Euler Characteristics of the Generalized Kloosterman sheaves for symplectic and orthogonal groups

In [HNY], generalizing Deligne’s and Katz’s constructions of the Kloosterman sheaves attached to families of Kloosterman sums, Heinloth, Ng$\widehat{\mathrm{o}}$, and Yun constructed Kloosterman sheaves $\mathrm{Kl}_{\widehat{G},\mathbf{I}}(\mathcal{K},\phi)$ for reductive groups $G$ and $\mathbf{I}$ the Iwahoric subgroup of $G$, which are $\widehat{G}$-local systems. The Kloosterman sheaves depend on $\mathcal{K}$ a character sheaf and $\phi$ a stable linear functional, which corresponds to simple supercuspidal representations of Gross-Reeder [GrossReeder] on the automorphic side.

Generalizing the construction in [HNY], Yun constructed in [YunEpipe] a family of generalized Kloosterman sheaves $\mathrm{Kl}_{\widehat{G},\mathbf{P}}(\mathcal{K},\phi)$ associated with parahoric subgroups $\mathbf{P}$. We recall some details of the construction below.

We fix a prime $\ell$ different from $\mathrm{char}(k)$. We consider constructible $\overline{\mathbb{Q}}_{\ell}$-complexes over various algebraic stacks over $k$ or $\overline{k}$. Let $(L_{\mathbf{P}},V_{\mathbf{P}})$ be $\theta$-groups studied by [Vinberg]. When $\mathrm{char}(k)$ is large, let $\mathbf{P}:=\mathbf{P}_{m}$, where $m$ is a regular elliptic number uniquely associated with the admissible parahoric $\mathbf{P}$ (see Section 2.2).

Let $\pi(\chi,\phi)$ be a cuspidal automorphic representation whose local component at $\infty$ is an epipelegic supercuspidal representation as realized in [YunEpipe], where $\chi$ is a character of $\widetilde{L_{\mathbf{P}}}$ and $\phi$ is a stable linear functional $V_{\mathbf{P}}\to k$.

By the global geometric Langlands correspondence, it is associated with the Kloosterman sheaf $\mathrm{Kl}_{{}^{L}G,\mathbf{P}}(\mathcal{K},\phi)$ (which is a ${}^{L}G$-local system), where $\mathcal{K}$ is the character sheaf associated with $\chi$ under sheaf-to-function dictionary. Let $X^{\circ}=\mathbb{P}_{k}^{1}-\{0,\infty\}$. The local system attached to the generalized Kloosterman sheaves and stardard representation $V$ of $\widehat{G}$ can be interpreted using the following diagram

Here $\mathfrak{G}_{\leq\lambda}$ is the variety corresponding to the dominant coweight $\lambda$ associated with $V$ and $\pi$ is the projection map. When $G=\mathrm{Sp}_{2n}$, $\mathfrak{G}_{\leq\lambda}=\mathrm{Sym}^{2}_{\leq 1}(M)$, where $M$ is $2n$-dimensinal symplectic space. The $\overline{\mathbb{Q}}_{\ell}$-local system $\mathrm{Kl}^{\mathrm{St}}_{\widehat{G},\mathbf{P}}(\mathbf{1},\phi)$ asssoicated to the Kloosterman sheaves $\mathrm{Kl}_{\widehat{G},\mathbf{P}}(\mathbf{1},\phi)$ (the $\widehat{G}$-local system descended from $\mathrm{Kl}_{{}^{L}G,\mathbf{P}}(\mathcal{K},\phi)$) with the trivial character sheaf $\mathcal{K}=\mathbf{1}$ and the standard representation of $G$ are

When $G$ is an orthogonal group, let $(M,q)$ be the associated quadratic spaces of dimension $2n$ or $2n+1$. The variety $\mathfrak{G}_{\leq\lambda}\subset\mathbb{P}(M)$ is defined by $Q(q)-\cup Q_{[i,m-i]}$, where $Q(q)$ is the quadric defined by $q=0$ and $Q_{[i,m-i]}$ is the quadric defined by $q=0$ when restricted to $M_{i}\oplus\cdots\oplus M_{m-i}$. We have

In this paper we prove the following main theorems:

Using the Grothendieck-Ogg-Shafarevich formula, $-\chi_{c}(\widetilde{X}^{\circ},\mathrm{Kl}^{\operatorname{st}}_{\widehat{G},\mathbf{P}_{m}}(\chi,\phi))$ is equivalent to the Swan conductor of $\mathrm{Kl}^{\operatorname{st}}_{\widehat{G},\mathbf{P}_{m}}(\chi,\phi)$ at $\infty$. Our main theorems then provide evidence for Reeder-Yu’s conjecture [ReederYu] about the Swan conductor of the Langlands parameter of epipelagic representations of $G$ in the function field case.

We set up the notations and review details about the epipelagic representations and the construction of the generalized Kloosterman sheaves in Section 2. We discuss the relation between the Euler characteristics of the local system and the Swan conductor of the conjectural Langlands parameter of epipelagic representations in Section 3. In Section 4 and Section 5, we prove the main theorems in the symplectic case and orthogonal case, respectively.

Let $\mathbb{G}$ be a a split reductive group over $k$ whose derived group is almost simple. Fix a pinning $\dagger=(\mathbb{B},\mathbb{T},\dots)$ of $\mathbb{G}$ for $\mathbb{B}$ a Borel subgroup and $\mathbb{T}\subset\mathbb{B}$ a split torus. Let $\mathbb{W}=N_{\mathbb{G}}(\mathbb{T})/\mathbb{T}$ be the Weyl group of $\mathbb{G}$ and let $\sigma\in\mathrm{Aut}^{\dagger}(\mathbb{G})$ be the image of $1$. Fix a cyclic subgroup $\mathbb{Z}/e\mathbb{Z}$ of the pinned automorphism group of $\mathbb{G}$. We assume $\mathrm{char}(k)$ is prime to $e$ and $k^{\times}$ contains $e$th roots of unity $\mu_{e}$.

Fix a $\mu_{e}$-cover $\widetilde{X}\to X$ which is totally ramified over $0$ and $\infty$. $\widetilde{X}$ is isomorphic to $\mathbb{P}_{k}^{1}$ with affine coordinate $t^{1/e}$. We denote

Let $\widehat{G}$ be the reductive group over $\overline{Q}_{\ell}$ whose root system is dual to that of $\mathbb{G}$. We define the Langlands dual group ${}^{L}G$ to be ${}^{L}G=\widehat{G}\rtimes\mu_{e}$.

We denote $K=F_{\infty}$ as the local field of $F$ at $\infty$. Let $\mathbf{P}\subset G(K)$ be a standard parahoric subgroup containing the standard Iwahori subgroup $\mathbf{I}$. Let $\mathfrak{A}$ be the apartment in the building of $G(K)$ corresponding to the maximal split torus of $G(K)$. We denote $\Psi_{\mathrm{aff}}$ as the set of affine roots of $G(K)$, which are certain affine functions on $\mathfrak{A}$. $\mathbf{P}$ determines a facet $\mathfrak{F}_{\mathbf{P}}$ in $\mathfrak{A}$, and let $\xi\in\mathfrak{A}$ denotes its barycenter. Let $m=m(\mathbf{P})$ be the smallest positive integer such that $\alpha(\xi)\in\frac{1}{m}\mathbb{Z}$ for all affine roots $\alpha\in\Psi_{\mathrm{aff}}$ (see [ReederYu, §3.3]).

Let $\mathbf{P}\supset\mathbf{P}^{+}\supset\mathbf{P}^{++}$ be the first three steps in the Moy-Prasad filtration of $\mathbf{P}$. Here $\mathbf{P}^{+}$ is the pro-unipotent radical of $\mathbf{P}$ and $L_{\mathbf{P}}:=\mathbf{P}/\mathbf{P}^{+}$ is the Levi factor of $\mathbf{P}$. We denote $V_{\mathbf{P}}=\mathbf{P}^{+}/\mathbf{P}^{++}$.

Let $\mathbb{W}^{\prime}=\mathbb{W}\rtimes\mu_{e}$. An element $w\in\mathbb{W}^{\prime}$ is $\mathbb{Z}$-regular if it permutes the roots freely and elliptic if $\mathbb{X}^{\ast}(\mathbb{T}^{\mathrm{ad}})^{w}=0$. The order of a $\mathbb{Z}$-regular elliptic element in $\mathbb{W}\sigma\subset\mathbb{W}^{\prime}$ is called a regular elliptic number of the pair $(\mathbb{W}^{\prime},\sigma)$.

$\mathbf{P}$ is called admissible if there exists a closed orbit of $L_{\mathbf{P}}$ on the dual space $V_{\mathbf{P}}^{\ast}$ with finite stabilizers. When $\mathbf{P}$ is admissible and $\mathrm{char}(k)$ is large, it was shown in [YunEpipe, §2.6] that there is a bijection between admissible parahoric subgroups $\mathbf{P}$ and regular elliptic numbers $m$ using [GLRY, Proposition 1],[Spr74, Proposition 6.4(iv)] and [ReederYu, Corollary 5.1].

Let $\psi:k\to\mathbb{Q}_{\ell}(\nu_{p})^{\times}$ be a nontrivial character and $\phi:V_{\mathbf{P}}\to k$ be a stable functional. As shown in [ReederYu, Proposition 2.4], the simple summands of the compact induction

are called epipelagic representations of $G(k)$ attached to the parahoric $\mathbf{P}$ and the stable functional $\phi$.

Let $\mathbf{I}_{0}\subset G(F_{0})$ be the Iwahori subgroup corresponding to the opposite Borel $\mathbf{B}^{\mathrm{opp}}$ of $G$, and let $\mathbf{P}_{0}\subset G(F_{0})$ be the parahoric subgroup containing $\mathbf{I}_{0}$ of the same type as $\mathbf{P}_{\infty}$. Let $\widetilde{\mathbf{P}}_{0}$ be the normalizer of $\mathbf{P}_{0}$ in $G(F_{0})$ and $\widetilde{L}_{\mathbf{P}}=\widetilde{\mathbf{P}}_{0}/\mathbf{P}_{0}^{+}$. Let $L_{\mathbf{P}}^{\mathrm{sc}}$ be the simply connected cover of the derived group of $L_{\mathbf{P}}$, and let $L_{\mathbf{P}}(k)^{\prime}=\mathrm{Im}(L_{\mathbf{P}}^{\mathrm{sc}}(k)\to L_{\mathbf{P}}(k))$. We fix a character $\chi:\widetilde{L}_{\mathbf{P}}(k)/L_{\mathbf{P}}(k)\to\overline{\mathbb{Q}}_{\ell}^{\times}$.

Let $\pi=\otimes^{\prime}_{x\in|X|}\pi_{x}$ of $G(\mathbb{A})$ be an automorphic representation whose local components satisfying the following conditions: $\pi_{\infty}$ is an epipelagic representation attached to $\mathbf{P}_{\infty}^{+}$ and $\phi$; $\pi_{0}$ has an eigenvector under $\widetilde{\mathbf{P}}_{0}$ on which it acts through $\chi$ via the quotient $\tilde{L}_{\mathbf{P}}(k)$; $\pi_{x}$ are unramified for $x\neq 0,\infty$.

It was shown in [YunEpipe, Proposition 2.11] that there is a unique cuspidal automorphic representation $\pi=\pi(\chi,\phi)$ of $G(\mathbb{A}_{F})$ satisfying the above conditions.

Let $\operatorname{Bun}=\operatorname{Bun}(\mathbf{P}_{0}^{opp},\mathbf{P}_{\infty}^{++})$ be the moduli stack of bundles with $\mathbf{P}^{opp}$-level structure at $0$ and $\mathbf{P}^{++}$-level structure at $\infty$. The Hecke correspondence for Bun classifies $(x,\mathcal{E},\mathcal{E}^{\prime},\tau)$ where $x\in\widetilde{X}^{\circ}$, $\mathcal{E}$, $\mathcal{E}^{\prime}\in\operatorname{Bun}$ and $\tau:\mathcal{E}|_{\widetilde{X}-x}\to\mathcal{E}^{\prime}|_{\widetilde{X}-x}$ is an isomorphism of $G$-torsors preserving the level structures at $0$ and $\infty$. The moduli stack $\operatorname{Bun}_{G}(\widetilde{\mathbf{P}}_{0},\mathbf{P}^{+}_{\infty})$ has a unique relevant point $\mathcal{E}$ with trivial automorphism group. Let $j:V_{\mathbf{P}}\hookrightarrow\operatorname{Bun}$ be the open embedding defined by $\mathcal{E}$ and AS_ψ the pullback along the canonical pairing $<-,->:V_{\mathbf{P}}\times V_{\mathbf{P}}^{*,st}\to\mathbb{G}_{a}$ of the Artin-Schreier sheaf on $\mathbb{G}_{a}$ corresponding to a fixed character $\psi:k\to\bar{\mathbb{Q}}_{\ell}$. According to [YunEpipe, Section 3], the Hecke eigensheaf can be described as

Let $\pi_{\infty}$ be the epipelagic supercuspidal representation of $G(k)$ as the local component at $\infty$ of an automorphic representation $\pi(\chi,\phi)$ as realized in section 2.3. As shown in [YunEpipe, Corollary 3.10], the ${}^{L}G$-local system $\mathrm{Kl}_{{}^{L}G,\mathbf{P}}(\mathcal{K},\phi)$ is the global Langlands parameter attached to $\pi(\chi,\phi)$. Here $\mathcal{K}$ is the character sheaf corresponding to $\chi$ under sheaf-to-function correspondence.

Let $\pi_{\infty}$ be as defined in the previous subsection. The local Langlands conjecture suggests that there should be a Galois representation $\rho_{\pi_{\infty}}:W(K^{s}/K)\to{}^{L}G(\overline{\mathbb{Q}}_{\ell})$ attached to $\pi_{\infty}$ as the conjectural Langlands parameter. Based on the conjecture relating adjoint gamma factors and formal degrees in [GrossReeder], Reeder and Yu predicted in [ReederYu, Section 7.1] that

We have the following lemma of the Swan conductor of the local system:

Thus, computing the Euler characteristic of the local system corresponding to the Kloosterman sheaves is equivalent to computing the swan conductor at $\infty$. Therefore, combining this with Section 3.1 and Section 3.3, the computation of $\chi_{c}(\tilde{X}^{\circ},\mathrm{Kl}^{\mathrm{St}}_{\widehat{G},\mathbf{P}_{m}}(\chi,\phi))$ gives evidence to Reeder-Yu’s prediction.

Let $(M,\omega)$ be a symplectic vector space of dimension $2n$ over $k$. One can extend $\omega$ linearly to a symplectic form on $M\otimes K$ and we denote $\mathbb{G}=\operatorname{Sp}(M,\omega)$ and $G=\operatorname{Sp}(M\otimes K,\omega)$. Since regular elliptic numbers $m$ of $\mathbb{W}$ in this case are in bijection with divisors $d|n$[Yun, 4.8], we have $m=2n/d$ and let $\ell=n/d$. We fix a decomposition

such that $dim_{M_{i}}=d$ and $\omega(M_{i},M_{j})\neq 0$ only if $i+j=m+1$. We identify $M_{j}$ with $M^{*}_{m+1-j}$.

Define the admissible parahoric subgroup $\mathbf{P}_{m}\subset G(K)$ to be the stabilizer of the lattice chain

where

and $\bar{\omega}$ is a uniformizer of $\mathcal{O}_{F}.$ It has Levi quotient denoted by $L_{m}=\prod_{i=1}^{\ell}GL(M_{i})$ where the $i$-th factor acts on $M_{i}$ by the standard representation and on $M^{*}_{i}=M_{m+1-i}$ by the dual of the standard representation. We have $\widetilde{L}_{m}=L_{m}$ and $\widetilde{L}^{ab}_{m}\cong\prod_{i=1}^{\ell}\mathbb{G}_{m}$ given by the determinants of the GL-factors. The vector space $V_{m}:=V_{\mathbf{P}_{m}}$ can be described as

We can arrange $M_{1},\cdots,M_{m}$ into a cyclic quiver

so that the involution $\tau$ sends $\left\{\psi_{i}:M_{i+1}\rightarrow M_{i}\right\}$ to $\left\{-\psi_{m-i}^{*}:M_{m-i}^{*}\rightarrow M_{m+1-i}^{*}\right\}$. Therefore, $V_{m}$ is the set of $\tau$-invariant cyclic quivers of the above shape. The dual space $V_{m}^{*}$ is the space of $\tau$-invariant cyclic quivers with all the arrows reversed, so we fit them within the same commutative diagram. Let $\phi_{i}:M_{i}\to M_{i+1}$ be the arrows and we view $\phi_{m}$ (resp. $\phi_{\ell}$) as a quadratic form on $M_{m}$ (resp. $M_{\ell}$). Then $\phi=\left(\phi_{1},\ldots,\phi_{m}\right)\in V_{m}^{*}$ is stable if and only if

• •

  All the maps $\phi_{i}$ are isomorphisms;

• •

  We have two quadratic forms on $M_{m}:\phi_{m}$ and the transport of $\phi_{\ell}$ to $M_{m}$ using the isomorphism $\phi_{\ell-1}\cdots\phi_{1}\phi_{m}:M_{m}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}M_{\ell}$. They are in general position in the same sense as explained in [YunEpipe, Section 6.2].

Let $\mathcal{E}=M\otimes\mathcal{O}_{X}$ be the trivial vector bundle of rank $2n$ over $X$ with a symplectic form (into $\mathcal{O}_{X}$ ) given by $\omega$. Define an increasing filtration of the fiber of $\mathcal{E}$ at $\infty$ by $F_{\leq i}\mathcal{E}_{\infty}=\oplus_{j=1}^{i}M_{j}$ and a decreasing filtration of the fiber of $\mathcal{E}$ at 0 by $F^{\geq i}\mathcal{E}_{0}=\oplus_{j=i}^{m}M_{j}$. The moduli stack $\operatorname{Bun}_{G}(\widetilde{\mathbf{P}_{0}},\mathbf{P}^{+}_{\infty})$ then classifies triples $(\mathcal{E},F_{\leq i}\mathcal{E}_{\infty},F^{\geq i}\mathcal{E}_{0})$. The group ind-scheme $\mathfrak{G}$ is the group of symplectic automorphisms of $\left.\mathcal{E}\right|_{X-\{1\}}$ preserving the filtrations $F_{*},F^{*}$ and acting by identity on the associated graded of $F_{*}$. Let $\lambda$ be the dominant short coroot. The subscheme $\mathfrak{G}_{\leq\lambda}$ consists of those $g\in\mathfrak{G}\subset G(F)$ whose entries have at most simple poles at $t=1$, and $\operatorname{Res}_{t=1}g$ has rank at most one. [YunEpipe, Lemma 7.4] shows that the subscheme $\mathfrak{G}_{\leq\lambda}$ can be embedded as an open subscheme of $\operatorname{Sym}^{2}_{\leq 1}(M)$, where $\operatorname{Sym}^{2}_{\leq 1}(M)\subset\operatorname{Sym}^{2}(M)$ is the subscheme of symmetric pure $2$-tensors. This is equivalent to say $u$ and $v$ are parallel vectors. We may write elements $u\cdot v\in\mathrm{Sym}^{2}(M)_{\leq 1}$ as $u=(u_{1},\dots,u_{m})$ and $v=(v_{1},\dots,v_{m})$ with $u_{i},v_{i}\in M_{i}$. Define $\gamma_{i}$ as follows

The definition is independent of the choice of $u,v$ expressing $u\cdot v$ therefore defines a regular function on $\mathrm{Sym}^{2}(M)_{\leq 1}.$ The following proposition gives an explicit description of $\operatorname{Kl}_{\widehat{G},\mathbf{P}}^{st}(\phi)$ that will be used later in the computation of the Euler characteristic.

By the same argument as in [Katz] the Swan conductor of $\mathrm{Kl}^{\operatorname{st}}_{\widehat{G},\mathbf{P}_{m}}(\chi,\phi)$ at $\infty$ does not depend on $\chi$, so does the Euler characteristic of $\mathrm{Kl}^{\operatorname{st}}_{\widehat{G},\mathbf{P}_{m}}(\chi,\phi)$. We make an inductive argument as follows.