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 $(\text {GL}_{N}, \vartheta )$ ([Reference Kottwitz and ShelstadKS, Section 2], [Reference ArthurA2, 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 $ ([Reference ArthurAl, 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 [Reference ArthurA2, 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 $(G,{}^{\vee }G^{\Gamma }, s, \text {St}_{G})$ ([Reference Kottwitz and ShelstadKS, p. 18], [Reference ArthurA2, p. 12]). The group $\mathrm {Out}_N(G)$ may be identified with the group of outer automorphisms of G ([Reference Kottwitz and ShelstadKS, 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 [Reference ArthurA2, 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 $ ([Reference ArthurA2, 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 [Reference Adams, Arancibia Robert and MezoAAM]. We prefer to use the first two here, in line with [Reference ArthurA2]. 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 [Reference MezoM]. 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 $C_{c}^{\infty }(\text {GL}_{N}(\mathbb {R} \rtimes \vartheta )) \rightarrow C_{c}^{\infty }(G(\mathbb {R},\delta ))$ [Reference ShelstadS3]. Since the image of geometric transfer lies in the $\mathrm {Out}_{N}(G)$ -invariant subspace of $C_{c}^{\infty }(G(\mathbb {R}, \delta ))$ ([Reference ArthurA2, 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 [Reference ArthurA2, p. 12] and [Reference Arancibia, Mœglin and RenardAMR, 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 $\psi _{G}^{}$ ([Reference ArthurA2, 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 ([Reference ArthurA2, 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) $$ \begin{align} \eta_{\psi_G}^{\mathrm{Ar}}(\delta_q)(\overline{s})= \mathrm{Trans}_{H(\mathbb R)}^{G(\mathbb R,\delta_q)}\left(\eta_{\psi_{H}}^{\mathrm{Ar}}(H(\mathbb R)\right) \end{align} $$

([Reference ArthurA2, 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 $H(\mathbb R)$ ([Reference Langlands and ShelstadLS], [Reference ShelstadS1] and [Reference ShelstadS2]). 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 [Reference ArthurA2, 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 [Reference ArthurA2, 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 [Reference ArthurA2, 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 [Reference Moeglin and RenardMR2] and [Reference Moeglin and RenardMR1], 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 ([Reference Moeglin and RenardMR1, 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 [Reference Moeglin and RenardMR2, 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 [Reference Moeglin and RenardMR2], 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 [Reference Adams, Barbasch and VoganABV] 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 ([Reference Adams, Barbasch and VoganABV, 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 ${^\vee }\mathcal {O} \subset {^\vee }\mathfrak {g}$ ([Reference Adams, Barbasch and VoganABV, 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 [Reference LanglandsL], 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 [Reference Adams, Barbasch and VoganABV], 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) ([Reference Adams, Barbasch and VoganABV, 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 ([Reference Adams, Arancibia Robert and MezoAAM, Section 2.2], [Reference Adams, Barbasch and VoganABV, 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 ([Reference Adams, Barbasch and VoganABV, 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 [Reference Adams, Barbasch and VoganABV, 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 $\xi =(S,\mathcal {V})$ [Reference Adams, Barbasch and VoganABV, 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 $X({^\vee }\mathcal {O},{^\vee }G^{\Gamma })$ ([Reference Bernstein and LuntsBL, Section 5]). This is an abelian category and, as explained in [Reference Adams, Barbasch and VoganABV, 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 [Reference Adams, Barbasch and VoganABV, Definition 15.11] using the alternative bases of standard representations and constructible sheaves. It is a deep result ([Reference Adams, Barbasch and VoganABV, 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 ([Reference Adams, Barbasch and VoganABV, Chapter 19], [Reference Borel, Grivel, Kaup, Haefliger, Malgrange and EhlersBGK+]), and are associated with ${^\vee }G$ -equivariant local systems on a conormal bundle of $X({^\vee }\mathcal {O},{^\vee }G^{\Gamma })$ ([Reference Adams, Barbasch and VoganABV, Section 24], [Reference Goresky and MacPhersonGM, Appendix 6.A]). The virtual characters associated by the pairing to the linear functionals $\chi ^{\mathrm {mic}}_S$ are stable ([Reference Adams, Barbasch and VoganABV, 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 [Reference Adams, Barbasch and VoganABV, 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 [Reference Adams, Barbasch and VoganABV, 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 })$ ([Reference Adams, Barbasch and VoganABV, 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

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

([Reference Adams, Barbasch and VoganABV, 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 $G(\mathbb R,\delta )$ ([Reference Adams, Barbasch and VoganABV, p. 289]).

It is argued on [Reference Adams, Barbasch and VoganABV] 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 [Reference Arancibia Robert and MezoARM1, 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 [Reference Arancibia Robert and MezoARM2, Section 11]. The proof for symplectic and special orthogonal groups follows [Reference Arancibia Robert and MezoARM2, 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 [Reference Adams, Barbasch and VoganABV]. 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 [Reference Adams, Barbasch and VoganABV, 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 $T^{\ast }_{{}^{\vee }G}\left ( X\left ({}^\vee \mathcal O,{}^\vee G^{\Gamma} \right )\right )$ ([Reference Adams, Barbasch and VoganABV, 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 [Reference Adams, Barbasch and VoganABV, 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 ([Reference Adams, Barbasch and VoganABV, 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 [Reference Adams, Barbasch and VoganABV, Lemma 24.3], $A_S^{\mathrm {mic}}$ is independent of the choice of $(y,\nu )$ . Furthermore, when $S = S_{\psi _{G}}$ as in (25)

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

([Reference Adams, Barbasch and VoganABV, 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 ([Reference Adams, Barbasch and VoganABV, 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 [Reference Adams, Arancibia Robert and MezoAAM]. Indeed, following [Reference Adams, Barbasch and VoganABV], a sheaf-theoretic version of the theory of twisted endoscopy is introduced in [Reference Christie and MezoCM]. This is used in [Reference Adams, Arancibia Robert and MezoAAM] 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 [Reference Adams, Barbasch and VoganABV, 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 [Reference Adams, Barbasch and VoganABV, 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 [Reference Adams, Barbasch and VoganABV, 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.

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 [Reference Adams, Barbasch and VoganABV, 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 $K\subset J$ ([Reference Adams, Barbasch and VoganABV, 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

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

([Reference AcharA, 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 [Reference Adams, Barbasch and VoganABV, 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 [Reference Adams, Arancibia Robert and MezoAAM] 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., $\mathrm {Lift}_{{G(\mathbb R,\delta _q)}}^{\text {GL}_N(\mathbb {R}) \rtimes \vartheta }=\mathrm {Trans}_{{G(\mathbb R,\delta _q)}}^{\text {GL}_N(\mathbb {R}) \rtimes \vartheta }$ ([Reference Adams, Arancibia Robert and MezoAAM]). Then it was proved that

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

([Reference Adams, Arancibia Robert and MezoAAM, 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 ([Reference Adams, Arancibia Robert and MezoAAM, 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 [Reference Adams, Arancibia Robert and MezoAAM, 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 [Reference ArthurA2, 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 [Reference ArthurA2, 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 [Reference ArthurA2, pp. 31, 36], the Arthur parameter $\psi _{H}^{}$ decomposes as a product $\psi _{H_1}^{} \times \psi _{H_2}^{}$ of Arthur parameters. By [Reference ArthurA2, 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 [Reference Adams, Arancibia Robert and MezoAAM, 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 [Reference Adams, Arancibia Robert and MezoAAM, Theorem 9.3]. Indeed, for $H_2$ a symplectic group, [Reference Adams, Arancibia Robert and MezoAAM, 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 [Reference Adams, Arancibia Robert and MezoAAM, 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) $$ \begin{align} \mathrm{Lift}_{H(\mathbb{R})}^{G(\mathbb{R},\delta)} \left( \left( \times_{i=1,2}\mathrm{Out}_{N_i}(H_i)\right) \cdot \eta_{\psi_{H}^{}}^{\mathrm{ABV}}(H(\mathbb R))\right) \ =\ \mathrm{Out}_N(G)\cdot \eta_{\psi_{G}}^{\mathrm{ABV}}(\delta)(\overline{s}). \end{align} $$

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.