2.1 Casselman–Wallach representation

Let G be a real reductive group and $\mathfrak {g}_{\mathbb {C}}$ be its complexified Lie algebra. Let $(\pi , V_1)$ be a representation of G and let $\mathcal {Z}(\mathfrak {g}_{\mathbb {C}})$ be the center of the universal enveloping algebra of $\mathfrak {g}_{\mathbb {C}}$ . The representation $(\pi , V_1)$ is called admissible if every irreducible representation of a maximal compact subgroup K of G has finite multiplicity in $V_1$ . The representation $(\pi , V_1)$ is called of Harish-Chandra type if it is admissible and $\mathcal {Z}(\mathfrak {g}_{\mathbb {C}})$ finite. The representation $(\pi , V_1)$ is called of moderate growth if the following condition holds: for every continuous seminorm $|\cdot |_{\mu }$ on $V_1$ , there exists a positive, moderate growth function $\phi $ on G, and a continuous seminorm $|\cdot |_{\nu }$ on $V_1$ such that

$$\begin{align*}|gv|_{\mu} \leq |\phi(g)|\ |v|_{\nu}, \quad \forall \ g \in G,\ v \in V_1. \end{align*}$$

The representation $(\pi , V_1)$ is called a Casselman–Wallach representation if the representation is Frechet, smooth, of moderate growth, and of Harish-Chandra type.

2.2 Nash groups, Nash manifolds, and Nash maps

In this section, we will define a Nash manifold over an Archimedean local field F of characteristics zero as discussed in [Reference Aizenbud and GourevitchAG08].

2.2.1 Semi-algebraic sets

A subset $A \subset F^n$ is called a semi-algebraic set if it satisfies the following: there exist finitely many polynomials $f_{ij}, g_{ik} \in F[x_{1}, x_{2}, \ldots x_{n}]$ such that

$$\begin{align*}A = \bigcup_{i=1}^{r}\left\{x \in F^n: f_{i1}(x)>0, f_{i2}(x)>0, \dots, f_{is_{i}} >0, g_{i1}(x)=0, g_{i2}(x)=0, \dots g_{it_{i}}=0 \right\}. \end{align*}$$

In other words, the semi-algebraic sets are those that can be written as the finite union of polynomial equations and inequalities. From the definition above, it is immediate that the collection of semi-algebraic sets is closed with respect to finite unions, finite intersections, and complements. A map $\nu $ between two semi-algebraic sets $A \subset F^{n},B \subset F^m$ is called semi-algebraic if the graph of the map $\nu $ is a semialgebraic subset of $F^{m+n}$ . The open semi-algebraic sets that define the topology can be realized in the following lemma.

Lemma 2.1 Let $X \subset F^n$ be a semi-algebraic set. Then, every open semi-algebraic subset of X can be presented as a finite union of sets of the form $\{x \in X|f_{i}(x)> 0,i = 1\ldots n\}$ , where $f_{i}$ are polynomials in n variables.

Let $U,V \subset F^{n}$ be two open semi-algebraic sets. A smooth, semi-algebraic map between U and V is called a Nash map. A Nash map which is bijective, whose inverse is also a Nash map is called a Nash diffeomorphism. A Nash submanifold of $F^n$ is a semi-algebraic, smooth submanifold of $F^n$ . A Nash group G is a Nash manifold such that the following is a Nash map:

$$\begin{align*}G \times G \longrightarrow G \end{align*}$$

$$\begin{align*}(g,h) \mapsto gh^{-1}. \end{align*}$$

Therefore, we will treat the groups $\operatorname {GPin}(V)$ and $\operatorname {GSpin}(V)$ as Nash groups in the subsequent sections of this article.

2.3 The groups $\operatorname {GSpin}(V)$ and $\operatorname {GPin}(V)$

In this section, we introduce the definitions of the groups $\operatorname {GSpin}(V)$ and $\operatorname {GPin}(V)$ and their properties. A reference for the material can be found in [Reference ScharlauSch85, Reference ShimuraShi04]. In the literature, the group which we refer to as $\operatorname {GPin}(V)$ is sometimes called the Clifford group and $\operatorname {GSpin}(V)$ is sometimes referred to as the special Clifford group, and are denoted by $\Gamma (V)$ and S $\Gamma (V)$ , respectively.

Here, V denotes a nondegenerate quadratic space over our Archimedean local field F. Let $\langle -, -\rangle $ be the corresponding bi-linear form. Let q be the quadratic norm defined over F. In this section, we also define involutions on $\operatorname {GPin}(V)$ , which is required to prove Theorem 1.2.

2.3.1 Clifford algebra

Let $T(V)= \bigoplus \limits _{l=0}^{\infty }V^{\otimes l}$ be the tensor algebra of V and we define the Clifford algebra $C(V)$ by the following quotient:

$$\begin{align*}C(V)= T(V)/\langle v \otimes v - q(v)\cdot 1: v \in V\rangle. \end{align*}$$

In $C(V),$ we have

$$\begin{align*}v\cdot v= q(v) \quad \forall v \in V. \end{align*}$$

We denote the image of $V^{\otimes l}$ in $C(V)$ as $C^{l}(V)$ . Denote by

$$\begin{align*}C^{+}(V)= \sum_{l \text{ even}} C^{l}(V) \qquad C^{-}(V)= \sum_{l \text{ odd}} C^{l}(V) \end{align*}$$

the even and odd Clifford algebras, respectively. Then, we have the following decomposition:

$$\begin{align*}C(V)=C^{+}(V) \oplus C^{-}(V). \end{align*}$$

The Clifford algebra is equipped with the natural involution $*$ by “reversing the indices” of $ v_{1}v_{2} \dots v_{l} \in C^{l}(V)$ , namely,

$$\begin{align*}(v_{1}v_{2} \dots v_{l})^{*}= v_{l}v_{l-1}\dots v_{1}. \end{align*}$$

The above involution is called the canonical involution, which preserves both $C^{+}(V)$ and $C^{-}(V)$ . We define a map

$$\begin{align*}\alpha : C(V) \longrightarrow C(V), \quad \alpha(x^{+}+x^{-})=x^{+}-x^{-}, \end{align*}$$

where $x^{+} \in C^{+}(V)$ and $x^{-} \in C^{-}(V)$ ; in other words, $\alpha $ acts on $C^{+}(V)$ as the identity and on $C^{-}(V)$ as the negative identity. For all $x \in C(V),$ the Clifford involution is

$$\begin{align*}\overline{x}=\alpha(x)^*=\alpha(x^{*}). \end{align*}$$

The map sending x to $\overline {x}$ is an involution on $C(V)$ and the Clifford norm is the map:

$$\begin{align*}N: C(V) \longrightarrow C(V), \quad N(x)=x\bar{x}. \end{align*}$$

2.3.2 $\operatorname {GPin}(V)$ and $\operatorname {GSpin}(V)$

We are now in a position to define the groups $\operatorname {GPin}(V)$ and $\operatorname {GSpin}(V)$ as follows:

$$\begin{align*}\operatorname{GPin}(V):=\{g \in C(V)^{\times}: \quad \alpha(g)V g^{-1}= V\};\end{align*}$$

$$\begin{align*}\operatorname{GSpin}(V):= \{g \in C^{+}(V)^{\times}: \quad gV g^{-1}= V\}, \end{align*}$$

and we call $\operatorname {GSpin}$ the general Spin group on V and $\operatorname {GPin}$ the general Pin group on V. We also define the projection map P of $\operatorname {GPin}$ groups onto the orthogonal groups as follows:

$$\begin{align*}P(g): V \longrightarrow V, \quad P(g)v= \alpha(g)vg^{-1}, \end{align*}$$

for all $g \in \operatorname {GPin}(V)$ . It is well known that P surjects onto $\operatorname {O}(V)$ because of the map $\alpha $ . This implies that we have the following commutative diagram:

2.3.3 Involution

Let

$$\begin{align*}\text{sign}: \operatorname{GPin}(V) \longrightarrow {\pm{1}}\end{align*}$$

be a homomorphism which sends the non-identity component to $-1$ . Therefore, the kernel of this map is $\operatorname {GSpin}(V)$ . We have the involution

$$\begin{align*}\sigma_V:\operatorname{GPin}(V)\longrightarrow\operatorname{GPin}(V),\quad \sigma_V(g)= \begin{cases}g^*,&\text{if } n=2k;\\ \operatorname{sign}(g)^{k+1}g^*,&\text{if } n=2k-1. \end{cases} \end{align*}$$

Note that $\sigma _V$ preserves the semisimple conjugacy classes of $\operatorname {GPin}(V)$ . We denote $\widetilde {\operatorname {GPin}}(V):=\langle g,\beta : g\in \operatorname {GPin}(V) \rangle $ , such that $g\beta =\beta g$ for all $g\in \operatorname {GPin}(V)$ and $\beta ^2=1$ . We now define the action of $\widetilde {\operatorname {GPin}}(V)$ on $\operatorname {GPin}(V)\times V$ by

(2.1) $$ \begin{align} \begin{aligned} g\cdot (h, v)&=(ghg^{-1}, P(g)v)\\ \beta\cdot (h, v)&=(\sigma_V(h), -v) \end{aligned} \end{align} $$

for $g\in \operatorname {GPin}(V)$ and $(h, v)\in \operatorname {GPin}(V)\times V$ . Note that the action of $\beta $ also preserves the semisimple conjugacy classes of $\operatorname {GPin}(V)$ .

We assume

$$\begin{align*}W=Span\{e_1,\dots,e_{n-1}\} \text{ so that } V=W \oplus Fe \text{ with } e:=e_n. \end{align*}$$

We then have $\operatorname {GPin}(V)_e=\operatorname {GPin}(W)$ and $ \widetilde {\operatorname {GPin}}(V)_e=\big \langle g,\, e\beta \;:\; g\in \operatorname {GPin}(W)\big \rangle .$ We define

$$\begin{align*}\widetilde{\operatorname{GPin}}(W):=\widetilde{\operatorname{GPin}}(V)_e. \end{align*}$$

Then, we also have a short exact sequence as follows:

$$\begin{align*}1\longrightarrow \operatorname{GPin}(W)\longrightarrow\widetilde{\operatorname{GPin}}(W)\xrightarrow{\;\,\chi\;\,}\{\pm1\}\longrightarrow 1, \end{align*}$$

where the surjection $\chi $ sends $e\beta $ to $-1$ , and $ \widetilde {\operatorname {GPin}}(W)\simeq \operatorname {GPin}(W)\rtimes \{1, e\beta \},$ where $e\beta $ acts on $\operatorname {GPin}(W)$ by conjugation viewed inside $\widetilde {\operatorname {GPin}}(V)$ .

We define an involution

(2.2) $$ \begin{align} \tau_W:\operatorname{GPin}(V)\longrightarrow\operatorname{GPin}(V),\quad \tau_W(g)=e\sigma_V(g)e^{-1} \end{align} $$

for $g\in \operatorname {GPin}(V)$ . This involution is precisely the action of the element $e\beta \in \widetilde {\operatorname {GPin}}(V)$ . Note that it is direct to show that $\tau _W(\operatorname {GPin}(W))=\operatorname {GPin}(W)$ (see [Reference Emory and TakedaET23, Lemma 2.10] for more details).

3.1 Reduction I: Elimination of W

In this section, we reduce Theorem 3.3 to the following vanishing assertion:

(3.1) $$ \begin{align} \mathcal{S}^*(\operatorname{GPin}(V)\times V)^{\widetilde{\operatorname{GPin}}(V),\chi}=0. \end{align} $$

The key ingredient is the following version of Frobenius descent [Reference Aizenbud and GourevitchAG09a, Theorem 2.5.7].

Lemma 3.4 (Frobenius descent)

Let G be a Nash group which is unimodular. Let X and Y be Nash manifolds on which G acts. In addition, assume that the action of G on Y is transitive. Suppose that we have a continuous G-equivariant Nash map

$$\begin{align*}\phi:X \rightarrow Y, \end{align*}$$

namely, $\varphi (g\cdot x)=g\cdot \varphi (x)$ for all $g\in G$ and $x\in X$ . Fix $y\in Y$ . Assume that the stabilizer $G_y\subseteq G$ of y is unimodular which implies that there exists a G-invariant measure on Y. Fix this measure. Let $\chi :G\to \mathbb {C}^1$ be a character of G. Then, there is a canonical isomorphism

$$\begin{align*}\mathcal{S}^*(X)^{G,\chi}\simeq \mathcal{S}^*(\phi^{-1}(y))^{G_y,\chi}. \end{align*}$$

We are now ready to prove the following.

Proof We follow the main argument of the proof in [Reference Emory and TakedaET23, Proposition 7.2] and adapt those to the Archimedean version. Briefly, we write the main ideas of the proof but, for completeness, we write in detail in case the proof is not certain in the Archimedean case.

Our goal is to prove

$$\begin{align*}\mathcal{S}^*(\operatorname{GPin}(V))^{\widetilde{\operatorname{GPin}}(W), \chi}\subseteq \mathcal{S}^*(\operatorname{GPin}(V)\times V)^{\widetilde{\operatorname{GPin}}(V), \chi}. \end{align*}$$

Let $X:=\{(g,v) \in \operatorname {GPin}(V)\times V \;:\; \langle v,v\rangle =\langle e,e \rangle \}.$ One can see that X is invariant under $\widetilde {\operatorname {GPin}}(V)$ and we have the following.

Lemma 3.6 X is closed in $\operatorname {GPin}(V)\times V$ .

Proof We have the quadratic form $q :V\rightarrow F$ and we know that $q(v)=v^2$ and ${v^2 \in F}$ . Hence, $\langle v,v \rangle = \langle e,e\rangle $ implies

$$\begin{align*}q(v+v)-q(v)-q(v)=q(e+e)-q(e)-q(e), \end{align*}$$

$$\begin{align*}4v^2-2v^2=4e^2-2e^2, \end{align*}$$

$$\begin{align*}v^2=e^2. \end{align*}$$

Since $v^2,e^2 \in F$ , this is a discrete set. Therefore, X is closed in $\operatorname {GPin}(V) \times V$ .

Therefore, we have

$$\begin{align*}\mathcal{S}^*(X)^{\widetilde{\operatorname{GPin}}(V), \chi}\subseteq\mathcal{S}^*(\operatorname{GPin}(V)\times V)^{\widetilde{\operatorname{GPin}}(V), \chi}. \end{align*}$$

Let $Y:=\{v\in V\;:\; \langle v,v\rangle =\langle e,e \rangle \}.$ One can also see that Y is invariant under the action of $\operatorname {GPin}(V)$ . By Witt’s theorem, we know that $\operatorname {O}(V)$ acts transitively on Y and hence $\widetilde {\operatorname {GPin}}(V)$ acts transitively on Y.

Now, consider the projection

$$\begin{align*}\phi:X\longrightarrow Y,\quad (g, v)\mapsto v, \end{align*}$$

which is $\widetilde {\operatorname {GPin}}(V)$ -equivariant. Recall that the stabilizer $\widetilde {\operatorname {GPin}}(V)_e=\widetilde {\operatorname {GPin}}(W)$ of e is unimodular. Hence, by the Frobenius descent (Lemma 3.4) applied to this $\phi $ , we obtain the canonical isomorphism

(3.2) $$ \begin{align} \mathcal{S}^*(X)^{\widetilde{\operatorname{GPin}}(V), \chi} \simeq \mathcal{S}^*(\phi^{-1}(e))^{\widetilde{\operatorname{GPin}}(V)_e,\chi}. \end{align} $$

By the obvious identification $\operatorname {GPin}(V)\times \{e\}\simeq \operatorname {GPin}(V)$ of sets, we have

$$\begin{align*}\mathcal{S}^*(\phi^{-1}(e))^{\widetilde{\operatorname{GPin}}(V)_e,\chi}\simeq\mathcal{S}^*(\operatorname{GPin}(V))^{\widetilde{\operatorname{GPin}}(W),\chi}. \end{align*}$$

Hence, we have

$$\begin{align*}\mathcal{S}^*(\operatorname{GPin}(V))^{\widetilde{\operatorname{GPin}}(W),\chi}\simeq \mathcal{S}^*(X)^{\widetilde{\operatorname{GPin}}(V), \chi}\subseteq \mathcal{S}^*(\operatorname{GPin}(V)\times V)^{\widetilde{\operatorname{GPin}}(V), \chi}. \end{align*}$$

The proposition is proven.

3.2 Reduction II: The semisimple orbits

The proof of our main theorem is now reduced to showing the vanishing assertion (3.1). In this section, we further reduce the vanishing assertion to the classical group scenarios as presented in [Reference Sun and ZhuSZ12] or [Reference WaldspurgerWal12].

The main idea is that any distribution in $\mathcal {S}^*({\operatorname {GPin}(V)}\times V)^{\widetilde {\operatorname {GPin}(V)},\chi }$ is supported in a smaller set through Harish-Chandra’s descent and Bernstein’s localization principle. We cite those theorems for the Archimedean case as follows.

Proposition 3.10 Define a map

by

where is the conjugacy class of the semisimple part of $P(g)$ under the Jordan decomposition.

If

$$\begin{align*}\mathcal{S}^*(\theta^{-1}(\gamma))^{\widetilde{\operatorname{GPin}}(V), \chi}=0 \end{align*}$$

for each semisimple conjugacy class , then

$$\begin{align*}\mathcal{S}^*(\operatorname{GPin}(V)\times V)^{\widetilde{\operatorname{GPin}}(V),\chi}=0. \end{align*}$$

3.3 Reduction III: $\operatorname {O}(V)$ situation

In this section, we further reduce the vanishing assertion of the hypothesis in Proposition 3.10 further to the orthogonal groups situation [Reference Sun and ZhuSZ12].

For a semisimple conjugacy class $\gamma \subseteq \operatorname {O}(V)_s$ of $P(g)$ , we cosnider the following map exactly as in [Reference Emory and TakedaET23]:

$$\begin{align*}\theta:\theta^{-1}(\gamma)\longrightarrow \gamma, \quad (g, v)\mapsto P(g)_s. \end{align*}$$

Applying the Frobenius descent to the above map, we have

$$\begin{align*}\mathcal{S}^{*}(\theta^{-1}(\gamma))^{\widetilde{\operatorname{GPin}}(V), \chi}\simeq\mathcal{S}^{*}(Z^\circ g\,\mathcal{U}_g\times V)^{\widetilde{\operatorname{GPin}}(V)_{g}, \chi} \end{align*}$$

since $\theta ^{-1}(P(g))=Z^\circ g\,\mathcal {U}_g\times V$ .

Furthermore, applying the Bernstein localization principle, we have the following.

Proof First, since $zg$ is semisimple for all $z\in Z^\circ $ and $\mathcal {U}_{zg}=\mathcal {U}_g$ , we have

$$\begin{align*}\mathcal{S}^{*}(zg\,\mathcal{U}_g\times V)^{\widetilde{\operatorname{GPin}}(V)_{g}, \chi}=0 \end{align*}$$

for all $z\in Z^\circ $ .

We then consider the following map:

$$\begin{align*}Z^\circ g\,\mathcal{U}_g\times V\longrightarrow Z^\circ,\quad zg\, \mathcal{U}_g\mapsto z. \end{align*}$$

Now, as each fiber $zg\,\mathcal {U}_g\times V$ is preserved by $\widetilde {\operatorname {GPin}}(V)_g$ , we may apply the Bernstein localization principle (Corollary 3.9) to arrive at the desired conclusion.

We finally reduce our main theorem to the $\operatorname {O}(V)$ situation. Recall that $\operatorname {GPin}(V)$ and $\operatorname {O}(V)$ have the same set of unipotent elements. This implies that for each semisimple $g,$ we have the bijection

$$\begin{align*}g\,\mathcal{U}_g\longrightarrow P(g\,\mathcal{U}_g) \end{align*}$$

induced by the canonical projection P. Furthermore, this map intertwines the actions of $\widetilde {\operatorname {GPin}}(V)_g$ and $P(\widetilde {\operatorname {GPin}}(V)_g)$ . Note that the kernel $Z_0$ of P acts trivially on the space $\mathcal {S}^{*}(g\,\mathcal {U}_g\times V)$ . Therefore, we have

$$\begin{align*}\mathcal{S}^{*}(g\,\mathcal{U}_g\times V)^{\widetilde{\operatorname{GPin}}(V)_{g}, \chi}\simeq\mathcal{S}^{*}(P(g\,\mathcal{U}_g)\times V)^{P(\widetilde{\operatorname{GPin}}(V)_g), \chi}. \end{align*}$$

Hence, to show our main theorem, it suffices to show the following $\operatorname {O}(V)$ situation [Reference Sun and ZhuSZ12, Theorem A]:

(3.3) $$ \begin{align} \mathcal{S}^{*}(P(g\,\mathcal{U}_g)\times V)^{P(\widetilde{\operatorname{GPin}}(V)_g), \chi} \subseteq \mathcal{S}^{*}(P(\operatorname{GPin}(V)_g)\times V)^{P(\widetilde{\operatorname{GPin}}(V)_g), \chi}=0 \end{align} $$

for all semisimple $g\in \operatorname {GPin}(V)$ .

3.4 End of proof

In case $P(\operatorname {GPin}(V)_g)=\operatorname {O}(V)_{P(g)}$ , the vanishing assertion (3.3) is equivalent to showing $\mathcal {S}^{*}(\operatorname {O}(V)_{P(g)}\times V)^{\operatorname {O}(V)_{P(g)}, \chi }=0$ , which is precisely the assertion proven in [Reference Sun and ZhuSZ12]. However, we do not always have $P(\operatorname {GPin}(V)_g)=\operatorname {O}(V)_{P(g)}$ as shown in [Reference Emory and TakedaET23, Lemma 3.11]. Accordingly, we need to modify [Reference Sun and ZhuSZ12]. The difference is that $P(\operatorname {GPin}(V)_g)$ might have a factor of $\operatorname {SO}$ as in [Reference Emory and TakedaET23, Lemma 3.11], for which we need the result of Sun–Zhu [Reference Sun and ZhuSZ12] for the $\operatorname {SO}$ case. The complete argument in the case of locally compact totally disconnected space is present in [Reference Emory and TakedaET23]. The same argument works in our case and we do not repeat the same argument.

In conclusion, we have shown the vanishing assertion of (1.1) which gives Theorem 1.1 for the groups $\operatorname {GPin} (V)$ and $\operatorname {GPin} (W)$ .

4.1 Vanishing of distribution

Analogously to the GPin case, we prove the following main technical result:

(4.3) $$ \begin{align} \mathcal{S}^*( \operatorname{GSpin}(V))^{\widetilde{\operatorname{GSpin}}(W), \chi}=0, \end{align} $$

where $\widetilde {\operatorname {GSpin}}(W)\simeq \operatorname {GSpin}(W)\times \{1, e_{n-1}^{k-1}e\beta \}$ acts on $\operatorname {GPin}(V)$ by restricting the actions (4.1). In particular, the element $e_{n-1}^{k-1}e\beta $ acts via the involution $\tau _W$ , which preserves $\operatorname {GSpin}(W)$ setwise. Recall that the action of $\widetilde {\operatorname {GSpin}}(V)$ on $\operatorname {GSpin}(V)\times V$ is defined in (4.1).

Proposition 4.3 We have a natural inclusion

$$\begin{align*}\mathcal{S}^*(\operatorname{GSpin}(V))^{\widetilde{\operatorname{GSpin}}(W), \chi}\subseteq \mathcal{S}^*(\operatorname{GSpin}(V)\times V)^{\widetilde{\operatorname{GSpin}}(V), \chi}. \end{align*}$$

Hence, if

$$\begin{align*}\mathcal{S}^*(\operatorname{GSpin}(V)\times V)^{\widetilde{\operatorname{GSpin}}(V), \chi}=0, \end{align*}$$

then $\mathcal {S}^*(\operatorname {GSpin}(V))^{\widetilde {\operatorname {GSpin}}(W), \chi }=0$ .

Proof This can be proven in the same way as Proposition 3.5. Namely, let

$$ \begin{align*} X&:=\{(g,v)\in\operatorname{GSpin}(V)\times V\;:\; \langle v, v\rangle=\langle e, e\rangle\}\\ Y&:=\{v\in V\;:\; \langle v, v\rangle=\langle e, e\rangle\}, \end{align*} $$

and consider the projection

$$\begin{align*}\phi:X\longrightarrow Y. \end{align*}$$

By Witt’s theorem, $\operatorname {GSpin}(V)$ acts transitively on Y. Moreover, using the actions listed in (4.1), one can show that $\phi (g \cdot (h,v)=g \cdot \phi (h,v)$ for all $g \in \widetilde {\operatorname {GSpin}}(V)$ and $(h,v) \in X$ and so this $\phi $ is a continuous $\widetilde {\operatorname {GSpin}}(V)$ -equivariant Nash map. Thus, by the Frobenius descent, we have

$$\begin{align*}\mathcal{S}^*(X)^{\widetilde{\operatorname{GSpin}}(V), \chi}\simeq \mathcal{S}^*(\operatorname{GSpin}(V)\times\{e\})^{\widetilde{\operatorname{GSpin}}(V)_e, \chi}, \end{align*}$$

where the left-hand side is a subspace of $\mathcal {S}^*(\operatorname {GSpin}(V)\times V)^{\widetilde {\operatorname {GSpin}}(V), \chi }$ . But clearly

$$\begin{align*}\mathcal{S}^*(\operatorname{GSpin}(V)\times\{e\})^{\widetilde{\operatorname{GSpin}}(V)_e, \chi}\simeq \mathcal{S}'(\widetilde{\operatorname{GSpin}}(V))^{\widetilde{\operatorname{GSpin}}(W), \chi}. \end{align*}$$

The proposition follows.

4.2 Reducing to classical group situation

It is now enough to show

$$\begin{align*}\mathcal{S}^*(\operatorname{GSpin}(V)\times V)^{\widetilde{\operatorname{GSpin}}(V), \chi}=0. \end{align*}$$

Arguing in the same way as the $\operatorname {GPin}$ case, this vanishing assertion reduces to

$$\begin{align*}\mathcal{S}^*(g\,\mathcal{U}_g\times V)^{\widetilde{\operatorname{GSpin}}(V)_g, \chi}=0 \end{align*}$$

for all semisimple $g\in \operatorname {GSpin}(V)$ . Since the canonical projection P is bijective on $g\,\mathcal {U}_g$ , we have the natural isomorphism

$$\begin{align*}\mathcal{S}^*(g\,\mathcal{U}_g\times V)^{\widetilde{\operatorname{GSpin}}(V)_g, \chi}\simeq\mathcal{S}'(P(g\,\mathcal{U}_g)\times V)^{P(\widetilde{\operatorname{GSpin}}(V)_g), \chi}. \end{align*}$$

Since we have

$$\begin{align*}\mathcal{S}^*(P(g\,\mathcal{U}_g)\times V)^{P(\widetilde{\operatorname{GSpin}}(V)_g), \chi} \subseteq\mathcal{S}'(P(\operatorname{GSpin}(V)_g)\times V)^{P(\widetilde{\operatorname{GSpin}}(V)_g), \chi}, \end{align*}$$

it is enough to show

$$\begin{align*}\mathcal{S}'(P(\operatorname{GSpin}(V)_g)\times V)^{P(\widetilde{\operatorname{GSpin}}(V)_g), \chi}=0. \end{align*}$$

We know $P(\operatorname {GSpin}(V)_g)$ is as in Lemma 4.2 and $P(\widetilde {\operatorname {GSpin}}(V)_g)$ is generated by $P(\operatorname {GSpin}(V)_g)$ and the element $\gamma \beta $ , where $\gamma =(\gamma _1,\dots ,\gamma _m,\gamma _+,\gamma _-)\in \operatorname {O}(V_1)\times \cdots \times \operatorname {O}(V_m)\times \operatorname {O}(V_+)\times \operatorname {O}(V_-) $ such that $\gamma h^{-1}\gamma ^{-1}=h$ , where $h:=P(g)\in \operatorname {SO}(V)$ as before (see also [Reference Emory and TakedaET23, Equation 8.1]. Note that since the orthogonal factor of $P(\operatorname {GSpin}(V)_g)$ is $\operatorname {SO}(V_+)\times \operatorname {SO}(V_-)$ , we always choose $\gamma _{\pm }=r_{e_{\pm }}^{k_{\pm }}$ . The remaining part of the proof is identical to the proof in the GPin case, and so the proof is complete.

In conclusion, we have shown the vanishing assertion of (1.1) which gives Theorem 1.1 for the groups $\operatorname {GSpin} (V)$ and $\operatorname {GSpin} (W)$ .