi}(\chi^{-1},\varpi^{-(n(\psi)+f(\chi))})=q^{-n/2}\sum_{\dot{t}\in(O_{F}/\mathfrak{p}_{F}^{f(\chi)})^{\times}}\chi(t)^{-1}\psi\left(-\varpi^{-(n(\psi)+f(\chi))}t\right)

is the Gauss sum.

Remark A.3.2

The definition of the Gauss sum is normalized so that

\displaystyle\left|G_{\psi}(\chi^{-1},-\varpi^{-(n(\psi)+f(\chi))})\right|=1.

We have

Proposition A.3.3

1)

Put $\displaystyle\psi_{a}(x)=\psi(ax)$ for $\displaystyle a\in F^{\times}$ . Then

$\displaystyle\varepsilon(\Phi,\psi_{a},d(x))=\det\Phi(a)\cdot|a|_{F}^{-\dim\Phi}\cdot\varepsilon(\Phi,\psi,d(x))$

where

$\displaystyle\det\Phi:F^{\times}\xrightarrow{\delta_{F}}W_{F}/\overline{[W_{F},W_{F}]}\xrightarrow{\det\circ\Phi}\mathbb{C}^{\times}.$

For any $\displaystyle s\in\mathbb{C}$

	$\displaystyle\displaystyle\varepsilon(\Phi,\psi,d(x),s)$	$\displaystyle\displaystyle=\varepsilon(\Phi\otimes\|\cdot\|_{F}^{s},\psi,d(x))$
		$\displaystyle\displaystyle=\varepsilon(\Phi,\psi,d(x))\cdot q^{-s(n(\psi)\cdot\dim\Phi+a(\Phi))}.$

Proposition A.3.4

If $\displaystyle n(\psi)=0$ and the Haar measure $\displaystyle d(x)$ is normalized so that

\displaystyle\int_{O_{F}}d(x)=1,

then

\displaystyle\varepsilon(\Phi,\psi,d(x))=w(\Phi)\cdot q^{a(\Phi)/2}=w(V)\cdot q^{a(V)/2}

with $\displaystyle w(\Phi)\in\mathbb{C}$ of absolute value one.

When $\displaystyle K/F$ is a finite tamely ramified Galois extension, the maximal unramified subextension $\displaystyle K_{0}=K\cap F^{\text{\rm ur}}$ is a cyclic extension of $\displaystyle F$ and $\displaystyle K/K_{0}$ is also cyclic extension. So, by means of Proposition A.3.1, we can give the explicit value of $\displaystyle\lambda(K/F,\psi)$ .

Let $\displaystyle\psi_{F}:F\to\mathbb{C}^{\times}$ be a continuous unitary character such that

\displaystyle\{x\in F\mid\psi_{F}(xOF)=1\}=\mathcal{D}(F/\mathbb{Q}_{p})^{-1}=\mathfrak{p}_{F}^{-d(F)}

and the Haar measure $\displaystyle d_{F}(x)$ on $\displaystyle F$ is normalized so that

\displaystyle\int_{O_{F}}d_{F}(x)=q^{-d(F)}.

Let $\displaystyle K/F$ be a tamely ramified finite Galois extension, and put $\displaystyle\psi_{K}=\psi_{F}\circ T_{K/F}$ . Put

\displaystyle e=e(K/F)=(K:K_{0}),\quad f=f(K/F)=(K_{0}:F)

where $\displaystyle K_{0}=K\cap F^{\text{\rm ur}}$ is the maximal unramified subextension of $\displaystyle K/F$ . Let

\displaystyle\left(\frac{\varepsilon}{K_{0}}\right)=\begin{cases}1&:\varepsilon\equiv\text{\rm square}\!\!\pmod{\mathfrak{p}_{K_{0}}},\\ -1&:\text{\rm otherwise}\end{cases}\qquad(\varepsilon\in O_{K_{0}}^{\times})

be the Legendre symbol of $\displaystyle K_{0}$ . Then we have

Proposition A.3.5

	$\displaystyle\displaystyle\lambda(K/F,\psi_{F})$	$\displaystyle\displaystyle=\lambda(K/F,\psi_{F},d_{F}(x),d_{K}(x))$
		$\displaystyle\displaystyle=\begin{cases}(-1)^{\frac{q^{f}-1}{e}\cdot\frac{e(e+2)}{8}}\cdot G_{\psi_{K_{0}}}(\left(\frac{\ast}{K_{0}}\right),\varpi_{0}^{-(d(K_{0})+1)})&:e=\text{\rm even},\\ (-1)^{(f-1)d(F)}&:e=\text{\rm odd}\end{cases}$

where $\displaystyle\varpi_{0}$ is a prime element of $\displaystyle K_{0}$ such that $\displaystyle\varpi_{0}\in N_{K/K_{0}}(K^{\times})$ .

Proposition A.3.6

If there exists an intermediate field $\displaystyle F\subset E\subset K$ such that $\displaystyle K/E$ is unramified quadratic extension, then $\displaystyle f=2f_{+}$ is even and

\displaystyle\lambda(K/F,\psi_{F})=\begin{cases}-(-1)^{\frac{q^{f_{+}}-1}{2}}&:\text{\rm$\displaystyle e$ is even},\\ (-1)^{d(F)}&:\text{\rm$\displaystyle e$ is odd}.\end{cases}

A.4 $\displaystyle\gamma$ -factors of admissible representations of Weil group

Definition A.4.1

The pair $\displaystyle(\Phi,V)$ is called an admissible representation of $\displaystyle W_{F}$ if

1)

$\displaystyle V$ is a finite dimensional complex vector space and $\displaystyle\Phi$ is a group homomorphism of $\displaystyle W_{F}$ to $\displaystyle GL_{\mathbb{C}}(V)$ ,
2)

$\displaystyle\text{\rm Ker}(\Phi)$ is an open subgroup of $\displaystyle W_{F}$ ,
3)

$\displaystyle\Phi(\widetilde{\text{\rm Fr}})\in GL_{\mathbb{C}}(V)$ is semisimple.

Let $\displaystyle(\Phi,V)$ be an admissible representation of $\displaystyle W_{F}$ . Since $\displaystyle I_{F}=\text{\rm Gal}(F^{\text{\rm alg}}/F^{\text{\rm ur}})$ is a normal subgroup of $\displaystyle W_{F}$ , $\displaystyle\Phi(\widetilde{\text{\rm Fr}})\in GL_{\mathbb{C}}(V)$ keeps

\displaystyle V^{I_{F}}=\{v\in V\mid\Phi(\sigma)v=v\;\forall\sigma\in I_{F}\}

stable. Then the $\displaystyle L$ -factor of $\displaystyle(\Phi,V)$ is defined by

\displaystyle L(\Phi,s)=L(V,s)=\det\left(1-q^{-s}\cdot\Phi(\widetilde{\text{\rm Fr}})|_{V^{I_{F}}}\right)^{-1}.

Since $\displaystyle\Phi:W_{F}\to GL_{\mathbb{C}}(V)$ is continuous group homomorphism, we have the $\displaystyle\varepsilon$ -factor $\displaystyle\varepsilon(\Phi,\psi,d(x),s)$ of $\displaystyle\Phi$ . Then the $\displaystyle\gamma$ -factor of $\displaystyle(\Phi,V)$ is defined by

\displaystyle\gamma(\Phi,\psi,d(x),s)=\gamma(V,\psi,d(x),s)=\varepsilon(\Phi,\psi,d(x),s)\cdot\frac{L(\Phi^{^},1-s)}{L(\Phi,s)}

where $\displaystyle\Phi^{^}$ is the dual representation of $\displaystyle\Phi$ .

A.5 Symmetric tensor representation of $\displaystyle SL_{2}(\mathbb{C})$

The complex special linear group $\displaystyle SL_{2}(\mathbb{C})$ acts on the polynomial ring $\displaystyle\mathbb{C}[X,Y]$ of two variables $\displaystyle X,Y$ by

\displaystyle g\cdot\varphi(X,Y)=\varphi((X,Y)g)\qquad(g\in SL_{2}(\mathbb{C}),\varphi(X,Y)\in\mathbb{C}[X,Y]).

Let

\displaystyle\mathcal{P}_{n}=\langle X^{n},X^{n-1}Y,\cdots,XY^{n-1},Y^{n}\rangle_{\mathbb{C}}

be the subspace of $\displaystyle\mathbb{C}[X,Y]$ consisting of the homogeneous polynomials of degree $\displaystyle n$ . The action of $\displaystyle SL_{2}(\mathbb{C})$ on $\displaystyle\mathcal{P}_{n}$ defines the symmetric tensor representation $\displaystyle\text{\rm Sym}_{n}$ of degree $\displaystyle n+1$ . The complex vector space $\displaystyle\mathcal{P}_{n}$ has a non-degenerate bilinear form defined by

\displaystyle\langle\varphi,\psi\rangle=\left.\varphi\left(-\frac{\partial}{\partial Y},\frac{\partial}{\partial X}\right)\psi(X,Y)\right|_{(X,Y)=(0,0)}\in\mathbb{C}

for $\displaystyle\varphi,\psi\in\mathcal{P}_{n}$ . This bilinear form is $\displaystyle SL_{2}(\mathbb{C})$ -invariant

\displaystyle\langle\text{\rm Sym}_{n}(g)\varphi,\text{\rm Sym}_{n}(g)\psi\rangle=\langle\varphi,\psi\rangle\quad(g\in SL_{2}(\mathbb{C}),\varphi,\psi\in\mathcal{P}_{n})

and

\displaystyle\langle\psi,\varphi\rangle=(-1)^{n}\langle\varphi,\psi\rangle\quad(\varphi,\psi\in\mathcal{P}_{n}).

So we have group homomorphisms

\displaystyle\text{\rm Sym}_{n}:SL_{2}(\mathbb{C})\to SO(\mathcal{P}_{n})\;\;\text{\rm if $\displaystyle n$ is even}

and

\displaystyle\text{\rm Sym}_{n}:SL_{2}(\mathbb{C})\to Sp(\mathcal{P}_{n})\;\;\text{\rm if $\displaystyle n$ is odd}.

A.6 Admissible representations of Weil-Deligne group

Fix a complex Lie group $\displaystyle\mathcal{G}$ such that the connected component $\displaystyle\mathcal{G}^{o}$ is a reductive complex algebraic linear group. Then the $\displaystyle\mathcal{G}^{o}$ -conjugacy class of the group homomorphisms

\displaystyle\varphi:W_{F}\times SL_{2}(\mathbb{C})\to\mathcal{G}

such that

1)

$\displaystyle I_{F}\cap\text{\rm Ker}(\varphi)$ is an open subgroup of $\displaystyle I_{F}$ ,
2)

$\displaystyle\varphi(\widetilde{\text{\rm Fr}})\in\mathcal{G}$ is semi-simple,
3)

$\displaystyle\varphi|_{SL_{2}(\mathbb{C})}:SL_{2}(\mathbb{C})\to\mathcal{G}^{o}$ is a morphism of complex linear algebraic group

corresponds bijectively the equivalence classes of the triples $\displaystyle(\rho,\mathcal{G},N)$ where $\displaystyle N\in\text{\rm Lie}(\mathcal{G})$ is a nilpotent element and

\displaystyle\rho:W_{F}\to\mathcal{G}

is a group homomorphism such that

1)

$\displaystyle\rho|_{I_{F}}:I_{F}\to\mathcal{G}$ is continuous,
2)

$\displaystyle\rho(\widetilde{\text{\rm Fr}})\in\mathcal{G}$ is semi-simple,

$\displaystyle\rho(g)N=|g|_{F}\cdot N$ for $\displaystyle\forall g\in W_{F}$ where

\displaystyle|\cdot|_{F}:W_{F}\xrightarrow{\text{\rm can.}}W_{F}/\overline{[W_{F},W_{F}]}\xrightarrow{\text{\rm l.c.f.t.}}F^{\times}\xrightarrow{q^{-\text{\rm ord}_{F}(\cdot)}}\mathbb{Q}^{\times}

by the relations

\displaystyle\rho|_{I_{F}}=\varphi|_{I_{F}},\quad\rho(\widetilde{\text{\rm Fr}})=\varphi(\widetilde{\text{\rm Fr}})\cdot\varphi\begin{pmatrix}q^{-1/2}&0\\ 0&q^{1/2}\end{pmatrix},\quad N=d\varphi\begin{pmatrix}0&1\\ 0&0\end{pmatrix}

(see [7, Prop.2.2]). Here two triples $\displaystyle(\rho,\mathcal{G},N)$ and $\displaystyle(\rho^{\prime},\mathcal{G},N^{\prime})$ is equivalent if there exists a $\displaystyle g\in\mathcal{G}$ such that $\displaystyle\rho^{\prime}=g\rho g^{-1}$ and $\displaystyle N^{\prime}=\text{\rm Ad}(g)N$ .

The couple $\displaystyle(\varphi,\mathcal{G})$ or the triple $\displaystyle(\rho,\mathcal{G},N)$ is called an admissible representation of the Weil-Deligne group.

Let $\displaystyle(r.V)$ be a continuous finite dimensional complex representation of $\displaystyle\mathcal{G}$ which is algebraic on $\displaystyle\mathcal{G}^{o}$ . Then the $\displaystyle L$ -factor associated with $\displaystyle(\varphi,\mathcal{G})$ and $\displaystyle(r,V)$ is defined by

\displaystyle L(\varphi,r,s)=\det\left(1-q^{-s}r\circ\rho(\widetilde{\text{\rm Fr}})|_{V_{N}^{I_{F}}}\right)^{-1},

where $\displaystyle V_{N}=\{v\in V\mid dr(N)v=0\}$ and

\displaystyle V_{N}^{I_{F}}=\{v\in V_{N}\mid r\circ\rho(\sigma)v=v\;\forall\sigma\in I_{F}\}.

The $\displaystyle\varepsilon$ -actor is defined by

\displaystyle\varepsilon(\varphi,r,\psi,d(x),s)=\varepsilon(r\circ\rho,\psi,d(x),s)\cdot\det\left(-q^{-s}r\circ\rho(\widetilde{\text{\r{F}r}})|_{V^{I_{F}}/V_{N}^{I_{F}}}\right)

where $\displaystyle\varepsilon(r\circ\rho,\psi,d(x),s)$ is the $\displaystyle\varepsilon$ -factor of the representation $\displaystyle(r\circ\rho,V)$ of $\displaystyle W_{F}$ defined in the subsection A.4. Finally the $\displaystyle\gamma$ -factor is defined by

\displaystyle\gamma(\varphi,r,\psi,d(x),s)=\varepsilon(\varphi,r,\psi,d(x),s)\cdot\frac{L(\varphi,r^{\vee},1-s)}{L(\varphi,r,s)}

where $\displaystyle r^{\vee}$ is the dual representation of $\displaystyle r$ .

Let $\displaystyle\text{\rm Sym}_{n}$ be the symmetric tensor representation of $\displaystyle SL_{2}(\mathbb{C})$ of degree $\displaystyle n+1$ . Then the $\displaystyle W_{F}\times SL_{2}(\mathbb{C})$ -module $\displaystyle V$ has a decomposition

\displaystyle V=\bigoplus_{n=0}^{\infty}V_{n}\otimes\text{\rm Sym}_{n}

where $\displaystyle V_{n}$ is a $\displaystyle W_{F}$ -module. Then we have

\displaystyle V_{N}^{I_{F}}=\bigoplus_{n=0}^{\infty}V_{n}^{I_{F}}\otimes\text{\rm Sym}_{n,N}

where $\displaystyle\text{\rm Sym}_{n,N}$ is the highest part of $\displaystyle\text{\rm Sym}_{n}$ . Since $\displaystyle r\circ\rho(\widetilde{\text{\rm Fr}})$ act on $\displaystyle V_{n}\otimes\text{\rm Sym}_{n,N}$ by $\displaystyle q^{-n/2}r\circ\varphi(\widetilde{\text{\rm Fr}})$ , we have

\displaystyle L(\varphi,r,s)=\prod_{n=0}^{\infty}\det\left(1-q^{-(s+n/2)}r\circ\varphi(\widetilde{\text{\rm Fr}})|_{V_{n}^{I_{F}}}\right)^{-1}.

If the Haar measure $\displaystyle d(x)$ on the additive group $\displaystyle F$ and the additive character $\displaystyle\psi:F\to\mathbb{C}^{\times}$ are normalized so that $\displaystyle\int_{O_{F}}d(x)=1$ and

\displaystyle\{x\in F\mid\psi(xO_{F})=1\}=O_{F},

then we have

\displaystyle\varepsilon(\varphi,r,\psi,d(x),s)=w(\varphi,r)\cdot q^{a(\varphi,r)(1/2-s)}

where

\displaystyle w(\varphi,r)=\prod_{n=0}^{\infty}w(V_{n})^{n+1}\cdot\prod_{n=1}^{\infty}\det\left(-\varphi(\widetilde{\text{\rm Fr}})|_{V_{n}^{I_{F}}}\right)^{n}

and

\displaystyle a(\varphi,r)=\sum_{n=0}^{\infty}(n+1)a(V_{n})+\sum_{n=1}^{\infty}n\cdot\dim V_{n}^{I_{F}}.

If $\displaystyle\varphi|_{SL_{2}(\mathbb{C})}=1$ , then $\displaystyle V_{n}=0$ for all $\displaystyle n>0$ and we have

\displaystyle w(\varphi,r)=w(r\circ\varphi)=w(r\circ\rho),\quad a(\varphi,r)=a(r\circ\varphi)=a(r\circ\rho).

Appendix B Symmetric or anti-symmetric forms on induced representations of Weil group

Let $\displaystyle K/F$ be a finite Galois extension of even degree. We will assume that the elements of $\displaystyle\Gamma=\text{\rm Gal}(K/F)$ of order two are central ²²2This is the case if $\displaystyle K/F$ is tamely ramified extension. See Proposition 3.3.1.. Fix an element $\displaystyle\tau\in\Gamma$ of order two. Let $\displaystyle K_{+}$ be the intermediate field of $\displaystyle K/F$ such that $\displaystyle\text{\rm Gal}(K/K^{+})=\langle\tau\rangle$ , and put

\displaystyle U_{K/K_{+}}=\{\varepsilon\in O_{K}^{\times}\mid N_{K/K_{+}}(\varepsilon)=1\}.

Take a continuous unitary character $\displaystyle\vartheta:U_{K/K_{+}}\to\mathbb{C}^{\times}$ and put $\displaystyle\widetilde{\vartheta}(x)=\vartheta(x^{1-\tau})$ ( $\displaystyle x\in K^{\times}$ ). The representation space $\displaystyle V_{\vartheta}=\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\vartheta}$ is the complex vector space of the $\displaystyle\mathbb{C}$ -valued functions $\displaystyle v$ on $\displaystyle\Gamma$ on which $\displaystyle(\sigma,x)\in W_{K/F}=\Gamma{\ltimes}_{\alpha_{K/F}}K^{\times}$ acts by

\displaystyle(x\cdot v)(\gamma)=\widetilde{\vartheta}(x^{\gamma})\cdot v(\gamma),\quad(\sigma\cdot v)(\gamma)=\widetilde{\vartheta}(\alpha_{K/F}(\sigma,\sigma^{-1}\gamma))\cdot v(\sigma^{-1}\gamma)

with the fundamental class $\displaystyle[\alpha_{K/F}]\in H^{2}(\Gamma,K^{\times})$ . The character $\displaystyle\chi_{\vartheta}$ of $\displaystyle V_{\vartheta}$ is

\displaystyle\chi_{\vartheta}(\sigma,x)=\begin{cases}0&:\sigma\neq 1,\\ \sum_{\gamma\in\Gamma}\widetilde{\vartheta}(x^{\gamma})&:\sigma=1\end{cases}

for $\displaystyle(\sigma,x)\in W_{K/F}$ , which is self-conjugate, that is $\displaystyle\overline{\chi}_{\vartheta}=\chi_{\vartheta}$ .

Let $\displaystyle\nu:W_{K/F}\to\mathbb{C}^{\times}$ be a continuous group homomorphism. We will look for the $\displaystyle\nu$ -invariant $\displaystyle\nu$ -symmetric bilinear form on $\displaystyle V_{\vartheta}$ , that is, the non-zero complex bilinear form $\displaystyle B$ on $\displaystyle V_{\vartheta}$ such that

1)

$\displaystyle B(g\cdot u,g\cdot v)=\nu(g)\cdot B(u,v)$ for all $\displaystyle g\in W_{K/F}$ ,
2)

$\displaystyle B(v,u)=\nu(\tau)\cdot B(u,v)$ for all $\displaystyle u,v\in V_{\vartheta}$ .

Note that, in this case, we have $\displaystyle\nu(\tau)=\pm 1$ .

If $\displaystyle\nu|_{K^{\times}}=1$ , then

\displaystyle B_{\nu}(u,v)=\sum_{\gamma\in\Gamma}\nu(\gamma)\cdot\widetilde{\vartheta}\left(\alpha_{K/F}(\gamma,\tau)\right)^{-1}\cdot u(\gamma)v(\gamma\tau)\quad(u,v\in V_{\vartheta})

is a non-degenerate $\displaystyle\nu$ -invariant $\displaystyle\nu$ -symmetric bilinear form on $\displaystyle V_{\vartheta}$ . For a $\displaystyle\rho\in\Gamma$ , define $\displaystyle w_{\rho}\in V_{\vartheta}$ by

\displaystyle w_{\rho}(\gamma)=\begin{cases}1&:\gamma=\rho,\\ 0&:\gamma\neq\rho\end{cases}

and $\displaystyle u_{\rho},v_{\rho}\in V_{\vartheta}$ by

\displaystyle u_{\rho}=\nu(\rho)^{-1}w_{\rho},\qquad v_{\rho}=\widetilde{\vartheta}\left(\alpha_{K/F}(\rho,\tau)\right)\cdot w_{\rho\tau}.

If we fix a complete system of representatives $\displaystyle\mathcal{S}$ of $\displaystyle\Gamma/\langle\tau\rangle$ , then $\displaystyle\{u_{\rho},v_{\rho}\}_{\dot{\rho}\in\mathcal{S}}$ is a $\displaystyle\mathbb{C}$ -basis of $\displaystyle V_{\vartheta}$ such that

\displaystyle B_{\nu}(u_{\rho},u_{\rho^{\prime}})=B_{\nu}(v_{\rho},v_{\rho^{\prime}})=0,\quad B_{\nu}(u_{\rho},v_{\rho^{\prime}})=\begin{cases}1&:\rho=\rho^{\prime},\\ 0&:\rho\neq\rho^{\prime}.\end{cases}

Proposition B.0.1

Assume that

1)

$\displaystyle\nu$ is of finite order,
2)

$\displaystyle\{\sigma\in\Gamma\mid\widetilde{\vartheta}(x^{\sigma})=\widetilde{\vartheta}(x)\;\forall x\in 1+\mathfrak{p}_{K}\}=\{1\}$ .

Then $\displaystyle V_{\vartheta}$ has $\displaystyle\nu$ -invariant $\displaystyle\nu$ -symmetric bilinear form if and only if $\displaystyle\nu|_{K^{\times}}=1$ . In this case, the form is a constant multiple of $\displaystyle B_{\nu}$ .

[Proof] Due to the second assumption and Remark 3.4.3, the induced representation $\displaystyle V_{\vartheta}=\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\widetilde{\vartheta}$ is irreducible. Since $\displaystyle\nu$ is of finite order, we can choose positive integers $\displaystyle s,t$ such that $\displaystyle M=\langle\varpi_{K}^{s}\rangle\times(1+\mathfrak{p}_{K}^{t})$ is a $\displaystyle\Gamma$ -subgroup of $\displaystyle K^{\times}$ on which $\displaystyle\vartheta$ and $\displaystyle\nu$ are trivial. Then the induced representation $\displaystyle\text{\rm Ind}_{K^{\times}}^{W_{K/F}}\vartheta$ and the character $\displaystyle\nu$ factor through the canonical morphism

\displaystyle W_{K/F}\to G=\Gamma{\ltimes}_{\alpha_{K/F}}K^{\times}/M.

So we will consider them on the finite group $\displaystyle G$ . The it is well-known that

\displaystyle\dim_{\mathbb{C}}\text{\rm Hom}_{G}(\overline{\nu},\text{\rm Hom}_{\mathbb{C}}(V_{\vartheta},V_{\vartheta}^{\ast}))=\begin{cases}1&:\nu\cdot\chi_{\vartheta}=\chi_{\vartheta},\\ 0&:\text{\rm otherwise},\end{cases}

where $\displaystyle V_{\vartheta}^{\ast}$ is the dual representation of $\displaystyle V_{\vartheta}$ . Since $\displaystyle T\in\text{\rm Hom}_{\mathbb{C}}(V_{\vartheta},V_{\vartheta}^{\ast})$ gives a complex bilinear form

\displaystyle B_{T}(u,v)=\langle u,Tv\rangle\qquad(u,v\in V_{\vartheta})

with the canonical pairing $\displaystyle\langle\,,\rangle:V_{\vartheta}\times V_{\vartheta}^{\ast}\to\mathbb{C}$ , and

\displaystyle\text{\rm Hom}_{\mathbb{C}}(V_{\vartheta},V_{\vartheta}^{\ast})=\text{\rm Sym}(V_{\vartheta},V_{\vartheta}^{\ast})\oplus\text{\rm Alt}(V_{\vartheta},V_{\vartheta}^{\ast}),

there exists $\displaystyle\nu$ -invarinat $\displaystyle\nu$ -symmetric bilinear form on $\displaystyle V_{\vartheta}$ if and only if $\displaystyle\nu\cdot\chi_{\vartheta}=\chi_{\vartheta}$ , and in this case

	$\displaystyle\displaystyle\dim_{\mathbb{C}}\text{\rm Hom}_{G}(\overline{\nu},\text{\rm Sym}(V_{\vartheta},V_{\vartheta}^{\ast}))$	$\displaystyle\displaystyle=\frac{1}{2}\left\{1+\frac{1}{\|G\|}\sum_{g\in G}\nu(g)\chi_{\vartheta}(g^{2})\right\},$
	$\displaystyle\displaystyle\dim_{\mathbb{C}}\text{\rm Hom}_{G}(\overline{\nu},\text{\rm Alt}(V_{\vartheta},V_{\vartheta}^{\ast}))$	$\displaystyle\displaystyle=\frac{1}{2}\left\{1-\frac{1}{\|G\|}\sum_{g\in G}\nu(g)\chi_{\vartheta}(g^{2})\right\},$

that is

\displaystyle\frac{1}{|G|}\sum_{g\in G}\nu(g)\chi_{\vartheta}(g^{2})=\nu(\tau).

Let us assume $\displaystyle\nu\cdot\chi_{\vartheta}=\chi_{\vartheta}$ . Then the prime element $\displaystyle\varpi_{K}$ of $\displaystyle K$ can be chosen so that $\displaystyle\nu(\varpi_{K})=1$ . In fact there exists a prime element $\displaystyle\varpi_{K}$ of $\displaystyle K$ such that $\displaystyle\varpi_{K}^{\tau}=\pm\varpi_{K}$ . Then

\displaystyle\widetilde{\vartheta}(\varpi_{K}^{\gamma})=\vartheta(\varpi_{K}^{\gamma(1-\tau)})=\vartheta(\varpi_{K}^{(1-\tau)\gamma}=\vartheta(\pm 1)

for all $\displaystyle\gamma\in\Gamma$ , and hence

\displaystyle\chi_{\vartheta}(1,\varpi_{K})=\sum_{\gamma\in\Gamma}\widetilde{\vartheta}(\varpi_{K}^{\gamma})=|\Gamma|\cdot\vartheta(\pm 1)\neq 0.

Then $\displaystyle\nu\cdot\chi_{\vartheta}=\chi_{\vartheta}$ implies $\displaystyle\nu(\varpi_{K})=1$ .

Note also that $\displaystyle\nu(x^{\gamma})=\nu(x)$ for all $\displaystyle x\in K^{\times}$ and $\displaystyle\gamma\in\Gamma$ , since $\displaystyle(\gamma,1)^{-1}(1,x)(\gamma,1)=(1,x^{\gamma})$ .

Since

\displaystyle\chi_{\vartheta}((\sigma,x)^{2})=\begin{cases}0&:\sigma^{2}\neq 1,\\ \sum_{\gamma\in\Gamma}\widetilde{\vartheta}\left(x^{(1+\sigma)\gamma}\alpha_{K/F}(\sigma,\sigma)^{\gamma}\right)&:\sigma^{2}=1,\end{cases}

we have

	$\displaystyle\displaystyle\sum_{g\in G}\nu(g)\chi_{\vartheta}(g^{2})$	$\displaystyle\displaystyle=\sum_{\stackrel{{\scriptstyle\scriptstyle\sigma,\gamma\in\Gamma}}{{\sigma^{2}=1}}}\sum_{\dot{x}\in K^{\times}/M}\nu(\sigma,x)\cdot\widetilde{\vartheta}\left(x^{(1+\sigma)\gamma}\alpha_{K/F}(\sigma,\sigma)^{\gamma}\right)$
		$\displaystyle\displaystyle=\sum_{\stackrel{{\scriptstyle\scriptstyle\sigma,\gamma\in\Gamma}}{{\sigma^{2}=1}}}\sum_{\dot{x}\in K^{\times}/M}\nu(\sigma,x^{\gamma^{-1}})\cdot\widetilde{\vartheta}\left(x^{(1+\sigma)}\alpha_{K/F}(\sigma,\sigma)^{\gamma}\right)$
		$\displaystyle\displaystyle=\sum_{\stackrel{{\scriptstyle\scriptstyle\sigma,\gamma\in\Gamma}}{{\sigma^{2}=1}}}\nu(\sigma)\cdot\widetilde{\vartheta}\left(\alpha_{K/F}(\sigma,\sigma)^{\gamma}\right)\sum_{\dot{x}\in K^{\times}/M}\nu(x)\cdot\widetilde{\vartheta}(x^{1+\sigma}).$

Since $\displaystyle\nu(\varpi_{K})=1$ and $\displaystyle\varpi_{K}^{1-\tau}=\pm 1$ , we have

\displaystyle\widetilde{\vartheta}(\varpi_{K}^{1+\sigma})=\vartheta\left(\varpi_{K}^{(1+\sigma)(1-\tau)}\right)=\vartheta\left((\pm 1)^{1+\sigma}\right)=1

for all $\displaystyle\sigma\in\Gamma$ , we have

\displaystyle\sum_{\dot{x}\in K^{\times}/M}\nu(x)\cdot\widetilde{\vartheta}(x^{1+\sigma})=s\sum_{\dot{x}\in(O_{K}/\mathfrak{p}_{K}^{t})^{\times}}\nu(x)\widetilde{\vartheta}(x^{1+\sigma}).

If $\displaystyle\nu(x)\widetilde{\vartheta}(x^{1+\sigma})=1$ for all $\displaystyle x\in O_{K}^{\times}$ , then we have

\displaystyle 1=\nu(x^{\tau})\widetilde{\vartheta}(x^{\tau(1+\sigma)})=\nu(x)\widetilde{\vartheta}(x^{1+\sigma})^{-1},

and hence

\displaystyle\widetilde{\vartheta}(x^{2\sigma})=\widetilde{\vartheta}(x^{-2})=\widetilde{\vartheta}(x^{2\tau})

for all $\displaystyle x\in O_{K}^{\times}$ . Since $\displaystyle x\mapsto x^{2}$ gives a surjection of $\displaystyle 1+\mathfrak{p}_{K}$ onto $\displaystyle 1+\mathfrak{p}_{K}$ , we have $\displaystyle\widetilde{\vartheta}(x^{\sigma})=\widetilde{\vartheta}(x^{\tau})$ for all $\displaystyle x\in 1+\mathfrak{p}_{K}$ , and hence $\displaystyle\sigma=\tau$ . Since

\displaystyle\alpha_{K/F}(\tau,\tau)^{\tau}\alpha_{K/F}(\tau,1)^{-1}\alpha_{K/F}(1,\tau)\alpha_{K/F}(\tau,\tau)^{-1}=1

and $\displaystyle\alpha_{K/F}(1,\tau)=\alpha_{K/F}(\tau,1)=1$ , we have

\displaystyle\widetilde{\vartheta}\left(\alpha_{K/F}(\tau,\tau)^{\gamma}\right)=\vartheta\left(\alpha_{K/F}(\tau,\tau)^{\gamma(1-\tau)}\right)=1

fro all $\displaystyle\gamma\in\Gamma$ . Then we have

	$\displaystyle\displaystyle\|G\|^{-1}\sum_{g\in G}\nu(g)\chi_{\vartheta}(g^{2})$	$\displaystyle\displaystyle=\left\|(O_{K}/\mathfrak{p}_{K}^{t})^{\times}\right\|^{-1}\nu(\tau)\sum_{\dot{x}\in(O_{K}/\mathfrak{p}_{K}^{t})^{\times}}\nu(x)$
		$\displaystyle\displaystyle=\begin{cases}0&:\nu\|_{O_{K}^{\times}}\neq 1,\\ \nu(\tau)&:\nu\|_{O_{K}^{\times}}=1.\end{cases}$

This completes the proof. $\displaystyle\blacksquare$

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Sendai 980-0845, Japan
Miyagi University of Education
Department of Mathematics

On certain supercuspidal representations of symplectic groups associated with tamely ramified extensions : the formal degree conjecture and the root number conjecture

1 Introduction

1.1

Conjecture 1.1.1

Conjecture 1.1.2

Conjecture 1.1.3

1.2

1.3

2 Supercuspidal representations of S​p2​n​(F)\displaystyle Sp_{2n}(F)

2.1 Regular irreducible characters of hyperspecial compact subgroup

Theorem 2.1.1

Proposition 2.1.2

Proposition 2.1.3

Remark 2.1.4

2.2 Symplectic spaces associated with tamely ramified extensions

Proposition 2.2.1

Proposition 2.2.2

Proposition 2.2.3

Proposition 2.2.4

Proposition 2.2.5

2.3 Construction of supercuspidal representations

Theorem 2.3.1

Proposition 2.3.2

Lemma 2.3.3

Proposition 2.3.4

3 Kaleta’s L\displaystyle L-parameter

3.1 Local Langlands correspondence of elliptic tori

Proposition 3.1.1

3.2 χ\displaystyle\chi-datum

3.3 Explicit value of c​(−1)\displaystyle c(-1)

Proposition 3.3.1

Proposition 3.3.2

Proposition 3.3.3

3.4 L\displaystyle L-parameters associated with characters of tame elliptic tori

Proposition 3.4.1

Proposition 3.4.2

Remark 3.4.3

4 Formal degree conjecture

4.1 γ\displaystyle\gamma-factor of adjoint representation

Theorem 4.1.1

Proposition 4.1.2

Proposition 4.1.3

4.2 γ\displaystyle\gamma-factor of principal parameter

4.3 Verification of formal degree conjecture

Theorem 4.3.1

5 Root number conjecture

5.1 Structure of adjoint representation

Theorem 5.1.1

5.1.1

5.1.2

5.1.3

5.2 Root number of adjoint representation

Theorem 5.2.1

5.2.1

5.2.2

5.2.3

5.2.4

Proposition 5.2.2

Proposition 5.2.3

5.3 Verification of root number conjecture

Theorem 5.3.1

6 The case of S​p4​(F)\displaystyle Sp_{4}(F)

6.1 Another candidate for the Langlands parameter

6.2 Formal degree conjecture

6.3 The root number conjecture

Appendix A Local factors

A.1 Weil group

A.2 Artin conductor of representations of Weil group

A.3 ε\displaystyle\varepsilon-factor of representations of Weil group

Proposition A.3.1

Remark A.3.2

Proposition A.3.3

Proposition A.3.4

Proposition A.3.5

Proposition A.3.6

A.4 γ\displaystyle\gamma-factors of admissible representations of Weil group

Definition A.4.1

A.5 Symmetric tensor representation of S​L2​(ℂ)\displaystyle SL_{2}(\mathbb{C})

A.6 Admissible representations of Weil-Deligne group

Appendix B Symmetric or anti-symmetric forms on induced representations of Weil group

On certain supercuspidal representations of symplectic groups associated with
tamely ramified extensions :
the formal degree conjecture and
the root number conjecture

2 Supercuspidal representations of $\displaystyle Sp_{2n}(F)$

3 Kaleta’s $\displaystyle L$ -parameter

3.2 $\displaystyle\chi$ -datum

3.3 Explicit value of $\displaystyle c(-1)$

3.4 $\displaystyle L$ -parameters associated with characters of tame elliptic tori

4.1 $\displaystyle\gamma$ -factor of adjoint representation

4.2 $\displaystyle\gamma$ -factor of principal parameter

6 The case of $\displaystyle Sp_{4}(F)$

A.3 $\displaystyle\varepsilon$ -factor of representations of Weil group

A.4 $\displaystyle\gamma$ -factors of admissible representations of Weil group

A.5 Symmetric tensor representation of $\displaystyle SL_{2}(\mathbb{C})$