Arthur packets for pure real forms of symplectic and special orthogonal groups

Abstract

Arthur packets have been defined for pure real forms of symplectic and special orthogonal groups following two different approaches. The first approach, due to Arthur, Moeglin, and Renard uses harmonic analysis. The second approach, due to Adams, Barbasch, and Vogan uses microlocal geometry. We prove that the two approaches produce essentially equivalent Arthur packets. This extends previous work of the authors and J. Adams for the quasisplit real forms.

1 Introduction

The first goal of this note is to review some packets of representations, conjectured by Arthur, in the setting of real symplectic and special orthogonal groups. We restrict our attention to pure real forms of these groups. There are two different approaches to defining these Arthur packets. One of them is due to Arthur himself for quasisplit forms of these groups [A2]. His analytic approach was extended to include pure real forms by Moeglin and Renard [MR1]. The other approach follows the work of Adams, Barbasch, and Vogan in microlocal geometry [ABV]. The principal goal of this note is to prove that the two approaches produce essentially equivalent Arthur packets. The equivalence of the Arthur packets was given in the quasisplit case in [AAM]. The only pure real form of a symplectic group is quasisplit, and so the truly new result here is the equivalence of Arthur packets for the pure real forms of special orthogonal groups.

We assume that the reader is somewhat familiar with the classification of real reductive groups, and the basic formalism of Arthur parameters and Arthur packets. Suppose $N \geq 2$ and G is either the complex group $\mathrm {Sp}_N$ or $\mathrm {SO}_N$ . Fix $\delta _{q}$ to be an antiholomorphic automorphism of G such that its fixed-point subgroup $G(\mathbb {R}, \delta _{q})$ is a quasisplit real form of G. Let $\Gamma = \mathrm {Gal}(\mathbb {C}/\mathbb {R})$ . Then $\delta _{q}$ determines an action of $\Gamma $ on G, and this action may be identified with the trivial cocycle in $H^{1}(\Gamma ,G) = H^{1}(\mathbb {R},G)$ . As we recall in Section 2, the remaining classes $\delta \in H^{1}(\Gamma , G)$ also determine real forms $G(\mathbb {R}, \delta )$ of G. We call the classes in $H^{1}(\Gamma , G)$ the pure real forms of G.

Let

(1) $$ \begin{align} \psi_G^{}:W_{\mathbb{R}}\times\mathrm{SL}_2\ \longrightarrow\ {}^{\vee}G^{\Gamma} \end{align} $$

be an Arthur parameter for G, where $^{\vee }G^{\Gamma }$ denotes the Galois form of the L-group of G. The centralizer $\text {Cent}(\psi _{G}(W_{\mathbb {R}}),{^\vee }G)$ of the image of $\psi _{G}$ in ${^\vee }G$ has component group denoted by

$$ \begin{align*}A_{\psi_{G}} = \text{Cent}(\psi_{G}(W_{\mathbb{R}}),{^\vee}G) \, / \, (\text{Cent}(\psi_{G}(W_{\mathbb{R}}),{^\vee}G))^{0}.\end{align*} $$

It is a finite 2-group ([A2, Equation (1.4.8)]). The Arthur parameter $\psi _{G}$ also determines an element $s_{\phi _{G}} \in A_{\psi _{G}}$ which is the coset of

(2) $$ \begin{align} \psi_{G}\left(1, \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \right) \in \text{Cent}(\psi_{G}(W_{\mathbb{R}}),{^\vee}G). \end{align} $$

In Arthur’s approach to defining the packet parameterized by $\psi _{G}$ , there is an additional object that comes into play. It is the group $\mathrm {Out}_{N}(G)$ , which is only non-trivial when N is even and $G =\mathrm {SO}_{N}$ . In the non-trivial case it is a group of order two, generated by an outer automorphism of $\mathrm {SO}_{N}$ . ([A2, p. 12]. See Section 3).

In [A2, Theorem 2.2.1], Arthur provides a stable distribution on the $\mathrm {Out}_{N}(G)$ -invariant test functions on $G(\mathbb {R},\delta _{q})$ . Although it is not immediately apparent from his notation, his distribution may be written as

(3) $$ \begin{align} \eta^{\mathrm{Ar}}_{\psi_{G}}(\delta_{q}) = \sum_{\tilde{\pi} \in \widetilde{\Pi}_{\psi_{G}}^{\mathrm{Ar}}(\delta_{q})} \mathrm{Tr}\left(\tau^{\mathrm{Ar}}_{\psi_{G}}(\tilde{\pi})(s_{\psi_{G}}) \right) \, \tilde{\pi} \end{align} $$

([A2, Equation (7.4.1)], [AAM, Section 10]). Here, $\widetilde {\Pi }_{\psi _{G}}^{\mathrm {Ar}}(\delta _{q})$ is a finite set of $\mathrm {Out}_{N}(G)$ -orbits of irreducible unitary representations of $G(\mathbb {R}, \delta _{q})$ . We call this set the Arthur packet of $\psi _{G}$ for $G(\mathbb {R},\delta _{q})$ . In (3) we identify $\tilde {\pi }$ with its distribution character, and $\tau _{\psi _{G}}^{\mathrm {Ar}}(\tilde {\pi })$ is a finite-dimensional representation of $A_{\psi _{G}}$ . The crux of Arthur’s theorem is that the distributions $\eta _{\psi _{G}}^{\mathrm {Ar}}(\delta _{q})$ satisfy both ordinary and twisted endoscopic identities as conjectured in [Al]. Arthur uses harmonic analysis in local and global settings to prove his theorem.

Continuing in the same vein, Moeglin and Renard extend Arthur’s results to pure real forms [MR1]. For every pure real form $\delta \in H^{1}(\Gamma , G)$ , there is a corresponding real form $G(\mathbb {R}, \delta )$ of G. Moeglin and Renard produce stable distributions

$$ \begin{align*}\eta_{\psi_{G}}^{\mathrm{Ar}}(\delta) = \sum_{\tilde{\pi} \in \widetilde{\Pi}_{\psi_{G}}^{\mathrm{Ar}}(\delta)} \mathrm{Tr}\left(\tau^{\mathrm{Ar}}_{\psi_{G}}( \tilde{\pi})(s_{\psi_{G}}) \right) \, \tilde{\pi}\end{align*} $$

whose terms mirror those of (3). In particular, the Arthur packet $\widetilde {\Pi }_{\psi _G^{}}^{\mathrm {Ar}}(\delta )$ is a finite set of $\mathrm {Out}_{N}(G)$ -orbits of irreducible unitary representations of $G(\mathbb {R},\delta )$ . The distributions $\eta _{\psi _{G}}^{\mathrm {Ar}}(\delta )$ also satisfy Arthur’s conjectured endoscopic identities, and the method of proof again relies on harmonic analysis.

Adams, Barbasch, and Vogan approach this subject following different methods, which are based on microlocal geometry and equivariant sheaf theory on a generalized flag variety. Using these methods they produce stable distributions of the form

$$ \begin{align*}\eta_{\psi_{G}}^{\mathrm{ABV}}(\delta)= \sum_{\pi \in \Pi_{\psi_{G}}^{\mathrm{ABV}}(\delta)} (-1)^{d S_{\pi} - d S_{\psi_{G}}}\ \mathrm{Tr}\left(\tau^{\mathrm{ABV}}_{\psi_{G}}(\pi) (1) \right) \, \pi.\end{align*} $$

The Arthur packet $\Pi _{\psi _{G}}^{\mathrm {ABV}}(\delta )$ is a finite set of irreducible representations of $G(\mathbb {R},\delta )$ . At the time of writing these irreducible representations are not known to be unitary, but they still come attached with finite-dimensional representations $\tau ^{\mathrm {ABV}}_{\psi _{G}}(\pi )$ of $A_{\psi _{G}}$ . In addition, the distributions $\eta _{\psi _{G}}^{\mathrm {ABV}}(\delta )$ satisfy Arthur’s conjectured endoscopic identities. There are notable advantages to this approach. One is that $\mathrm {Out}_{N}(G)$ -orbits do not appear. Furthermore, the approach works for any connected reductive algebraic group G and all of its real forms.

One cannot expect a literal equality between $\eta _{\psi _{G}}^{\mathrm {ABV}}(\delta )$ and $\eta _{\psi _{G}}^{\mathrm {Ar}}(\delta )$ since the latter is expressed in terms of $\mathrm {Out}_{N}(G)$ -orbits and the former is not. Apart from this wrinkle, one would expect the two distributions to agree. In the quasisplit setting, i.e., $\delta = \delta _{q}$ , the precise relationship between $\eta _{\psi _{G}}^{\mathrm {ABV}}(\delta _{q})$ and $\eta _{\psi _{G}}^{\mathrm {Ar}}(\delta _{q})$ is given in [AAM, Theorem 9.3]. The main result of this note is the extension of this theorem to the pure real forms, namely

$$ \begin{align*}\eta_{\psi_G^{}}^{\mathrm{Ar}}(\delta) = \mathrm{Out}_N(G)\cdot\left(\eta_{\psi_G^{}}^{\mathrm{ABV}}(\delta)\right), \quad \widetilde{\Pi}_{\psi_{G}}^{\mathrm{Ar}}(\delta) = \mathrm{Out}_N(G)\cdot \Pi_{\psi_{G}}^{\mathrm{ABV}}(\delta)\end{align*} $$

(Theorem 3.4). In fact, we prove a more refined identity in which $\eta _{\psi _G^{}}^{\mathrm {Ar}}(\delta )$ and $\eta _{\psi _G^{}}^{\mathrm {ABV}}(\delta )$ are regarded as representations of $G(\mathbb {R}, \delta ) \times A_{\psi _{G}}$ . This allows us to extract information about the finite-dimensional representations $\tau _{\psi _{G}}^{\mathrm {Ar}}(\tilde {\pi })$ and $\tau _{\psi _{G}}^{\mathrm {ABV}}(\pi )$ appearing in each distribution, and to conclude that

$$ \begin{align*}\tau^{\mathrm{Ar}}_{\psi_G^{}}(\widetilde{\pi})\, =\, \tau^{\mathrm{ABV}}_{\psi_G^{}}(\pi)\end{align*} $$

where $\widetilde {\pi }$ is the $\mathrm {Out}_N(G)$ -orbit of $\pi \in \Pi _{\psi _{G}}^{\mathrm {ABV}}(\delta )$ . Along the way, we work through several exercises proposed in [AAM, Section 10].

Our work is part of a broader comparison of Arthur packets. Similar comparisons have been made for real unitary groups in [ARM2], for p-adic general linear groups [CR], and low rank p-adic symplectic and special orthogonal groups [CFM+].

2 Pure real forms and their representations

In this section G may be any connected complex reductive algebraic group. We fix an antiholomorphic involutive automorphism $\delta _{q}$ of G ([ABV, Equation (2.1)(b)]) such that the fixed-point subgroup $G(\mathbb {R}, \delta _{q})$ is a quasisplit group ([S4, Equation 3.2]). Suppose $\delta $ is a one-cocyle representing a class in $H^{1}(\Gamma , G)$ as in the introduction, and let $\sigma $ be the non-identity element in $\Gamma $ . Then $\delta (\sigma )$ is an element in G and the fixed-point subgroup $G(\mathbb {R},\delta )$ of $\mathrm {Int}(\delta (\sigma )) \delta _{q}$ is a real form of G ([S5, Equation 12.3.7]). We call $\delta $ a pure real form of G. We often blur the distinction between $\delta $ , its class in $H^{1}(\Gamma , G)$ , and (the isomorphism class of) its real form $G(\mathbb {R}, \delta )$ . The quasisplit real form corresponds to the trivial cocycle of $H^{1}(\Gamma , G)$ which we shall also denote by $\delta _{q}$ .

We note in passing that it is possible, and often preferable, to recast the definition of pure real forms so that $\delta _{q}$ is an algebraic automorphism, instead of an antiholomorphic automorphism ([AT, Corollary 4.7]). In addition, pure real forms are special instances of the strong real forms of [ABV, Definition 2.13] and the rigid inner twists of ([KI, Section 5.2]).

When G is the stabilizer of a bilinear form, as is the case for symplectic or orthogonal groups, $H^{1}(\Gamma , G)$ is in bijection with the real isomorphism classes of the complex bilinear form ([KMRT, Proposition 29.1] Proposition 29.1). If we take $G = \mathrm {Sp}_{N}$ then the quasisplit real form $G(\mathbb {R},\delta _{q}) = \mathrm {Sp}_{N}(\mathbb {R})$ is split. It is well-known that a split real symplectic group corresponds to a non-degenerate alternating bilinear form, and that this bilinear form has only one real isomorphism class ([J] Theorem 6.3). Hence, $H^{1}(\Gamma , \mathrm {Sp}_{N})$ is trivial and the only pure real form of $\mathrm {Sp}_{N}$ is the split form $\mathrm {Sp}_{N}(\mathbb {R})$ . The non-split real forms $\mathrm {Sp}(p,q)$ are not pure.

On the other hand, the orthogonal group $\mathrm {O}_{N}$ is the stabilizer of a non-degenerate symmetric bilinear form, and its real isomorphism classes are parameterized by $(p,q)$ where $p+q=N$ ([J] Theorem 6.8). Here, p represents the $p \times p$ identity matrix, q represents the negative of the $q \times q$ identity matrix, and $(p,q)$ corresponds to the real form $\mathrm {O}(p,q)$ . Fix $(p',q')$ corresponding to a quasisplit orthogonal group $\mathrm {O}(p',q')$ and take $G = \mathrm {SO}_{N}$ . Then we may identify $(p',q')$ with $\delta _{q}$ and write $G(\mathbb {R}, \delta _{q}) = \mathrm {SO}(p',q')$ . The pure real forms relative to $\mathrm {SO}(p',q')$ are given by those $(p,q)$ for which q has the same parity as $q'$ ([KMRT, Equation (29.29)]). If N is even there are exactly two inequivalent choices for $\delta _{q} = (p',q')$ : one corresponding to the split form $\mathrm {SO}(N/2,N/2)$ , and the other for the quasisplit (but not split) form $\mathrm {SO}((N/2)+1, (N/2)-1)$ . If N is odd, then the only choice for $\delta _{q}$ up to equivalence is the one corresponding to the split form $\mathrm {SO}((N+1)/2, (N-1)/2)$ . Going through these possibilities, the conclusion is that the pure real forms of $\mathrm {SO}_{N}$ correspond to the groups $\mathrm {SO}(p,q)$ where $p+q = N$ . For even N, the real form $\mathrm {SO}^{*}(N)$ ([K2, Equation (1.141)]) is not pure. We therefore avoid the difficulties presented by this real form in [A2, Section 9.1].

Let $\delta \in H^{1}(\mathbb {R},G)$ be a pure real form. We define a representation of $G(\mathbb {R}, \delta )$ to be an admissible group representation in the sense of [V, Definition 1.1.5]. We let $\Pi (G(\mathbb {R}, \delta ))$ be the set of infinitesimal equivalence classes of irreducible representations of $G(\mathbb {R},\delta )$ . As customary, we will not distinguish between a representation and its equivalence class. Every representation in $\Pi (G(\mathbb {R},\delta ))$ has an infinitesimal character, which we may identify with a ${^\vee }G$ -orbit of a semisimple element in ${^\vee }\mathfrak {g}$ ([ABV, Lemma 15.4]). For a fixed semisimple ${^\vee }G$ -orbit ${^\vee }\mathcal {O} \subset {^\vee }\mathfrak {g}$ , define $\Pi ({^\vee }\mathcal {O}, G(\mathbb {R}, \delta ))$ to be the subset of elements in $\Pi (G(\mathbb {R}, \delta ))$ with infinitesimal character ${^\vee }\mathcal {O}$ . Let

$$ \begin{align*}\Pi({^\vee}\mathcal{O}, G/\mathbb{R}) = \coprod_{\delta \in H^{1}(\Gamma, G)} \Pi({^\vee}\mathcal{O}, G(\mathbb{R}, \delta))\end{align*} $$

be the disjoint union of the (equivalence classes) of irreducible representations of all pure real forms relative to a fixed quasisplit form $\delta _{q}$ .

3 Arthur packets

We return to G being a symplectic or special orthogonal group as in the introduction. In the following two sections we revisit the two approaches to defining Arthur packets and their underlying distributions. The summaries are meant to provide enough detail to be able to connect with the work of [AAM] and [ARM2], where similar comparisons of the two approaches have been established. In the final section we prove the precise identities relating the two kinds of Arthur packets.

3.1 The approach of Arthur, Moeglin, and Renard

We give a review of Arthur’s construction of $\eta _{\psi _{G}}^{\mathrm {Ar}}(\delta _{q})$ in (3), and Moeglin and Renard’s extension $\eta _{\psi _{G}}^{\mathrm {Ar}}(\delta )$ to any pure real form $\delta $ . The starting point of Arthur’s approach is to express G as a twisted endoscopic group of $(\text {GL}_{N}, \vartheta )$ ([KS, Section 2], [A2, Section1.2]). In this pair, $\vartheta $ is the outer automorphism of $\text {GL}_{N}$ defined by

$$ \begin{align*}\vartheta(g) = \tilde{J} \, (g^{-1})^{\intercal} \, \tilde{J}^{-1}, \quad g \in \text{GL}_{N}, \end{align*} $$

where $\tilde {J}$ is the anti-diagonal matrix

$$ \begin{align*} \tilde{J} = \scriptsize \begin{bmatrix}0 & & & 1\\ & &-1 & \\ & \unicode{x22F0} & & \\ (-1)^{N-1} & & & 0\end{bmatrix}\normalsize. \end{align*} $$

The semidirect product $\text {GL}_{N} \rtimes \langle \vartheta \rangle $ is a disconnected algebraic group with non-identity component $\text {GL}_{N} \rtimes \vartheta $ . The group G is attached to the pair $(\text {GL}_{N}, \vartheta )$ through an element $s \vartheta \in \text {GL}_{N} \rtimes \vartheta $ whose fixed-point set $({^\vee }\text {GL}_{N})^{s \vartheta }$ contains ${^\vee }G$ as an open subgroup. This gives us an inclusion ${}^{\vee }G\hookrightarrow {^\vee }\text {GL}_N$ , which can be extended into an inclusion

$$ \begin{align*} \text{St}_{G} : {{}^\vee}G^{\Gamma} \hookrightarrow {^\vee}\text{GL}_{N}^{\Gamma}, \end{align*} $$

that makes $(G,{}^{\vee }G^{\Gamma },s, \text {St}_{G})$ a twisted endoscopic datum of $({}^{\vee }\text {GL}_{N}, {}^{\vee }\vartheta )$ .

The inclusion allows us to extend the Arthur parameter $\psi _G$ of (1) into an Arthur parameter

(4) $$ \begin{align} \psi = \text{St}_{G} \circ \psi_{G} \end{align} $$

for $\text {GL}_{N}$ . There is a Langlands parameter $\varphi _{\psi }$ associated with $\psi $ ([Al, Section 4]). It is defined by

(5) $$ \begin{align} \varphi_{\psi}^{}(w)=\psi \left(w, \begin{bmatrix} |w|^{\frac12}&0\\0&|w|^{-\frac12} \end{bmatrix} \right), \quad w\in W_{\mathbb R}. \end{align} $$

The local Langlands correspondence attaches to $\varphi _\psi $ an irreducible representation $\pi _\psi $ of $\text {GL}_{N}(\mathbb R)$ . This representation satisfies

$$ \begin{align*}\pi_\psi\circ\vartheta \cong \pi_\psi,\end{align*} $$

and may therefore be extended to a representation $\pi ^{\thicksim }_{\psi }$ of $\text {GL}_N\rtimes \left <\vartheta \right>$ . The extension $\pi ^{\thicksim }_{\psi }$ is not unique. We choose the extension following [A2, pp. 62–63] by fixing a Whittaker datum. Let $\mathrm {Tr}_{\vartheta }(\pi ^{\thicksim }_{\psi })$ be the twisted character of $\pi ^{\thicksim }_{\psi }$ , i.e., the distribution

$$ \begin{align*} \mathrm{Tr}_{\vartheta}(\pi^{\thicksim}_{\psi}):\ C_{c}^{\infty}(\text{GL}_{N} (\mathbb{R})\rtimes \vartheta)\ &\longrightarrow\ \mathbb C\\ f \ &\longmapsto\ \mathrm{Tr} \int_{\text{GL}_{N}(\mathbb{R})} f(x\vartheta) \, \pi^{\thicksim}_{\psi}(x) \, \pi^{\thicksim}_{\psi}(\vartheta) \, dx. \end{align*} $$

In the same manner, one may define a twisted character $\mathrm {Tr}(\pi ^{\thicksim })$ for any extension $\pi ^{\thicksim }$ of an irreducible representation $\pi $ of $\text {GL}_{N}(\mathbb {R})$ satisfying $\pi \circ \vartheta \cong \pi $ . Let $K \Pi (\text {GL}_{N}(\mathbb {R}) \rtimes \vartheta )$ be the $\mathbb {Z}$ -module generated by all such twisted characters. For any pure real form $\delta $ of G, we define $K \Pi (G(\mathbb {R}, \delta ))$ to be the $\mathbb {Z}$ -module generated by the distribution characters of $\pi \in \Pi (G(\mathbb {R}, \delta ))$ . The module $K\Pi (G(\mathbb {R}, \delta ))$ is isomorphic to the Grothendieck group of finite-length representations of $G(\mathbb {R}, \delta )$ , and we identify the two modules.

Before introducing Arthur’s definition of $\eta _{\psi _{G}}^{\mathrm {Ar}}(\delta _{q})$ , we should return to the group $\mathrm {Out}_N(G)$ mentioned in the introduction. The group $\mathrm {Out}_N(G)$ is defined as the quotient

$$ \begin{align*}\mathrm{Out}_N(G)\, =\, \mathrm{Aut}_N(G)/\mathrm{Int}(G), \end{align*} $$

where $\mathrm {Aut}_N(G)$ is the group of automorphisms of the endoscopic datum $(G,{}^{\vee }G^{\Gamma }, s, \text {St}_{G})$ ([KS, p. 18], [A2, p. 12]). The group $\mathrm {Out}_N(G)$ may be identified with the group of outer automorphisms of G ([KS, Equation (2.1.8)]). When G is a symplectic group or an odd rank special orthogonal group it follows that $\mathrm {Out}_N(G)$ is trivial. When $G = \mathrm {SO}_{N}$ and N is even, then $\mathrm {Out}_N(G) \cong \mathrm {O}_{N}/ \, \mathrm {SO}_{N}$ , a group of order two. In this case, we choose a canonical representative

$$ \begin{align*}w = \tilde{w}(N) \in \mathrm{O}_{N}\end{align*} $$

as in [A2, p. 10], so that

(6) $$ \begin{align} \mathrm{Out}_N(G)\, \cong\, \mathrm{O}_{N} / \mathrm{SO}_{N}\, \cong\, \left<w\right>. \end{align} $$

The group $G(\mathbb {R}, \delta )$ is preserved by $\mathrm {Out}_{N}(G)$ for any pure real form $\delta $ ([A2, Section 9.1]). Let $\widetilde {\Pi }(G(\mathbb R,\delta ))$ be the set of orbits of $\mathrm {Out}_N(G)$ in $\Pi (G(\mathbb R,\delta ))$ . By definition, $\widetilde {\Pi }(G(\mathbb R,\delta ))={\Pi }(G(\mathbb R,\delta ))$ , unless G is an even rank special orthogonal group.

Suppose G is an even rank special orthogonal group. Then $\widetilde {\Pi }(G(\mathbb R,\delta ))$ contains orbits of cardinality one or two. The restriction of the space of distributions $K \Pi (G(\mathbb {R}, \delta ))$ to the $\mathrm {Out}_{N}(G)$ -invariant subspace of $C_{c}^{\infty }(G(\mathbb {R}, \delta ))$ is the module of $\mathrm {Out}_{N}(G)$ -coinvariants

(7) $$ \begin{align} K\Pi(G,\mathbb{R},\delta)/ (1-w) \cdot K\Pi(G,\mathbb{R},\delta). \end{align} $$

Here, w acts on a distribution character $\pi \in \Pi (G(\mathbb {R}, \delta ))$ by the transpose action

$$ \begin{align*}(w \cdot \pi)(f) = \pi\left(\mathrm{Int}(w) \cdot f \right), \quad f \in C_{c}^{\infty}(G(\mathbb{R},\delta)),\end{align*} $$

where

$$ \begin{align*}\mathrm{Int}(w)\cdot f(x) = f(\mathrm{Int}(w)(x)) = f(wxw^{-1}), \quad x \in G(\mathbb{R},\delta).\end{align*} $$

The actual representation corresponding to the distribution $w \cdot \pi $ is the usual w-conjugate representation. The module (7) is isomorphic to $K \widetilde {\Pi }(G(\mathbb R,\delta ))$ , the $\mathbb {Z}$ -module generated by the $\mathrm {Out}_{N}(G)$ -orbits $\widetilde {\Pi }(G(\mathbb R,\delta ))$ . We identify these two modules. Actually we are more interested in the complex vector space

$$ \begin{align*}K_{\mathbb{C}} \widetilde{\Pi}(G(\mathbb R,\delta)) = \mathbb{C} \otimes_{\mathbb{Z}} K \widetilde{\Pi}(G(\mathbb R,\delta)).\end{align*} $$

Generally, we shall abbreviate the $\mathbb {C}$ -tensor product of any of our $\mathbb {Z}$ -modules with $K_{\mathbb {C}}$ . There are isomorphisms between the vector spaces

$$ \begin{align*} K_{\mathbb{C}} \Pi(G(\mathbb{R},\delta))_{|C_{c}^{\infty}(G(\mathbb{R},\delta))^{w}} & \cong K_{\mathbb{C}}\widetilde{\Pi}(G(\mathbb{R},\delta)) \\ & \cong K_{\mathbb{C}} \Pi(G(\mathbb{R},\delta)/ (1-w) \cdot K_{\mathbb{C}} \Pi(G(\mathbb{R}, \delta))\\ & \cong (1/2) (1+w) \cdot K_{\mathbb{C}} \Pi (G(\mathbb{R},\delta)), \end{align*} $$

and we identify all of them with $K_{\mathbb {C}}\widetilde {\Pi }(G(\mathbb {R},\delta ))$ . Given a distribution $\pi \in \Pi (G(\mathbb {R},\delta )),$ we define

(8) $$ \begin{align} \mathrm{Out}_{N}(G)\cdot \pi \in K_{\mathbb{C}} \widetilde{\Pi}(G(\mathbb{R},\delta)) \end{align} $$

to mean any of the following equivalent operations

• • Restriction of $\pi $ to the $\mathrm {Out}_{N}(G)$ -invariant subspace $C_{c}^{\infty }(G(\mathbb {R},\delta ))^{w}$ ,

• • The $\mathrm {Out}_{N}(G)$ -orbit of $\pi $ ,

• • $(1/2) (1+w) \cdot \pi $ .

We extend (8) to $K_{\mathbb {C}} \Pi (G(\mathbb {R},\delta ))$ linearly. The final interpretation of (8) was used in [AAM]. We prefer to use the first two here, in line with [A2]. When G is not an even rank special orthogonal group then $\mathrm {Out}_{N}(G)$ is trivial and we define (8) to be $\pi $ .

Arthur defines the virtual character $\eta ^{\mathrm {Ar}}_{\psi _G}(\delta _q)$ as the solution of the twisted endoscopic transfer identity

(9) $$ \begin{align} \mathrm{Tr}_{\vartheta}(\pi^{\thicksim}_{\psi})=\mathrm{Trans}_{G(\mathbb R,\delta_q)}^{\text{GL}_N(\mathbb{R}) \rtimes \vartheta} ( \eta^{\mathrm{Ar}}_{\psi_{G}}(\delta_q)), \end{align} $$

with

(10) $$ \begin{align} \mathrm{Trans}_{G(\mathbb R,\delta_q)}^{\text{GL}_N \rtimes \vartheta}: K_{\mathbb{C}} \Pi(G(\mathbb R,\delta_q))^{st}\longrightarrow K_{\mathbb{C}} \Pi(\text{GL}_N(\mathbb R)\rtimes \vartheta), \end{align} $$

being the endoscopic transfer map studied in [M]. The transfer map $\mathrm {Trans}_{G(\mathbb R,\delta _q)}^{\text {GL}_N \rtimes \vartheta }$ is defined on the space of stable distributions $K\Pi (G(\mathbb R,\delta _q))^{st}$ of $G(\mathbb {R},\delta _q)$ . It is dual to Shelstad’s geometric transfer map $C_{c}^{\infty }(\text {GL}_{N}(\mathbb {R} \rtimes \vartheta )) \rightarrow C_{c}^{\infty }(G(\mathbb {R},\delta ))$ [S3]. Since the image of geometric transfer lies in the $\mathrm {Out}_{N}(G)$ -invariant subspace of $C_{c}^{\infty }(G(\mathbb {R}, \delta ))$ ([A2, Corollary 2.1.2]), the endoscopic transfer map (10) passes to the $\mathrm {Out}_{N}(G)$ -coinvariants of $K_{\mathbb {C}}\Pi (G(\mathbb R, \delta _q))^{st}$ . As earlier, we may regard this space as the image of $\frac {1}{2}(1+w)$ on $K_{\mathbb {C}} \Pi (G(\mathbb R,\delta _q))^{st}$ , or as a subspace of $K_{\mathbb {C}} \tilde {\Pi }(G(\mathbb {R}, \delta _{q}))$ . By [A2, p. 12] and [AMR, Proposition 9.1], the map $\mathrm {Trans}_{G(\mathbb R,\delta _q)}^{\text {GL}_N \rtimes \vartheta }$ is injective on this space. Consequently, equation (9) characterizes $\eta ^{\mathrm {Ar}}_{\psi _G}(\delta _q)$ uniquely, and the Arthur packet $\widetilde {\Pi }_{\psi _G}^{\mathrm {Ar}}(\delta _q)$ is defined as the $\mathrm {Out}_N(G)$ -orbits of irreducible representations occurring in $\eta ^{\mathrm {Ar}}_{\psi _G}(\delta _q)$ .

As things stand, the distribution $\eta ^{\mathrm {Ar}}_{\psi _G}(\delta _q)$ is a complex linear combination of irreducible unitary representations. It turns out that it is in fact a character value of a unitary representation of

$$ \begin{align*}G(\mathbb{R}, \delta_{q}) \times A_{\psi_{G}}.\end{align*} $$

To see this, we consider the family of endoscopic groups in the image of $\psi _{G}^{}$ ([A2, pp. 35–37]). Each endoscopic group in this family corresponds to an element $\bar {s} \in A_{\psi _{G}}$ as follows. For each $\bar {s}\in A_{\psi _G^{}}$ choose a semisimple representative $s \in {^\vee }G$ , and let $H(\mathbb {R})$ be a quasisplit group whose dual group ${^\vee }H$ is the identity component of the centralizer in ${^\vee }G$ of s. Then there is a natural embedding ([A2, Equation (1.4.10)])

(11) $$ \begin{align} \epsilon:{^{\vee}}H^{\Gamma}\ \hookrightarrow\ {^{\vee}}G^{\Gamma}, \end{align} $$

and an Arthur parameter $\psi _{H}$ for H such that

$$ \begin{align*}\psi_{G}^{} = \epsilon \circ \psi_{H}^{}.\end{align*} $$

The group H is a product of symplectic and special orthogonal groups. For a more precise description see (38) below. Using Equation (9), Arthur attaches to each factor in this product a distribution and defines the virtual character $\eta _{\psi _{H}}^{\mathrm {Ar}}(H(\mathbb R))$ of H to be their product (Equation (39)). For each $\bar {s}\in A_{\psi _G}$ , the virtual character $\eta _{\psi _G}^{\mathrm {Ar}}(\delta _q)( \bar {s})$ of $G(\mathbb R,\delta _q)$ is defined by

(12) $$ \begin{align} \eta_{\psi_G}^{\mathrm{Ar}}(\delta_q)(\overline{s})= \mathrm{Trans}_{H(\mathbb R)}^{G(\mathbb R,\delta_q)}\left(\eta_{\psi_{H}}^{\mathrm{Ar}}(H(\mathbb R)\right) \end{align} $$

([A2, Equation (2.2.6)]), where

(13) $$ \begin{align} \mathrm{Trans}_{H(\mathbb R)}^{G(\mathbb R,\delta_q)}:\, K_{\mathbb{C}} \Pi(H(\mathbb R))^{st}\, \longrightarrow\, K_{\mathbb{C}} \Pi(G(\mathbb R,\delta_{q})), \end{align} $$

is the ordinary endoscopic transfer map on the space of stable virtual characters of $H(\mathbb R)$ ([LS], [S1] and [S2]). We remind the reader that (13) is pinned down by a choices of Whittaker data for $H(\mathbb {R})$ and $G(\mathbb {R}, \delta _{q})$ . This defines a map

$$ \begin{align*}\overline{s}\ \longmapsto\ \eta_{\psi_G}^{\mathrm{Ar}}(\delta_{q})(\overline{s}), \quad \bar{s} \in A_{\psi_{G}} \end{align*} $$

whose values are complex linear combinations of irreducible unitary representations of $G(\mathbb R,\delta _q)$ . One can see that $\eta _{\psi _G}^{\mathrm {Ar}}(\delta _q)(\cdot )$ is the character of a finite-length unitary representation of $G(\mathbb R,\delta _q)\times A_{\psi _G^{}}$ in the following manner. In [A2, Equation (7.1.2)], the virtual character $\eta _{\psi _{G}}^{\mathrm {Ar}}(\delta _q)(\bar {s})$ is written as

(14) $$ \begin{align} \eta_{\psi_{G}}^{\mathrm{Ar}}(\delta_q)(\bar{s})\, =\, \sum_{\sigma \in \tilde{\Sigma}_{\psi_{G}}} < s_{\psi_{G}}\bar{s}, \sigma> \sigma, \end{align} $$

where $s_{\psi _{G}}$ is given in (2). Here, $\tilde {\Sigma }_{\psi _{G}}$ is a finite set of non-negative integral linear combinations

$$ \begin{align*}\sigma = \sum_{\widetilde{\pi}} m(\sigma, \widetilde{\pi}) \, \widetilde{\pi}\end{align*} $$

of $\mathrm {Out}_{N}(G)$ -orbits of irreducible unitary representations of $ G(\mathbb {R}, \delta _q)$ . As explained in [A2, pp. 385–386], there is an injective map from $\tilde {\Sigma }_{\psi _{G}}$ into the set of characters of $A_{\psi _{G}}$ which are trivial on the centre of ${^\vee }G$ . The injection is denoted by

$$ \begin{align*}\sigma \, \longmapsto\, < \cdot ,\, \sigma>.\end{align*} $$

Following [A2, Proposition 7.4.3 and Equation (7.4.1)], we may rewrite (14) as

(15) $$ \begin{align} \eta_{\psi_{G}}^{\mathrm{Ar}}(\delta_{q})(\bar{s}) = \sum_{\widetilde{\pi} \in \widetilde{\Pi}_{\psi_{G}}^{\mathrm{Ar}}(\delta_{q})} \left( \sum_{\sigma \in \tilde{\Sigma}_{\psi_{G}}} m(\sigma, \widetilde{\pi}) \, <s_{\psi_{G}} \bar{s}, \sigma>\right) \widetilde{\pi}. \end{align} $$

By defining a finite-dimensional representation

(16) $$ \begin{align} \tau_{\psi_{G}}^{\mathrm{Ar}} ( \widetilde{\pi}) = \bigoplus_{\sigma \in \tilde{\Sigma}_{\psi_{G}}} m(\sigma, \widetilde{\pi}) \, <\cdot, \sigma>, \end{align} $$

Equation (15) becomes

(17) $$ \begin{align} \eta_{\psi_G^{}}^{\mathrm{Ar}}(\delta_q)(\bar{s})= \sum_{\widetilde{\pi} \in \widetilde{\Pi}_{\psi_{G}^{}}^{\mathrm{Ar}}(G(\mathbb R,\delta_q))} \mathrm{Tr}\left(\tau_{\psi_{G}^{}}^{\mathrm{Ar}}( \widetilde{\pi} )(s_{\psi_{G}} \bar{s})\right)\, \widetilde{\pi}. \end{align} $$

In particular, for $\bar {s}=1$ we see that

$$ \begin{align*}\eta_{\psi_G}^{\mathrm{Ar}}(\delta_q)\, =\, \eta_{\psi_G}^{\mathrm{Ar}}(\delta_q)(1)\, =\, \sum_{ \widetilde{\pi} \in \widetilde{\Pi}_{\psi_{G}^{}}^{\mathrm{Ar}}(G(\mathbb R,\delta_q))} \mathrm{Tr}\left(\tau_{\psi_{G}^{}}^{\mathrm{Ar}}( \widetilde{\pi})(s_{\psi_{G}})\right)\, \widetilde{\pi}.\end{align*} $$

Identity (17) is called the ordinary endoscopic transfer identity, and is one of the main points in Arthur’s local conjectures.

In [MR2] and [MR1], Moeglin and Renard prove Arthur’s local conjectures for all pure real forms $\delta $ of G. Following Arthur’s approach, for each $\bar {s}\in A_{\psi _G^{}}$ they define $\eta _{\psi _G}^{\mathrm {Ar}}(\delta )(\overline {s})$ as in (12). The only difference is that Kottwitz’s sign $e(\delta )$ appears in their definition ([MR1, Section 2.1])

(18) $$ \begin{align} \eta_{\psi_G}^{\mathrm{Ar}}(\delta)(\bar{s})\ =\ e(\delta)\, \mathrm{Trans}_{H(\mathbb R)}^{G(\mathbb R,\delta)} \left( \eta_{\psi_{H}}^{\mathrm{Ar}}(H(\mathbb{R}) \right). \end{align} $$

In [MR2, Theorem 9.3] they prove the ordinary endoscopic transfer identity (17) with $\delta _q$ replaced by $\delta $ . Thus, $\eta _{\psi _{G}}^{\mathrm {Ar}}(\delta _{q})$ and the definition of Arthur packets are extended to any pure real form $\delta $ of G. We point out that, similar to what happens in the quasisplit case, the distribution $\eta _{\psi _G}^{\mathrm {Ar}}(\delta )(\overline {s})$ is only defined up to the action of $\mathrm {Out}_{N}(G)$ . We also notice that for $\bar {s}=1$ the distribution

$$ \begin{align*}\eta_{\psi_G}^{\mathrm{Ar}}(\delta)\, :=\, \eta_{\psi_G}^{\mathrm{Ar}}(\delta)(1)\, =\, \mathrm{Trans}_{G(\mathbb R,\delta_{q})}^{G(\mathbb R,\delta)} \left( \eta_{\psi_{G}}^{\mathrm{Ar}}(\delta_{q}) \right)\end{align*} $$

is stable, since $\mathrm {Trans}_{G(\mathbb R,\delta _{q})}^{G(\mathbb R,\delta )}$ carries stable virtual characters to stable virtual characters. The Arthur packet $\widetilde {\Pi }_{\psi _G^{}}^{\mathrm {Ar}}(\delta )$ is the set of $\mathrm {Out}_N(G)$ -orbits of irreducible representations occurring in $\eta _{\psi _G}^{\mathrm {Ar}}(\delta )$ . In [MR2], Moeglin and Renard use cohomological and parabolic induction to actually give a description of the representations in each Arthur packet.

3.2 The approach of Adams, Barbasch, and Vogan

In this section, we give a quick review of Adams, Barbasch, and Vogan’s solution $\eta _{\psi _{G}}^{\mathrm {ABV}}(\delta )$ to Arthur’s conjectures. The results in [ABV] apply to any complex connected reductive group G. However, we continue by assuming that G is symplectic or special orthogonal, or any finite product of these groups. We continue by writing $\delta $ for any pure real form of G.

Let $P({^\vee }G^{\Gamma })$ be the set of quasiadmissible homomorphisms $\varphi :W_{\mathbb {R}} \rightarrow {^\vee }G^{\Gamma }$ of G ([ABV, Definition 5.2]), i.e., the set of L-homomorphisms for a quasisplit form of G. Associated with any $\varphi \in P({^\vee }G^{\Gamma })$ , there is an infinitesimal character ${^\vee }\mathcal {O} \subset {^\vee }\mathfrak {g}$ ([ABV, Proposition 5.6]). Let

(19) $$ \begin{align} P({^\vee} \mathcal{O}, {^\vee}G^{\Gamma} ), \end{align} $$

be the subset of $P({^\vee }G^{\Gamma })$ consisting of homomorphisms with infinitesimal character ${^\vee }\mathcal {O}$ . The group ${^\vee }G$ acts on $P({^\vee } \mathcal {O},{^\vee }G^{\Gamma })$ by conjugation. It is to the set of ${^\vee }G$ -orbits

(20) $$ \begin{align} P({^\vee}\mathcal{O}, {^\vee}G^{\Gamma})/ {^\vee}G \end{align} $$

that the Langlands correspondence, in its original form [L], assigns L-packets of representations of $G(\mathbb R,\delta )$

$$ \begin{align*}\varphi\, \longleftrightarrow\, \Pi_\varphi(\delta) \end{align*} $$

( $\Pi _\varphi (\delta )$ is empty when $\varphi $ is not relevant to $\delta $ ). A great innovation of [ABV], which is central to their construction of Arthur packets, is to describe the L-packets of $G(\mathbb {R},\delta )$ in terms of an appropriate geometry on ${}^{\vee }G^{\Gamma }$ . This is done through the introduction of the complex variety $X({^\vee }\mathcal {O}, {^\vee }G^{\Gamma })$ of geometric parameters, which lies between (19) and (20) ([ABV, Definition 6.9]). It may be regarded as a set of equivalence classes in $P({^\vee }\mathcal {O}, {^\vee }G^{\Gamma })$ upon which ${^\vee }G$ still acts by conjugation with finitely many orbits ([AAM, Section 2.2], [ABV, Proposition 6.16]). The map to equivalence classes

$$ \begin{align*}P({^\vee}\mathcal{O}, {^\vee}G^{\Gamma}) \longrightarrow X({^\vee}\mathcal{O}, {^\vee}G^{\Gamma})\end{align*} $$

passes to a bijection at the level of ${^\vee }G$ -orbits ([ABV, Proposition 6.17]). Thus, the local Langlands correspondence may supplemented by

(21) $$ \begin{align} S_\varphi \longleftrightarrow\varphi\longleftrightarrow \Pi_{\varphi}(\delta), \end{align} $$

where $S_{\varphi } \subset X\left ({}^\vee \mathcal{O},{}^\vee G^{\Gamma} \right )$ is the ${^\vee }G$ -orbit corresponding to the ${^\vee }G$ -orbit of $\varphi $ . One motivation for introducing $ X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )$ is that the closure relations between ${}^{\vee }G$ -orbits imply relationships between the representations of corresponding L-packets. Moreover, in [ABV, Theorem 10.4] the local Langlands correspondence is refined from a bijection between L-packets and L-parameters, to a bijection between (equivalence classes of) irreducible representations and what they refer to as complete geometric parameters. More precisely, the authors supplement each ${}^{\vee }G$ -orbit $S\subset X({^\vee }\mathcal {O},{^\vee }G^{\Gamma })$ with an irreducible ${}^{\vee }G$ -equivariant local system $\mathcal {V}$ of vector spaces on S, and define the set of (pure) complete geometric parameters

$$ \begin{align*}\Xi({^\vee}\mathcal{O},{^\vee}G^{\Gamma})\end{align*} $$

for $X({^\vee }\mathcal {O}, {^\vee }G^{\Gamma })$ as the set of pairs $\xi =(S,\mathcal {V})$ [ABV, Definition 7.6]. The set of ${^\vee }G$ -equivariant local systems on S are in bijection with the representations in an extended L-packet

$$ \begin{align*}\Pi_{S}(G/\mathbb R) \supset \Pi_{\varphi}(\delta),\end{align*} $$

where $S = S_{\varphi }$ as in (21). More symbolically, there is a bijection

(22) $$ \begin{align} \pi(\xi)\longleftrightarrow \xi=(S,\mathcal{V}), \end{align} $$

where $\pi (\xi )$ runs over all irreducible representations with infinitesimal character ${^\vee }\mathcal {O}$ of pure real forms of G including $G(\mathbb {R},\delta )$ . As usual, the bijection is pinned down by a choice of Whittaker datum.

A striking feature of (22) is that the pair $\xi =(S, \mathcal {V})$ determines a perverse sheaf $P(\xi )$ on $X({^\vee }\mathcal {O},{^\vee }G^{\Gamma })$ . Let $\mathcal {P}(X({^\vee }\mathcal {O},{^\vee }G^{\Gamma }))$ be the category of $\, {^\vee }G$ -equivariant perverse sheaves of complex vector spaces on $X({^\vee }\mathcal {O},{^\vee }G^{\Gamma })$ ([BL, Section 5]). This is an abelian category and, as explained in [ABV, Equation (7.10)(d)], its simple objects are parameterized by the set of complete geometric parameters

$$ \begin{align*}\xi\longmapsto P(\xi),\quad \xi \in \Xi({^\vee}\mathcal{O}, {^\vee}G^{\Gamma}).\end{align*} $$

Hence, we may extend (22) to a one-to-one correspondence

(23) $$ \begin{align} \pi(\xi)\longleftrightarrow \xi\longleftrightarrow P(\xi). \end{align} $$

We write $K\Pi ({^\vee }\mathcal {O},G/\mathbb R)$ and $KX({^\vee }\mathcal {O},{^\vee }G^{\Gamma })$ for $\mathbb {Z}$ -modules generated by $\Pi ({^\vee }\mathcal {O},G/\mathbb R)$ and $\mathcal {P}(X({^\vee }\mathcal {O},{^\vee }G^{\Gamma }))$ , respectively. These two groups are Grothendieck groups of finite-length objects and have respective bases

$$ \begin{align*}\{\pi(\xi) : \xi \in \Xi({^\vee}\mathcal{O},{^\vee}G^{\Gamma})\} \quad\text{and}\quad\{P(\xi) : \xi \in \Xi({^\vee}\mathcal{O},{^\vee}G^{\Gamma})\}.\end{align*} $$

Correspondence (23) expresses a duality between irreducible representations and irreducible perverse sheaves, in the form a perfect pairing

(24) $$ \begin{align} \langle \cdot, \cdot \rangle_{G} : K \Pi({^\vee}\mathcal{O},G/\mathbb R) \times K X\left({}^\vee\mathcal O,{}^\vee G^{\Gamma}\right) \longrightarrow \mathbb{Z}. \end{align} $$

The pairing is defined in [ABV, Definition 15.11] using the alternative bases of standard representations and constructible sheaves. It is a deep result ([ABV, Theorem 15.12]), that this pairing satisfies

$$ \begin{align*}\langle \pi(\xi), P(\xi') \rangle_{G} = e(\xi) \, (-1)^{dS_\xi} \, \delta_{\xi, \xi'}, \quad \xi,\xi' \in \Xi({^\vee}\mathcal{O},{^\vee} G^{\Gamma}),\end{align*} $$

with $dS_\xi $ the dimension of $S_\xi $ , and $\delta _{\xi , \xi '}$ the Kronecker delta. The sign $e(\xi )$ is the Kottwitz sign of the real form upon which $\pi (\xi )$ is defined.

Using pairing (24), we may regard virtual characters as $\mathbb {Z}$ -valued linear functionals on $K X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )$ . The theory of microlocal geometry provides a family of linear functionals

$$ \begin{align*}\chi^{\mathrm{mic}}_{S} : K X\left({}^\vee\mathcal O,{}^\vee G^{\Gamma}\right) \longrightarrow \mathbb{Z} \end{align*} $$

parameterized by the ${^\vee }G$ -orbits $S \subset X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )$ . The linear functional $\chi _{S}^{\mathrm {mic}}$ is called the microlocal multiplicity along S. The microlocal multiplicities appear in the theory of characteristic cycles ([ABV, Chapter 19], [BGK+]), and are associated with ${^\vee }G$ -equivariant local systems on a conormal bundle of $X({^\vee }\mathcal {O},{^\vee }G^{\Gamma })$ ([ABV, Section 24], [GM, Appendix 6.A]). The virtual characters associated by the pairing to the linear functionals $\chi ^{\mathrm {mic}}_S$ are stable ([ABV, Corollary 1.26, Theorems 1.29 and 1.31]).

The microlocal multiplicity is the last ingredient needed for the definition of the Arthur packets in [ABV, Theorem 26.25]. Suppose $\psi _G$ is an Arthur parameter of G, and let $\varphi _{\psi _G^{}}$ be the L-parameter defined by $\psi _G^{}$ through (5). Let ${^\vee }\mathcal {O}$ be the infinitesimal character of $\varphi _{\psi _{G}}$ . Then by bijection (21), the parameter $\varphi _{\psi _G^{}}$ corresponds to a ${}^{\vee }G$ -orbit

(25) $$ \begin{align} \varphi_{\psi_G^{}} \longleftrightarrow S_{\psi_{G}} \end{align} $$

in $ X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )$ . The authors define $\eta ^{\mathrm {mic}}_{\psi _G}$ to be the virtual character associated with $\chi _{S_{\psi _{G}}}^{\mathrm {mic}}$ using pairing (24). That is, $\eta ^{\mathrm {mic}}_{\psi _G}$ is the unique virtual character satisfying

$$ \begin{align*}\langle \eta^{\mathrm{mic}}_{\psi_{G}}, \mu \rangle_{G} = \chi_{S_{\psi_{G}}}^{\mathrm{mic}}(\mu), \quad \mu \in K X\left({}^\vee\mathcal O,{}^\vee G^{\Gamma}\right).\end{align*} $$

The stable virtual character $\eta ^{\mathrm {mic}}_{\psi _{G}}$ can be expressed as a linear combination of irreducible representations of pure real forms of G which include $\delta $ . Let

(26) $$ \begin{align} \eta^{\mathrm{ABV}}_{\psi_{G}}(\delta) \end{align} $$

be the summand of $\eta ^{\mathrm {mic}}_{\psi _{G}}$ coming from the representations in $\Pi ({}^{\vee }\mathcal {O},G(\mathbb {R},\delta ))$ . The Arthur packet $\Pi _{\psi _G}^{\mathrm {ABV}}(\delta )$ is then defined as the set of irreducible representations occurring in $\eta ^{\mathrm {ABV}}_{\psi _{G}}(\delta )$ . More explicitly,

(27) $$ \begin{align} \Pi^{\mathrm{ABV}}_{\psi_{G}}(\delta) \ =\ \{ \pi(\xi) : \xi \in \Xi({^\vee}\mathcal{O}, {^\vee}G^{\Gamma}), \chi^{\mathrm{mic}}_{S_{\psi_{G}}}(P(\xi)) \neq 0, \pi(\xi)\in \Pi(G(\mathbb{R},\delta)) \}. \end{align} $$

It is shown in [ABV, Corollary 6.21] that $\eta _{\psi _{G}}^{\mathrm {ABV}}(\delta )$ satisfies the ordinary spectral transfer identities (17). In proving (17), the authors introduce sheaf-theoretic versions of the identities and assert that they are equivalent to their analytic counterparts. They begin by giving a sheaf-theoretic version of the endoscopic transfer map (13), denoted by Lift. It is defined as follows. As in the previous section, for each $\bar {s}\in A_{\psi _G^{}}$ choose a semisimple representative $s \in {^\vee }G$ , and let $H(\mathbb {R})$ be its (quasisplit) endoscopic group. Then inclusion (18) induces an embedding of varieties $ \epsilon :X({}^{\vee }\mathcal {O},{^\vee }H^{\Gamma })\hookrightarrow X({}^{\vee }\mathcal {O},{^\vee }G^{\Gamma })$ ([ABV, Equation (26.17)(e)]). The inverse image functor of $\epsilon $ ,

$$ \begin{align*}\epsilon^{\ast}: KX({}^{\vee}\mathcal{O},{^\vee}G^{\Gamma}) \longrightarrow KX({}^{\vee}\mathcal{O},{^\vee}H^{\Gamma}),\end{align*} $$

and the pairing (24), allow one to define a map

$$ \begin{align*}{\epsilon_{\ast}: K_{\mathbb{C}} \Pi({}^{\vee}\mathcal{O},H/\mathbb R) \longrightarrow K_{\mathbb{C}} \Pi({}^{\vee}\mathcal{O}, G/\mathbb R)}\end{align*} $$

([ABV, Definition 26.18]). The Lift map is defined from $\epsilon _{\ast }$ in two steps. First, set $\text {Lift}_0$ to be the restriction of $\epsilon _{\ast }$ to the subspace $K_{\mathbb {C}}\Pi ({}^{\vee }\mathcal {O},H(\mathbb R))^{st}\subset K_{\mathbb {C}} \Pi ({}^{\vee }\mathcal {O},H/\mathbb {R})$ generated by the stable virtual characters in the Grothendieck group $K\Pi ({}^{\vee }\mathcal {O},{H}(\mathbb {R}))$ of representations of $H(\mathbb {R})$ . The Lift map

$$ \begin{align*}\text{Lift}_{H(\mathbb R)}^{G(\mathbb R, \delta)}:K\Pi({}^{\vee}\mathcal{O},H(\mathbb R))^{st}\longrightarrow K\Pi({}^{\vee}\mathcal{O},G(\mathbb R,\delta))\end{align*} $$

is then the projection of $\text {Lift}_0$ to the Grothendieck group of representations of $G(\mathbb R,\delta )$ ([ABV, p. 289]).

It is argued on [ABV] that for the quasisplit real form $G(\mathbb {R},\delta _{q})$ the sheaf-theoretic and analytic endoscopic lifting maps are the same,

$$ \begin{align*}\mathrm{Lift}_{H(\mathbb R)}^{G(\mathbb R,\delta_q)}\, =\, \mathrm{Trans}_{H(\mathbb R)}^{G(\mathbb R,\delta_q)}.\end{align*} $$

In the next section, we will need this identity for any pure real form $G(\mathbb {R},\delta )$ , not just the quasisplit form $G(\mathbb {R}, \delta _{q})$ . The identity for pure real forms is a consequence of [ARM1, Theorem 1.1] which applies to any pure real form of an arbitrary group G (including the quasisplit form). The details for unitary groups are given in [ARM2, Section 11]. The proof for symplectic and special orthogonal groups follows [ARM2, Section 11] mutatis mutandis. We therefore have the following result.

Lemma 3.1 For all pure real forms $\delta $ of G

$$ \begin{align*} \mathrm{Lift}_{H(\mathbb R)}^{G(\mathbb R,\delta)}=\mathrm{Trans}_{H(\mathbb R)}^{G(\mathbb R,\delta)}. \end{align*} $$

Let us go back to the proof of the ordinary endoscopic transfer identities in [ABV]. The next step is to introduce a sheaf-theoretic version of the representation $\tau _{\psi _{G}}^{\mathrm {Ar}}(\tilde {\pi })$ given in (16). This is done using deep theorems in microlocal analysis which allow one to express $\chi ^{\mathrm {mic}}_{S_{\psi _{G}}}$ as the rank of a local system on a conormal bundle of $ X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )$ . More precisely, it is proven in [ABV, Theorem 24.8] that to each perverse sheaf P on $ X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )$ there is attached an ${}^{\vee }G$ -equivariant local system

(28) $$ \begin{align} Q^{\mathrm{mic}}(P) \end{align} $$

of complex vector spaces on a subset of the conormal bundle to the ${}^{\vee }G$ -action $T^{\ast }_{{}^{\vee }G}\left ( X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )\right )$ ([ABV, Equation (19.1)(d)]). Furthermore, for each $^{\vee }G$ -orbit $S \subset X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )$ the rank of $Q^{\mathrm {mic}}(P)$ at any non-degenerate point $(y,\nu )$ of $T^{\ast }_{S}\left ( X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )\right )$ is equal to the microlocal multiplicity of P along S, i.e.,

$$ \begin{align*}\dim \left(Q^{\mathrm{mic}}(P)_{y,\nu}\right)\, =\, \chi^{\mathrm{mic}}_{S}(P). \end{align*} $$

As explained in [ABV, Corollary 24.9], the restriction of $Q^{\mathrm {mic}}(P)$ to $T^{\ast }_{S}( X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right ))$ may be represented by a finite-dimensional representation $\tau ^{\mathrm {mic}}_{S}(P)$ of the micro-component group ([ABV, Definition 24.7])

$$ \begin{align*}A_S^{\mathrm{mic}}={}^{\vee}G_{y,\nu}/\left({}^{\vee}G_{y,\nu}\right)_0,\quad (y,\nu)\in T^{\ast}_{S}\left( X\left({}^\vee\mathcal O,{}^\vee G^{\Gamma}\right)\right), \end{align*} $$

verifying

(29) $$ \begin{align} \dim \left(\tau_{\psi_G}^{\mathrm{mic}}(P)\right)\ =\ \chi_{S_{\psi_G}}^{\mathrm{mic}}(P). \end{align} $$

We point out that by [ABV, Lemma 24.3], $A_S^{\mathrm {mic}}$ is independent of the choice of $(y,\nu )$ . Furthermore, when $S = S_{\psi _{G}}$ as in (25)

$$ \begin{align*}A_{\psi_G}=A_{S_{\psi_G}}^{\mathrm{mic}}, \end{align*} $$

([ABV, Definition 24.7]). This permits us to define the representation

(30) $$ \begin{align} \tau_{\psi_{G}}^{\mathrm{ABV}}(\pi(\xi)) = \tau_{S_{\psi_G}}^{\mathrm{mic}}(P(\xi)), \quad \xi \in \Xi({}^{\vee}\mathcal{O},{}^{\vee}G^{\Gamma}). \end{align} $$

With this notation, Equation (29) and pairing (24) combine to produce the following decomposition of the virtual character $\eta ^{\mathrm {ABV}}_{\psi _{G}}(\delta )$ of (26)

$$ \begin{align*} \eta_{\psi_{G}}^{\mathrm{ABV}}(\delta) = \sum_{\pi(\xi) \in \Pi_{\psi_{G}}^{\mathrm{ABV}}(\delta)} (-1)^{dS_\xi - dS_{\psi_{G}}} \ \dim\left(\tau^{\mathrm{ABV}}_{\psi_{G}}(\pi(\xi)) \right) \, \pi(\xi). \end{align*} $$

Here, $dS=\dim (S)$ for each ${}^{\vee }G$ -orbit S. In addition, for each $\bar {s}\in A_{\psi _{G}^{}}$ there is a virtual character

(31) $$ \begin{align} \eta_{\psi_{G}}^{\mathrm{ABV}}(\delta)(\bar{s})\ =\ \sum_{\pi(\xi) \in \Pi_{\psi_{G}}^{\mathrm{ABV}}(\delta)} (-1)^{dS_\xi - dS_{\psi_{G}}} \ \mathrm{Tr}\left(\tau^{\mathrm{ABV}}_{\psi_{G}^{}}(\pi(\xi))(\bar{s}) \right) \, \pi(\xi), \end{align} $$

which resembles $\eta _{\psi _{G}}^{\mathrm {Ar}}(\delta _{q})(\bar {s})$ in (17). The ordinary endoscopic transfer identity ([ABV, Theorem 26.25]*) takes the form

(32) $$ \begin{align} \mathrm{Lift}_{H(\mathbb R)}^{G(\mathbb R,\delta)}\left(\eta_{\psi_{H}}^{\mathrm{ABV}}(H(\mathbb R))\right) = \eta_{\psi_{G}}^{\mathrm{ABV}}(\delta)(\bar{s}). \end{align} $$

It is natural to seek a sheaf-theoretic analog of the twisted endoscopic transfer identity (9) as well. This is proven in [AAM]. Indeed, following [ABV], a sheaf-theoretic version of the theory of twisted endoscopy is introduced in [CM]. This is used in [AAM] to give a sheaf-theoretic version of the twisted transfer map (10)

$$ \begin{align*}\text{Lift}_{G(\mathbb R,\delta_q)}^{\text{GL}_N(\mathbb R)\rtimes \vartheta}: K_{\mathbb{C}} \Pi(G(\mathbb R,\delta_q))^{st} \longrightarrow K_{\mathbb{C}} \Pi(\text{GL}_N(\mathbb R)\rtimes \vartheta).\end{align*} $$

The twisted endoscopic transfer identity takes the form

$$ \begin{align*}\mathrm{Lift}_{{G(\mathbb R,\delta_q)}}^{\text{GL}_N(\mathbb R)\rtimes \vartheta}\left(\eta_{\psi_{G}}^{\mathrm{ABV}}(\delta_q)\right) = \mathrm{Tr}_{\vartheta} \left( \eta_{\psi}^{\mathrm{ABV}}(\delta_q)^{{\thicksim}} \right), \end{align*} $$

where $\psi $ is as in (4), and $\eta _{\psi }^{\mathrm {ABV}}(\delta _q)^{{\thicksim }}$ is the virtual character obtained from $\eta _{\psi }^{\mathrm {ABV}}(\delta _q)$ after canonically extending each $\pi \in \Pi _{\psi }^{\mathrm {ABV}}(\text {GL}_N(\mathbb R))$ to a representation of $\text {GL}_N(\mathbb R)\times \left <\vartheta \right>$ .

We end this section by assuming that $G= \mathrm {SO}_{N}$ , N even, and describing the effect of the action of $\mathrm {Out}_N(G)$ on the objects involved in the definition of the Arthur packets in [ABV, Equation (134)]. As in the previous section, $\mathrm {Out}_N(G)\cong \left <w\right>$ with $w \in \mathrm {O}_{N}$ as in (6). The outer automorphism $\mathrm {Int}(w)$ on G induces a natural bijection

(33) $$ \begin{align} \mathrm{Int}({w}):\, X\left({}^\vee\mathcal O, {^\vee}G^{\Gamma}\right) \, \longrightarrow\, X\left({w}\cdot {}^\vee\mathcal O, {^\vee}G^{\Gamma}\right), \end{align} $$

which by [ABV, Proposition 7.15(a)], induces a bijection of complete geometric parameters

$$ \begin{align*}w^{\ast}: \Xi\left({w}\cdot {}^\vee\mathcal O, {^\vee}G^{\Gamma}\right)\, \longrightarrow\, \Xi\left({}^\vee\mathcal O, {^\vee}G^{\Gamma}\right). \end{align*} $$

Moreover, by [ABV, Proposition 7.15(b)] the inverse image functor attached to (33)

$$ \begin{align*}\mathrm{Int}({w})^{\ast}:\mathcal{P}\left(X\left({w}\cdot {}^\vee\mathcal O, {^\vee}G^{\Gamma}\right)\right)\, \longrightarrow\, \mathcal{P}\left(X\left({}^\vee\mathcal O, {^\vee}G^{\Gamma}\right)\right) \end{align*} $$

is a fully faithful exact functor, satisfying

(34) $$ \begin{align} \mathrm{Int}({w})^{\ast}(P(\xi))=P(w^{\ast}\xi). \end{align} $$

Two more maps are induced by (33). The differential $\mathrm {Ad}(w)$ of (33) defines a homeomorphism between the tangent bundles

$$ \begin{align*}\mathrm{Ad}({w}) : T \left( X\left({}^\vee\mathcal O, {^\vee}G^{\Gamma}\right) \right) \, \longrightarrow \, T \left( X\left({w}\cdot {}^\vee\mathcal O,{^\vee}G^{\Gamma}\right) \right),\end{align*} $$

and duality gives us a homeomorphism between the corresponding conormal bundles

$$ \begin{align*}\mathrm{Ad}^{*}({w}):T_{{^\vee}{G}}^{\ast} ( X (w\cdot{}^\vee\mathcal O,{^\vee}G^{\Gamma} ) ) \, \longrightarrow \, T_{{}^{\vee}{G}}^{\ast} \left( X\left({}^\vee\mathcal O,{^\vee}G^{\Gamma}\right) \right)\end{align*} $$

such that for any ${}^{\vee }G$ -orbit $S \subset X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )$ , we have

$$ \begin{align*}\mathrm{Ad}^{*}({w}) \left(T_{\mathrm{Int}(w) S}^{\ast} \left( X\left({w}\cdot{}^\vee\mathcal O_{G}, {^\vee}G^{\Gamma}\right)\right) \right)= T_S^{\ast} \left( X\left({}^\vee\mathcal O_{G},{^\vee}G^{\Gamma}\right) \right).\end{align*} $$

We can now describe the effect of $\mathrm {Out}(G)$ on ABV-packets.

Proposition 3.2 Suppose $\psi _G$ is an Arthur parameter of $G = \mathrm {SO}_{N}$ , and N is even. Let w be as in (6). Then

1. (a) $\tau ^{\mathrm {ABV}}_{\mathrm {Int}({w})\,\circ \,\psi _{G}^{}}(\pi (w^{\ast }\xi ))\, =\, \tau ^{\mathrm {ABV}}_{\psi _{G}^{}}(\pi (\xi ))\circ \mathrm {Int}({w}), \quad \xi \in \Xi \left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )$

2. (b)

   $$ \begin{align*} \Pi_{\mathrm{Int}({w})\,\circ\,\psi_{G}^{}}^{\mathrm{ABV}}(\delta)\, &=\, \left\{\pi(w^{\ast}\xi)\ :\ \xi \in \Xi\left({}^\vee\mathcal O,{}^\vee G^{\Gamma}\right),\ \pi(\xi)\in \Pi_{\psi_{G}^{}}^{\mathrm{ABV}}(\delta)\right\}\nonumber\\ \, &=\, \left\{\pi(\xi)\circ \mathrm{Int}({w})\ :\ \xi \in \Xi\left({}^\vee\mathcal O,{}^\vee G^{\Gamma}\right),\ \pi(\xi)\in \Pi_{\psi_{G}^{}}^{\mathrm{ABV}}(\delta)\right\} \end{align*} $$

3. (c) $w \cdot \eta _{\psi _{G}}^{\mathrm {ABV}}(\delta )(\overline {s}) = \eta _{\mathrm {Int}(w) \circ \psi _{G}}^{\mathrm {ABV}} (\delta ) (\mathrm {Int}(w)(\overline {s})).$

Proof Let $\xi \in \Xi ({^\vee }\mathcal {O},{^\vee }G^{\Gamma })$ and let $Q^{\mathrm {mic}}(P(\xi ))$ be the ${}^{\vee }G$ -equivariant local system on a subset of $T^{\ast }_{{}^{\vee }G}\left ( X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )\right )$ as in (28). The stalks of $ Q^{\mathrm {mic}}(P(\xi ))$ at a point $(y,\nu )\in T^{\ast }_{S}\left ( X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )\right )$ are given in [ABV, Equation (24.10)(b)] by the relative hypercohomology

$$ \begin{align*}Q^{\mathrm{mic}}(P(\xi))_{y,\nu}=H^{-\dim S}(J,K;P(\xi)). \end{align*} $$

Here, $(J,K)$ is a pair of compact subspaces of $ X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )$ with $K\subset J$ ([ABV, Equation (24.10)(a)]). Let $h:\, J-K\, \hookrightarrow \, J$ be the inclusion and let $P(\xi )_{|J}$ be the restriction of $P(\xi )$ to J. By definition

$$ \begin{align*}H^{\ast}\left(J,K;P(\xi)\right)= H^{\ast}\left(J;Rh_{!}\, h{^!}P(\xi)_{|J} \right). \end{align*} $$

We wish to transfer these objects using the homeomorphisms induced by $w= w^{-1}$ . Let us write $w\cdot h \cdot w\, =\, \mathrm {Int}(w)\circ h \circ \mathrm {Int}(w)$ for short. Then clearly $w \cdot h \cdot w$ is the inclusion of the w-conjugate of $J-K$ into the w-conjugate of J. Together with (34), we deduce

$$ \begin{align*} & Q^{\mathrm{mic}}(P(w^{\ast}\xi))_{\mathrm{Int}(w)(y),\mathrm{Ad}^{*}(w)(\nu)}\\&\quad =Q^{\mathrm{mic}} \left( \mathrm{Int}(w)^{\ast}P(\xi) \right)_{\mathrm{Int}(w)(y),\mathrm{Ad}^{*}(w)(\nu)}\\&\quad = H^{-\dim ({\mathrm{Int}(w)S})} \left( \mathrm{Int}(w)(J), \mathrm{Int}(w)(K); \mathrm{Int}(w)^{\ast}P(\xi)_{|J} \right)\\&\quad = H^{-\dim ({\mathrm{Int}(w)S})} \left( \mathrm{Int}(w)(J); R(w\cdot h\cdot w)_{!}(w\cdot h\cdot w){^!} \mathrm{Int}(w)^{\ast}P(\xi)_{|J} \right). \end{align*} $$

In order to simplify this expression we make some observations about the functors which appear in it. Since $\mathrm {Int}(w)$ is a homeomorphism satisfying $\left (\mathrm {Int}(w)\right )^2=\mathrm {Id}$ , it follows that

$$ \begin{align*}\mathrm{Int}(w)_{!} = \mathrm{Int}(w)_{*},\quad \mathrm{Int}(w)^{*} = \mathrm{Int}(w)^{!},\quad \ R\mathrm{Int}(w)_{*}\, \mathrm{Int}(w)^{*} = \mathrm{Id}\end{align*} $$

as long as one keeps track of the domains and codomains of the functors. Additionally, since both h and $\mathrm {Int}(w)$ are open embeddings

$$ \begin{align*}R(w\cdot h \cdot w)_{!} = R\mathrm{Int}(w)_!\circ Rh_! \circ R\mathrm{Int}(w)_! =\mathrm{Int}(w)_\ast \circ Rh_! \circ \mathrm{Int}(w)_\ast\end{align*} $$

and

$$ \begin{align*}(w\cdot h \cdot w)^! = \mathrm{Int}(w)^!\circ h^! \circ \mathrm{Int}(w)^! \, =\, \mathrm{Int}(w)^\ast\circ h^! \circ \mathrm{Int}(w)^\ast. \end{align*} $$

([A, Proposition 1.3.7]). Hence,

$$ \begin{align*} & H^{-\dim ({\mathrm{Int}(w)S})} \left(\mathrm{Int}(w)(J); R(w\cdot h\cdot w)_{!}(w\cdot h\cdot w){^!} \mathrm{Int}(w)^{\ast}P(\xi)_{|J} \right)\\ &= H^{-\dim ({\mathrm{Int}(w)S})} \left( \mathrm{Int}(w)(J); R\mathrm{Int}(w)_\ast Rh_!h^!P(\xi)_{|J} \right)\\ &\cong H^{-\dim ({\mathrm{Int}(w)S})}\left(J; Rh_! h^!P(\xi)_{|J} \right)\\ &= H^{-\dim (S)}(J,K; P(\xi))\\ &=Q^{\mathrm{mic}}(P(\xi))_{y,\nu}. \end{align*} $$

This implies that

(35) $$ \begin{align} (\mathrm{Ad}^{*}(w))^{\ast} Q^{\mathrm{mic}}(P(\xi))\, =\, Q^{\mathrm{mic}}(P(w^{\ast}\xi)), \end{align} $$

where $(\mathrm {Ad}^{*}(w))^{\ast }$ is the inverse image functor of $\mathrm {Ad}^{*}(w)$ . The representation $\tau _{\psi _{G}}^{\mathrm {ABV}}(\pi (\xi ))$ of $A_{\psi _{G}}$ is determined by the local system $Q^{\mathrm {mic}}(P(\xi ))$ on the left. An application of [ABV, Proposition 7.18] to (35) tells us that the representation of $A_{\mathrm {Int}(w)\circ \psi _G}$ determined by the local system $Q^{\mathrm {mic}}(P(w^{\ast }\xi ))$ is $\tau ^{\mathrm {ABV}}_{\psi _{G}^{}}(\pi (\xi ))\circ \mathrm {Int}(w)$ . This proves (a).

For part (b) we observe that

$$ \begin{align*} \chi^{\mathrm{mic}}_{S_{\mathrm{Int}(w)\circ\psi_G}}(P(w^{\ast}\xi))\ &=\ \dim \left( Q^{\mathrm{mic}}(P(w^{\ast}\xi))_{\mathrm{Int}(w)(y),\mathrm{Ad}^{*}(w)(\nu)} \right)\\ &= \ \dim \left(Q^{\mathrm{mic}}(P(\xi))_{y,\nu}\right) \\ &=\ \chi^{\mathrm{mic}}_{S_{\psi_G}}(P(\xi)). \end{align*} $$

Assertion (b) now follows from definition (27).

Finally, by parts (a) and (b), we may write

$$ \begin{align*} & w \cdot \eta_{\psi_{G}}^{\mathrm{ABV}}(\delta)(\overline{s})\\& \quad = \sum_{\pi(\xi) \in \Pi_{\psi_{G}}^{\mathrm{ABV}}(\delta)} (-1)^{dS_\xi - dS_{\psi_{G}}} \ \mathrm{Tr}\left(\tau^{\mathrm{ABV}}_{\psi_{G}^{}}(\pi(\xi))\circ \mathrm{Int}(w)\left(\mathrm{Int}(w)(\bar{s})\right) \right) \, w \cdot \pi(\xi)\\& \quad = \sum_{\pi(\xi) \in \Pi_{\psi_{G}}^{\mathrm{ABV}}(\delta)} (-1)^{dS_{w^\ast\xi} - dS_{\mathrm{Int}(w)\circ \psi_G}} \ \mathrm{Tr} \left( \tau^{\mathrm{ABV}}_{\mathrm{Int}(w)\,\circ\,\psi_{G}^{}} (\pi(w^{\ast}\xi))\left(\mathrm{Int}(w)(\overline{s}) \right) \right) \, \pi(w^{\ast}\xi)\\& \quad = \eta_{\mathrm{Int}(w)\circ \psi_G}^{\mathrm{ABV}}(\delta)(\mathrm{Int}(w)(\overline{s})).\\[-31pt] \end{align*} $$

3.3 The comparison of the approaches

We wish to compare $\eta ^{\mathrm {Ar}}_{\psi _G^{}}(\delta )(\bar {s})$ , defined in (18), with $\eta ^{\mathrm {ABV}}_{\psi _G^{}}(\delta )(\bar {s})$ , defined in (31). When G is an even rank special orthogonal group, the former distribution is only defined on $\mathrm {Out}_{N}(G)$ -orbits (6), and so the most we can hope for is an identity of the form

(36) $$ \begin{align} \eta^{\mathrm{Ar}}_{\psi_G^{}}(\delta)(\bar{s}) = \mathrm{Out}_{N}(G) \cdot \eta^{\mathrm{ABV}}_{\psi_G^{}}(\delta)(\bar{s}). \end{align} $$

The proof of this identity is the goal of this section. Once this is established, the identity

$$ \begin{align*}\widetilde{\Pi}_{\psi_G^{}}^{\mathrm{Ar}}(\delta)\, =\, \mathrm{Out}_N(G)\cdot \Pi_{\psi_G^{}}^{\mathrm{ABV}}(\delta)\end{align*} $$

is immediate. It is then also not difficult to prove

$$ \begin{align*}\quad \tau^{\mathrm{Ar}}_{\psi_G^{}}(\widetilde{\pi})= \tau^{\mathrm{ABV}}_{\psi_G^{}}(\pi), \quad \pi \in \Pi_{\psi_G^{}}^{\mathrm{ABV}}(\delta),\end{align*} $$

where $\widetilde {\pi } = \mathrm {Out}_{N}(G) \cdot \pi $ (cf. (16) and (30)).

The main objective of [AAM] was to prove these identities in the case of $\delta = \delta _q$ a quasisplit pure real form of G. This was done by first proving that the two versions of twisted endoscopic transfer agreed (on $\mathrm {Out}_{N}(G)$ -orbits), i.e., $\mathrm {Lift}_{{G(\mathbb R,\delta _q)}}^{\text {GL}_N(\mathbb {R}) \rtimes \vartheta }=\mathrm {Trans}_{{G(\mathbb R,\delta _q)}}^{\text {GL}_N(\mathbb {R}) \rtimes \vartheta }$ ([AAM]). Then it was proved that

$$ \begin{align*}\mathrm{Lift}_{{G(\mathbb R,\delta_q)}}^{\text{GL}_N(\mathbb R)\rtimes \vartheta} \left(\eta_{\psi_{G}}^{\mathrm{ABV}}(\delta_q)\right) = \mathrm{Tr}_{\vartheta} \left( \eta_{\psi}^{\mathrm{ABV}}(\delta_q)^{\thicksim} \right) = \mathrm{Tr}_{\vartheta} \left( \pi_{\psi}^{\thicksim} \right) = \mathrm{Lift}_{{G(\mathbb R,\delta_q)}}^{\text{GL}_N(\mathbb R)\rtimes \vartheta} \left(\eta_{\psi_{G}}^{\mathrm{Ar}}(\delta_q)\right)\end{align*} $$

([AAM, Corollary 7.10]). By using the injectivity of $\mathrm {Lift}_{{G(\mathbb R,\delta _q)}}^{\text {GL}_N(\mathbb R)\rtimes \vartheta }$ on $\mathrm {Out}_{N}(G)$ -orbits one obtains (36) for $\delta = \delta _{q}$ and $\bar {s} = 1$ . The extension to arbitrary $\bar {s} \in A_{\psi _{G}}$ follows from a comparison of ordinary endoscopy ([AAM, Proposition 6.3, Theorem 9.3]). In what follows, we will review the comparison of ordinary endoscopy and continue using it to achieve identity (36) for arbitrary pure real forms $\delta $ of G.

Let $\bar {s}\in A_{\psi _G^{}}$ and fix a semisimple representative $s\in {}^{\vee }G$ of $\bar {s}$ . As in the previous sections, we write $H(\mathbb {R})$ for the quasisplit endoscopic group whose dual ${^\vee }H$ is the identity component of the centralizer in ${^\vee }G$ of s. Recall that using (11) there is an Arthur parameter $\psi _{H}$ for H such that $\psi _{G}^{} = \epsilon \circ \psi _{H}^{}.$

An intermediate step toward to proving (36) is to prove analogous identities for the endoscopic group $H(\mathbb {R})$ . More precisely, we wish to prove an identity of the form

(37) $$ \begin{align} \eta_{\psi_H}^{\mathrm{Ar}}(H(\mathbb R,\delta_q)) \,=\, \left( \times_{i=1,2}\mathrm{Out}_{N_i}(H_i) \right)\cdot \eta_{\psi_H}^{\mathrm{ABV}}(H(\mathbb{R})), \end{align} $$

with $H_i,\ i=1,2$ defined in Equation (38) below. In [AAM, Section 10] this identity is verified in the case of $G = \mathrm {SO}_N$ , N odd. The other cases were left as exercises. We complete these exercises here.

Suppose N is even and G equals to $\mathrm {Sp}_N$ or $\mathrm {SO}_N$ . The explicit description in [A2, Equation (1.4.8)] of the centralizer in ${^\vee }G$ of the image of $\psi _{G}$ makes it clear that every element $\bar {s} \in A_{\psi _{G}}$ has a diagonal representative s in the centralizer with eigenvalues $\pm 1$ . Hence, as mentioned in [A2, pp. 13–14], the quasisplit endoscopic group $H(\mathbb {R})$ determined by s is a direct product

$$ \begin{align*}H_1(\mathbb{R}) \times H_2(\mathbb{R}),\end{align*} $$

in which

(38) $$ \begin{align} H_1\, =\, \mathrm{SO}_{N_1},\quad H_2\, =\, \left\{ \begin{array}{@{}cl} \mathrm{Sp}_{N_2}&\text{if }G=\mathrm{Sp}_N,\\ \mathrm{SO}_{N_2}&\text{if }G=\mathrm{SO}_N, \end{array} \right.\quad \ N_1,\, N_2\, \text{ even},\ N_1+N_2=N. \end{align} $$

The following additional conditions hold:

• • For $G(\mathbb R,\delta _q)=\mathrm {Sp}_N(\mathbb R,\delta _q)$ , the quasisplit group $H_1(\mathbb {R})= \mathrm {SO}_{N_1}(\mathbb R,\delta _q)$ can be either split or non-split.

• • For $G(\mathbb R,\delta _q)=\mathrm {SO}_N(\mathbb R,\delta _q)$ split, the quasisplit groups $H_i(\mathbb {R})= \mathrm {SO}_{N_i}(\mathbb R,\delta _q)$ , $i=1,2,$ are both split or both non-split.

• • For $G(\mathbb R,\delta _q)=\mathrm {SO}_N(\mathbb R,\delta _q)$ non-split, one of the two quasisplit groups $H_i(\mathbb {R})= \mathrm {SO}_{N_i}(\mathbb R,\delta _q)$ , $i=1,2,$ is split and the other is non-split.

Following [A2, pp. 31, 36], the Arthur parameter $\psi _{H}^{}$ decomposes as a product $\psi _{H_1}^{} \times \psi _{H_2}^{}$ of Arthur parameters. By [A2, Theorem 2.2.1(a)] the stable virtual character $\eta _{\psi _{H}}^{}(H(\mathbb R))$ is defined as the tensor product

(39) $$ \begin{align} \eta_{\psi_{H}}^{\mathrm{Ar}}(H(\mathbb{R})) = \eta_{\psi_{H_{1}}}^{\mathrm{Ar}} (H_{1}(\mathbb{R})) \otimes \eta_{\psi_{H_{2}}}^{\mathrm{Ar}}(H_{2}(\mathbb{R})), \end{align} $$

where we recall that $\eta _{\psi _{H_i}}^{\mathrm {Ar}}(H_i(\mathbb R))$ , for $H_{i}$ an even rank special orthogonal group, is defined as an orbit under $\mathrm {Out}_{N_i}(H_i)\cong \mathrm {O}_{N_i}/\mathrm {SO}_{N_i}$ . This is to say, that $\eta _{\psi _{H_i}}^{\mathrm {Ar}}(H_i(\mathbb R))$ is invariant under the action of the outer automorphisms induced by the orthogonal group $\mathrm {O}_{N_i}$ . We recall also that $\mathrm {Out}_{N_2}(H_2)$ is the trivial group for $H_2$ a symplectic group.

Let us move to the right-hand side of (37). An argument similar to the one implemented in the proof of [AAM, Corollary 6.2] permits us to obtain a decomposition

(40) $$ \begin{align} \eta_{\psi_{H}}^{\mathrm{ABV}}(H(\mathbb R))\, =\, \eta_{\psi_{H_1}^{}}^{\mathrm{ABV}}(H_1(\mathbb R)) \otimes \eta_{\psi_{H_2}^{}}^{\mathrm{ABV}}(H_2(\mathbb R)). \end{align} $$

Identity (37) follows from the previous two identities and [AAM, Theorem 9.3]. Indeed, for $H_2$ a symplectic group, [AAM, Theorem 9.3(a)] tells us that

$$ \begin{align*}\eta_{\psi_{H_2}^{}}^{\mathrm{Ar}}(H_2(\mathbb R)) \,=\, \eta_{\psi_{H_2}^{}}^{\mathrm{ABV}} (H_2(\mathbb R)).\end{align*} $$

For $H_i,\, i=1,2$ an even rank special orthogonal group, Proposition 3.2 (c) with $\overline {s}=1$ tells us that

$$ \begin{align*}\mathrm{Out}_{N_i}(H_i)\cdot \eta_{\psi_{H_i}^{}}^{\mathrm{ABV}}(H_i(\mathbb R))\ =\ \left\{ \eta_{\psi_{H_i}^{}}^{\mathrm{ABV}}(H_i(\mathbb R)), \eta_{{^{\vee}}\mathrm{Int}(w)\circ \psi_{H_i}^{}}^{\mathrm{ABV}}(H_i(\mathbb R)) \right\} \end{align*} $$

and from [AAM, Theorem 9.3(b)] we conclude

$$ \begin{align*}\eta_{\psi_{H_i}^{}}^{\mathrm{Ar}}(H_i(\mathbb R)) \,=\, \mathrm{Out}_{N_i}(H_i)\cdot \eta_{\psi_{H_i}^{}}^{\mathrm{ABV}}(H_i(\mathbb R)). \end{align*} $$

Identity (37) is now immediate from the decompositions of $\eta _{\psi _{H}^{}}^{\mathrm {Ar}}(H(\mathbb R))$ and $\eta _{\psi _{H}^{}}^{\mathrm {ABV}}(H(\mathbb R))$ given in equations (39) and (40), respectively.

The comparison of distributions in (37) for endoscopic groups can be lifted to a comparison of distributions for $G(\mathbb {R},\delta )$ by using ordinary endoscopic transfer. The details are given in the following lemma.

Lemma 3.3 Let $\psi _G$ be an Arthur parameter for G. For any $\overline {s}\in A_{\psi _G}$ , choose H and $\psi _H$ as in the beginning of this section. Then

(41) $$ \begin{align} \mathrm{Lift}_{H(\mathbb{R})}^{G(\mathbb{R},\delta)} \left( \left( \times_{i=1,2}\mathrm{Out}_{N_i}(H_i)\right) \cdot \eta_{\psi_{H}^{}}^{\mathrm{ABV}}(H(\mathbb R))\right) \ =\ \mathrm{Out}_N(G)\cdot \eta_{\psi_{G}}^{\mathrm{ABV}}(\delta)(\overline{s}). \end{align} $$

Proof The proof is done case-by-case, depending on the form of G.

1. a. For $G=\mathrm {SO}_N$ , N odd, the groups $H_{1}$ , $H_{2}$ are odd rank special orthogonal groups ([A2, p. 13]). Therefore, $\times _{i=1,2}\mathrm {Out}_{N_i}(H_i)$ and $\mathrm {Out}_{N}(G)$ are trivial, and (41) reduces to Equation (32).

2. b. Let $G=\mathrm {Sp}_N$ . Then $\times _{i=1,2}\mathrm {Out}_{N_i}(H_i)$ is a group of order two whose generator may be represented by the element ${w}= {w}_1\times \mathrm {I_{N_2}} \in \mathrm {O}_{N_1} \times \mathrm {Sp}_{N_2}, $ with $w_1$ as in (6). Similarly, the automorphism ${}^{\vee }\mathrm {Int}(w)$ dual to $\mathrm {Int}(w)$ may be represented by the element ${}^{\vee }{w}= {w}_1\times \mathrm {(-I_{N_2+1})} \in \mathrm {O}_{N_1}\times \mathrm {O}_{N_2+1}$ . Now, by Proposition 3.2 (c) with $\overline {s}=1$

   $$ \begin{align*}\eta^{\mathrm{ABV}}_{\psi_{H_1}^{}}(H_1(\mathbb R)) \circ \mathrm{Int}(w_1) = \eta^{\mathrm{ABV}}_{\mathrm{Int}({}^{\vee}w_1) \circ \psi_{H_1}}(H_1(\mathbb R)) \end{align*} $$
   and so decomposition (40) permits us to write
   $$ \begin{align*}\eta^{\mathrm{ABV}}_{\psi_{H}^{}}(H(\mathbb R)) \circ \mathrm{Int}(w) = \eta^{\mathrm{ABV}}_{\mathrm{Int}({}^{\vee}w) \circ \psi_{H}}(H(\mathbb R)). \end{align*} $$
   Moreover, for $G=\mathrm {Sp}_{N}$ the element ${}^{\vee }{w}$ belongs to ${^{\vee }}{G}=\mathrm {SO}_{N+1}(\mathbb {C})$ . Consequently the Arthur parameters
   $$ \begin{align*}\psi_G\, =\, \epsilon \circ \psi_{H}\quad\text{and}\quad \mathrm{Int}\left({{}^{\vee}w}\right)\circ\psi_G\ \ =\ \epsilon \circ \left( \mathrm{Int}\left({{}^{\vee}w} \right) \circ \psi_{H}\right)\end{align*} $$
   are in the same ${^{\vee }}G$ -orbit, and the ${^\vee }G$ -conjugate elements $\overline {s}$ and $\mathrm {Int}({}^{\vee }w)(\bar {s})$ correspond to the same endoscopic group $H(\mathbb {R})$ . The ordinary endoscopic transfer (32) therefore implies
   $$ \begin{align*} \mathrm{Lift}_{H(\mathbb R)}^{G(\mathbb R,\delta)}\left(\eta_{\mathrm{Int}({}^{\vee}w) \circ \psi_{H}}^{\mathrm{ABV}}(H(\mathbb R))\right) \ &=\ \eta_{\mathrm{Int}({}^{\vee}w)\circ\psi_{G}}^{\mathrm{ABV}}(\delta) (\mathrm{Int}({}^{\vee}w)(\overline{s}))\\ \ &=\ \eta_{\psi_{G}}^{\mathrm{ABV}}(\delta)(\bar{s}) \\ \ &=\ \mathrm{Lift}_{H(\mathbb R)}^{G(\mathbb R,\delta)} \left( \eta_{\psi_{H}}^{\mathrm{ABV}}(H(\mathbb R))\right). \end{align*} $$
   Since $\mathrm {Out}_N(G)$ is trivial in this case, we have proved (41).

3. c. Let $G=\mathrm {SO}_N$ where N is even. We follow the same reasoning as in b. Now $\times _{i=1,2}\mathrm {Out}_{N_i}(H_i)$ is a group of order four whose generators may be represented by elements

   $$ \begin{align*}w={w}_1\times \mathrm{Id}_{N_2} \in \mathrm{O}_{N_1} \times \mathrm{SO}_{N_2}\ \text{ and }\ w'=\mathrm{Id}_{N_1}\times {w}_2 \in \mathrm{SO}_{N_1}\times \mathrm{O}_{N_2}, \end{align*} $$
   where $w_{1}$ and $w_{2}$ are as in (6). The same pair of elements can be used as generators for the dual automorphisms. Proposition 3.2 (c) with $\overline {s}=1$ gives us
   $$ \begin{align*}\eta^{\mathrm{ABV}}_{\psi_{H_i}^{}}(H_i(\mathbb R)) \circ \mathrm{Int}(w_i) \, =\, \eta^{\mathrm{ABV}}_{\mathrm{Int}(w_i) \circ \psi_{H_i}}(H_i(\mathbb R)), \end{align*} $$
   and through decomposition (40) we are able to deduce
   $$ \begin{align*} \eta^{\mathrm{ABV}}_{\psi_{H}}(H(\mathbb R)) \circ \mathrm{Int}(ww') \, &=\, \eta^{\mathrm{ABV}}_{\mathrm{Int}({ww'}) \circ \psi_{H}}(H(\mathbb R)), \quad \text{and}\\ \eta^{\mathrm{ABV}}_{\mathrm{Int}({w'}) \circ \psi_{H}}(H(\mathbb R)) \circ \mathrm{Int}(ww')\, &=\, \eta^{\mathrm{ABV}}_{\mathrm{Int}({w}) \circ \psi_{H}}(H(\mathbb R)). \end{align*} $$
   In addition, the product $ww'$ belongs to ${^{\vee }}{G}=\mathrm {SO}_N(\mathbb {C})$ and
   $$ \begin{align*}\mathrm{Int}\left({w}\right)\, \circ\, \psi_G = \mathrm{Int}\left({w} w'\right) \circ \left( \mathrm{Int}\left({w'}\right)\, \circ\, \psi_G\right). \end{align*} $$
   Hence, the Arthur parameters
   $$ \begin{align*}\psi_G\, =\, \epsilon \, \circ\, \psi_{H} \quad \mbox{ and } \quad \mathrm{Int}({w} w') \circ \psi_{G}, =\, \epsilon \circ\left(\mathrm{Int}({w} w') \circ \psi_{H}\right) \end{align*} $$
   belong to the same ${}^{\vee }G$ -orbit, and
   $$ \begin{align*}\mathrm{Int}\left({w}\right)\, \circ\, \psi_G\, =\, \epsilon\, \circ\, \left( \mathrm{Int}( {w}) \circ \psi_{H} \right) \quad \mbox{ and } \quad \mathrm{Int}\left({w'}\right)\, \circ\, \psi_G =\, \epsilon\, \circ\, \left( \mathrm{Int}({w'}) \circ \psi_{H} \right) \end{align*} $$
   belong to the same ${}^{\vee }G$ -orbit. As in case b. the ${^\vee }G$ -conjugate elements, $\bar {s}$ and $\mathrm {Int}(w w')(\bar {s})$ , correspond to the same endoscopic group $H(\mathbb {R})$ . Similarly, the ${^\vee }G$ -conjugate elements, $\mathrm {Int}(w)(\bar {s})$ and $\mathrm {Int}(w')(\bar {s})$ , correspond to the same endoscopic group $H(\mathbb {R})$ . Therefore, the endoscopic transfer map (32) yields
   $$ \begin{align*} \mathrm{Lift}_{H(\mathbb R)}^{G(\mathbb R,\delta)}\left(\eta_{\mathrm{Int}(w w') \circ \psi_{H}}^{\mathrm{ABV}}(H(\mathbb R))\right) \ &=\ \eta_{\mathrm{Int}(w w') \circ \psi_{G}}^{\mathrm{ABV}}(\delta)(\mathrm{Int}(w w')(\overline{s}))\\ \ &=\ \eta_{\psi_{G}}^{\mathrm{ABV}}(\delta)(\overline{s})\\ &=\ \mathrm{Lift}_{H(\mathbb R)}^{G(\mathbb R,\delta)}\left(\eta_{\psi_{H}}^{\mathrm{ABV}}(H(\mathbb R))\right),\ \end{align*} $$
   and likewise
   $$ \begin{align*}\mathrm{Lift}_{H(\mathbb R)}^{G(\mathbb R,\delta)}\left(\eta_{\mathrm{Int}(w) \circ \psi_{H}}^{\mathrm{ABV}}(H(\mathbb R))\right) = \mathrm{Lift}_{H(\mathbb R)}^{G(\mathbb R,\delta)}\left(\eta_{\mathrm{Int}(w') \circ \psi_{H}}^{\mathrm{ABV}}(H(\mathbb R))\right).\end{align*} $$

   Finally, by Proposition 3.2 (c), we have

   $$ \begin{align*}w \cdot \eta_{\psi_{G}}^{\mathrm{ABV}}(\delta)(\overline{s})= \eta_{\mathrm{Int}(w) \circ\psi_{G}}^{\mathrm{ABV}}(\delta)(\mathrm{Int}(w)(\overline{s})). \end{align*} $$

   Since $\mathrm {Out}_N(G)$ is a group of order two generated by w, the desired identity (41) follows.

We can now lift Lemma 3.3 to our main theorem.

Theorem 3.4 Let $\psi _G^{}$ be an Arthur parameter for G. Then for any pure real form $\delta $ of G

$$ \begin{align*}\, \eta_{\psi_G^{}}^{\mathrm{Ar}}(\delta)(\overline{s})\ =\ \mathrm{Out}_N(G)\cdot\left(\eta_{\psi_G^{}}^{\mathrm{ABV}}(\delta) (\overline{s})\right),\quad \overline{s}\in A_{\psi_G^{}}. \end{align*} $$

In particular,

$$ \begin{align*}\widetilde{\Pi}_{\psi_G^{}}^{\mathrm{Ar}}(\delta)\, =\, \mathrm{Out}_N(G)\cdot \Pi_{\psi_G^{}}^{\mathrm{ABV}}(\delta). \end{align*} $$

In addition, for any $\pi \in \Pi _{\psi _G^{}}^{\mathrm {ABV}}(\delta )$

$$ \begin{align*} \tau^{\mathrm{Ar}}_{\psi_G^{}}(\widetilde{\pi})\, =\, \tau^{\mathrm{ABV}}_{\psi_G^{}}(\pi), \quad \tau^{\mathrm{Ar}}_{\psi_{G}^{}}(\widetilde{\pi})(s_{\psi_{G}^{}})\, =\, (-1)^{dS_\pi - dS_{\psi_{G}^{}}}, \end{align*} $$

with $\widetilde {\pi }$ the $\mathrm {Out}_N(G)$ -orbit of $\pi $ , and $S_\pi $ the ${}^{\vee }G$ -orbit in $ X({}^\vee G^{\Gamma} )$ corresponding to the L-parameter of $\pi $ .

Proof Let $\bar {s}\in A_{\psi _G^{}}$ . By Lemma 3.1, Equation (37), and Lemma 3.3, we deduce

$$ \begin{align*} \eta_{\psi_G^{}}^{\mathrm{Ar}}(\delta)(\overline{s})& = \mathrm{Trans}_{H(\mathbb R)}^{G(\mathbb R,\delta)}(\eta_{\psi_{H}}^{\mathrm{Ar}}(H(\mathbb R)))\\& = \mathrm{Lift}_{H(\mathbb R)}^{G(\mathbb R,\delta)}(\eta_{\psi_{H}}^{\mathrm{Ar}}(H(\mathbb R)))\\& = \mathrm{Lift}_{H(\mathbb R)}^{G(\mathbb R,\delta)}\left(\left(\times_{i=1,2}\mathrm{Out}_{N_i}(H_i)\right)\cdot\eta_{\psi_{H}}^{\mathrm{ABV}}(H(\mathbb R))\right)\\& = \mathrm{Out}_N(G)\cdot \eta_{\psi_G^{}}^{\mathrm{ABV}}(\delta)(\bar{s}). \end{align*} $$

This proves the first assertion, and the identity of the Arthur packets follows immediately. Let $\widetilde {\pi } = \mathrm {Out}_N(G) \cdot \pi $ . Then we may write

$$ \begin{align*} \sum_{\widetilde{\pi} \in \widetilde{\Pi}_{\psi_{G}^{}}^{\mathrm{Ar}}(\delta)} \mathrm{Tr} \left( \tau^{\mathrm{Ar}}_{\psi_{G}^{}} (\widetilde{\pi}) (s_{\psi_{G}}\bar{s}) \right)\, \widetilde{\pi} \ =\ \sum_{ \widetilde{\pi} \in \mathrm{Out}_N(G) \cdot \Pi_{\psi_{G}^{}}^{\mathrm{ABV}}(\delta)} (-1)^{dS_\pi - dS_{\psi_{G}^{}}} \ \mathrm{Tr}\left(\tau^{\mathrm{ABV}}_{\psi_{G}^{}}(\pi)(\bar{s}) \right) \, \widetilde{\pi}. \end{align*} $$

Since $\widetilde {\Pi }(G(\mathbb {R},\delta ))$ is a basis of $K_{\mathbb {C}} \widetilde {\Pi }(G(\mathbb {R},\delta ))$ , this equation implies

$$ \begin{align*}\mathrm{Tr} \left(\tau^{\mathrm{Ar}}_{\psi_{G}}(\widetilde{\pi})(s_{\psi_{G}} \bar{s})\right) = (-1)^{dS_{\pi} - dS_{\psi_{G}}} \ \mathrm{Tr} \left(\tau^{\mathrm{ABV}}_{\psi_{G}}(\pi)(\bar{s}) \right).\end{align*} $$

This may be regarded as an equality between virtual characters on $A_{\psi _G}$ , and so the linear independence of these characters implies the last assertions of the theorem.