Shelstad’s character identity from the point of view of index theory

The authors are grateful to Kuok Fai Chao for background information about the Langlands program. The second author was supported by the Australian Research Council, through Discovery Early Career Researcher Award DE160100525.

We recall the definition of inner forms, as discussed for example in [ABV]. Let $G_{\mathbb{C}}$ be a connected reductive complex algebraic group. A real form of $G_{\mathbb{C}}$ is an involutive automorphism $\sigma\colon G_{\mathbb{C}}\rightarrow G_{\mathbb{C}}$ which is antiholomorphic; i.e., $T_{e}\sigma(iX)=-iT_{e}\sigma(X)$ for all $X\in\mathfrak{g}_{\mathbb{C}}$. The involution $\sigma$ is a generalised notion of complex conjugation. The group of real points of $\sigma$

is called the real form of $G_{\mathbb{C}}$ associated to $\sigma.$

The set of inner forms for $G$ can be identified bijectively to the first Galois cohomology $H^{1}(\Gamma,\operatorname{Inn}(G_{\mathbb{C}}))$ where $\Gamma$ is the Galois group $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$ of two elements, where the complex conjugation (the generator) takes $\eta\in\operatorname{Aut}(G_{\mathbb{C}})$ to $\sigma\circ\eta\circ\sigma^{-1}\in\operatorname{Aut}(G_{\mathbb{C}})$, and $\operatorname{Inn}(G_{\mathbb{C}})$ is the subgroup of inner automorphisms of $G_{\mathbb{C}}$ respecting the action of $\Gamma$. See [B].

If $G$ and $G^{\prime}$ are inner forms to each other in a complex group $G_{\mathbb{C}}$, then any Cartan subgroup $H<G$ is conjugate in $G_{\mathbb{C}}$ to a Cartan subgroup $H^{\prime}<G^{\prime}$. See Lemma 2.1 of [L3] and [S2]. We will use this to identify elements $h\in H$ to corresponding elements $h^{\prime}\in H^{\prime}$.

Inner forms are closely related to the Langlands program. Let $G$ be a real reductive algebraic group, and let $G^{\vee}$ denote the Langlands dual group of $G$. Consider the $L$-group ${}^{L}\!G=G^{\vee}\rtimes\operatorname{Gal}(\mathbb{C}/\mathbb{R})$. Let $W_{\mathbb{R}}$ be the Weil group of real numbers, i.e., the group $\mathbb{C}^{\times}\cup j\mathbb{C}^{\times}$ subject to relations

Then the irreducible admissible representations of $G$ are parametrised by $L$-parameters. These are group homomorphisms

satisfying certain conditions. In fact, the local Langlands correspondence gives rise to a surjective map

where $f^{-1}(\{\phi\})$ is finite for every $\phi\in\Phi(G).$ The finite set

is called the $L$-packet associated to the $L$-parameter $\phi$. In other words, two admissible irreducible representations $\pi_{1},\pi_{2}$ are said to be in the same $L$-packet if and only if they have the same $L$-parameter. Representations having the same $L$-parameter are indistinguishable in the sense of Langlands.

When $G$ and $G^{\prime}$ are inner forms to each other, they share the same $L$-group. Groups with the same $L$-group are crucial in the sense that the representation theory of these groups are closely related and can be studied globally. The character identity in Theorem 2.4 below, regarding inner forms is an instance of this philosophy. In fact, by the local Langlands correspondence, the $L$-packets of admissible representations for $G$ and $G^{\prime}$ are related by their corresponding $L$-parameters. Hence, a character identity involving a common $L$-packet can be expected. For more details of Langlands program and character identity we refer to Langlands [L1, L2, L3] and Shelstad [S1, S2].

Let $\phi$ be an $L$-parameter of admissible representations of $G$. According to Langlands [L3], all admissible representations in the same $L$-packet of a tempered representation are tempered. Hence the $L$-parameter $\phi$ of a tempered representation is called tempered. Let $G^{\prime}$ be an inner form of $G$. Suppose $G$ is quasi-split, meaning that $G$ has a $\sigma$-invariant Borel subgroup. In that case, we have $\Phi(G^{\prime})\subset\Phi(G)$. Suppose $\phi\in\Phi(G^{\prime})$. We write $\phi^{\prime}$ for $\phi$, when we view it as an $L$-parameter of $G^{\prime}$. Then $\phi^{\prime}$ is tempered if $\phi$ is. The same line of statements remains true with “tempered’ replaced by “discrete series”.

For any irreducible admissible representation $\pi$, we denote its global character by $\Theta_{\pi}$. Shelstad’s character identity is stated as follows.

This is Theorem 6.3 in [S2]. In this paper, we will prove Shelstad’s character identity (2.3) using $K$-theory and index theory when $\Pi_{\phi}$ and $\Pi_{\phi^{\prime}}$ are $L$-packets of discrete series representations.

We now assume in addition that the group $G$ is semisimple, and has discrete series representations. Harish-Chandra has the following equivalent statements for existence of discrete series. See for example Theorem 22.1 in [T].

As a result, all but the second case in Example 2.2 admit discrete series and hence they have compact Cartan subgroups. Assume $G$ has discrete series and $T$ is a compact Cartan subgroup from now on.

So an inner form $G^{\prime}$ of $G$ also has discrete series representations.

For discrete series, we have the following property, of $L$-packets, which can be found for example in [Lab].

Let $K<G$ be a maximal compact subgroup containing $T$. Let $W_{G}$ and $W_{K}<W_{G}$ be the Weyl groups of the root systems of $(\mathfrak{g}^{\mathbb{C}},\mathfrak{t}^{\mathbb{C}})$ and $(\mathfrak{k}^{\mathbb{C}},\mathfrak{t}^{\mathbb{C}})$, respectively. Then two discrete series representations of $G$ have the same infinitesimal character if and only if their Harish-Chandra parameters are in the same $W_{G}$-orbit. They are equivalent if and only if they are in the same $W_{K}$-orbit, so every discrete series $L$-packet can be identified with $W_{G}/W_{K}$.

In this paper, we will give a geometric proof of a special case of Theorem 2.4

This case of Theorem 2.4 can be proved by explicitly writing out Harish–Chandra’s character formula for the discrete series and rearranging terms. Our proof does not involve character formulas, and will show how this case of the character identities is related to the geometry of the space $G/T$, and also how character identities are related to $K$-theory and index theory.

Let $G$ be a connected, real semisimple Lie group with finite centre. Suppose $G$ acts properly and isometrically on a Riemannian manifold $M$. Suppose $M/G$ is compact. Let $E\to M$ be a $\mathbb{Z}_{2}$-graded, $G$-equivariant, Hermitian vector bundle. Let $D$ be a $G$-equivariant, elliptic, self-adjoint first order differential operator on $E$ that is odd with respect to the grading. The reduced group $C^{*}$-algebra $C^{*}_{r}G$ of $G$ is the closure in the operator norm of the algebra of all convolution operators on $L^{2}(G)$ by functions in $L^{1}(G)$. Let $K_{0}(C^{*}_{r}G)$ be its even $K$-theory. Then we have the equivariant index of $D$

which is the image of the class defined by $D$ in the equivariant $K$-homology group of $M$ under the analytic assembly map. See [BCH] for details.

Let $g\in G$ be a semisimple element, and $Z<G$ its centraliser. The orbital integral map

on $C_{c}(G)$, defines a trace map

See Section 2.1 in [HW2].

Theorem 2.1 in [HW2] is a fixed point formula for the number

We will not use this fixed point formula, but only the following localisation or excision property of the index. This property follows directly from Theorem 2.1 in [HW2], but is in fact a step in its proof: see Proposition 4.6 in [HW2].

Let $K<G$ be maximal compact. Let $T<K$ be a maximal torus and suppose that $T$ is a Cartan subgroup of $G$; i.e. $G$ has a discrete series. Let

be the root system of $(\mathfrak{g}^{\mathbb{C}},\mathfrak{t}^{\mathbb{C}})$. Choose a positive system $R^{+}\subset R$, and let $\rho$ be half the sum of the elements of $R^{+}$. Let $\lambda\in i\mathfrak{t}^{*}$ be regular, and dominant with respect to $R^{+}$. Suppose $\lambda-\rho$ is integral. Let $\pi^{G}_{\lambda}$ be the discrete series representation of $G$ with Harish-Chandra parameter $\lambda$. The values of its character $\Theta_{\pi^{G}_{\lambda}}$ on the regular elements of $T$ can be realised in terms of index theory.

Consider the $G$-manifold $G/T$, equipped with the $G$-invariant Riemannian metric defined by a $K$-invariant inner product on $\mathfrak{g}$. Consider the $G$-invariant complex structure $J^{G/T}_{R^{+}}$ on $G/T$ such that, as complex vector spaces,

Explicitly, this isomorphism is given by the inclusion $\mathfrak{g}/\mathfrak{t}\hookrightarrow(\mathfrak{g}/\mathfrak{t})^{\mathbb{C}}$ followed by projection onto the positive root spaces. Similarly, if $w\in W_{G}$, then we will write $J^{G/T}_{wR^{+}}$ for the complex structure defined as above, with $R^{+}$ replaced by $wR^{+}$.

For any integral element $\nu\in i\mathfrak{t}^{*}$, we have the $G$-equivariant line bundle

Here we write $\mathbb{C}_{\nu}$ for the vector space $\mathbb{C}$ on which $T$ acts with weight $\nu$. Let $\bar{\partial}_{L^{G}_{\lambda-\rho}}$ be the Dolbeault operator on $G/T$ coupled to $L^{G}_{\lambda-\rho}$. Let $\bar{\partial}_{L^{G}_{\lambda-\rho}}^{*}$ be its formal adjoint with respect to the $L^{2}$-inner product defined by the Riemannian metric on $G/T$ and the natural Hermitian metric on $L^{G}_{\lambda-\rho}$.

For a proof, see Propositions 5.1 and 5.2 in [HW2]. This proof is based on the fact that the natural class $[\pi_{\lambda}^{G}]\in K_{0}(C^{*}_{r}G)$ (see e.g.  [Laf]) equals

The geometric proof of Theorem 2.8 is based on suitable identifications of neighbourhoods of the fixed point sets of the actions by $T$ on $G/T$ and $G_{c}/T$.

Let $w\in N_{G_{c}}(T)$. Consider the $T$-invariant complex structure on $\mathfrak{g}/\mathfrak{t}$ defined by ${w^{-1}R^{+}}$ and the one on $\mathfrak{g}_{c}/\mathfrak{t}$ defined by ${R^{+}}$. Let $\mathfrak{t}^{\perp}\subset\mathfrak{g}$ be the orthogonal complement to $\mathfrak{t}$ in $\mathfrak{g}$ with respect to an $\operatorname{Ad}(K)$-invariant inner product. Similarly, let $\mathfrak{t}^{\perp_{c}}\subset\mathfrak{g}_{c}$ be the orthogonal complement to $\mathfrak{t}$ in $\mathfrak{g}_{c}$. Then we have a complex-linear isomorphism

Note that $\operatorname{Ad}(w)$ preserves $\mathfrak{t}^{\perp_{c}}$.

The map $\eta\colon\mathfrak{t}^{\perp}\to G/T$ mapping $X\in\mathfrak{t}^{\perp}$ to $\exp(X)T$ is $T$-equivariant, and a local diffeomorphism near $0\in\mathfrak{t}^{\perp}$. Let $\tilde{U}\subset\mathfrak{t}^{\perp}$ be an $\operatorname{Ad}(T)$-invariant open neighbourhood of $0$ on which this map defines a diffeomorphism onto its image $U\subset G/T$. The tangent map of $\eta$ at $0$ is the identification $\mathfrak{t}^{\perp}\cong T_{eT}G/T$, which is complex-linear by definition. So $\eta^{*}(J^{G/T}_{w^{-1}R^{+}}|_{U})$ is homotopic to $J_{w^{-1}R^{+}}|_{\tilde{U}}$ as $T$-invariant complex structures, if we choose $\tilde{U}$ small enough. So $\eta$ is holomorphic up to a homotopy of $T$-invariant almost complex structures, which is as good as being holomorphic for index theory purposes.

Let $\eta_{c}\colon\mathfrak{t}^{\perp_{c}}\to G_{c}/T$ be defined by $\eta_{c}(Y)=\exp(Y)T$ for $Y\in\mathfrak{t}^{\perp_{c}}$. This is a $T$-equivariant map, and a local diffeomorphism near $0$. Choose $\tilde{U}$ small enough, so that $\eta_{c}|_{\tilde{\psi}_{w}(\tilde{U})}$ is a diffeomorphism onto its image $U_{c}^{w}$, which is a neighbourhood of $eT$ in $G_{c}/T$. Again, this diffeomorphism is holomorphic up to a $T$-equivariant homotopy of complex structures if we choose $\tilde{U}$ small enough. For later use, we define the map $\eta_{c}^{w}\colon\mathfrak{t}^{\perp_{c}}\to G_{c}/T$ by $\eta_{c}^{w}(Y)=\exp(Y)wT$, for $Y\in\mathfrak{t}^{\perp_{c}}$. This map is $T$-equivariant because $w$ normalises $T$.

The maps $\psi_{w}$, $\eta$ and $\eta_{c}$ combine into a diffeomorphism

holomorphic up to a homotopy of complex structures. The map $\psi_{w}$ is not $T$-equivariant, so neither is $\varphi_{w}$ in general.

Let $W_{K}:=N_{K}(T)/T$ and $W_{G}:=N_{G_{c}}(T)/T$. Then $W_{G}$ is the Weyl group of the root system $R$, which explains the notation $W_{G}$. And $W_{K}$ is the subgroup of $W_{G}$ generated by reflections defined by compact roots. Note that

(see Lemma 6.7 in [HW1]), and

Hence we have an inclusion

and

Here we fix representatives $w\in W_{G}$ of all classes $[w]\in W_{G}/W_{K}$ once and for all.

Consider the manifold

equipped with the action by $G$ on the first factor. Consider the $G$-invariant complex structure on $M$ such that, for every $[w]\in W_{G}/W_{K}$, the restricted complex structure on $G/T\times\{[w]\}$ is $J^{G/T}_{w^{-1}R^{+}}$.

We have $T$-invariant neighbourhoods

of $(G/T)^{T}$ and

of $(G_{c}/T)^{T}$. We choose the set $\tilde{U}$ small enough, the sets $w_{K}U$ and $wU_{c}^{w}$ are all disjoint, for $w_{K}\in W_{K}$ and $w\in W_{G}$. Consider the map

given by

for $w_{K}\in W_{K}$, $w\in W_{G}$ and $xT\in U$.

The arguments in this section lead to the following conclusion.

See Figure 1 for an example of the map $\varphi$.

Consider the line bundle $L\to M$ such that for every $[w]\in W_{G}/W_{K}$,

We saw in the proof of Lemma 3.2 that

So the pullback of the left hand side of (3.3) along $\eta$ equals