Contragredients and a multiplicity one theorem for general spin groups

Abstract

Each orthogonal group ${\text {O}}(n)$ has a nontrivial ${\text {GL}}(1)$-extension, which we call ${\text {GPin}}(n)$. The identity component of ${\text {GPin}}(n)$ is the more familiar ${\text {GSpin}}(n)$, the general Spin group. We prove that the restriction to ${\text {GPin}}(n-1)$ of an irreducible admissible representation of ${\text {GPin}}(n)$ over a nonarchimedean local field of characteristic zero is multiplicity free and also prove the analogous theorem for ${\text {GSpin}}(n)$. Our proof uses the method of Aizenbud, Gourevitch, Rallis and Schiffman, who proved the analogous theorem for ${\text {O}}(n)$, and of Waldspurger, who proved that for ${\text {SO}}(n)$. We also give an explicit description of the contragredient of an irreducible admissible representation of ${\text {GPin}}(n)$ and ${\text {GSpin}}(n)$, which is needed to apply their method to our situations.

Appendices

Appendix A: Centralizer of semisimple element

In this appendix, we reproduce the proof of Proposition 3.2, which gives the explicit description of the centralizer ${\text {O}}(V)_h$ of a semisimple element $h\in {\text {O}}(V)$. Though this is well-known already from the 60’s [14], we reproduce the proof in detail because we have not been able to locate a proof in the literature to the precision we need. The beginning part of our proof is borrowed from [8, p.79-82].

Let $p(x)\in F[x]$ be the minimum polynomial of h, and let

$$\begin{aligned} A:=F[x]/(p(x)). \end{aligned}$$

Since h is invertible, p(x) has a nonzero constant term, which means x is invertible in A. Hence we have the natural isomorphism

$$\begin{aligned} A=F[x]/(p(x))\simeq F[x, x^{-1}]/(p(x)), \end{aligned}$$

where on the right-hand side by (p(x)) we actually mean the ideal $p(x)F[x, x^{-1}]$. On $F[x, x^{-1}]$ we have the involution defined by $x\mapsto x^{-1}$. Since p(x) is a minimum polynomial of an element h in the orthogonal group ${\text {O}}(V)$, one can see that $p(x^{-1})=a x^mp(x)$ for some $a\in F$, where m is the degree of p. (To see this, consider the eigenvalues of p(x) over the algebraic closure.) Namely the involution preserves the ideal $p(x)F[x, x^{-1}]$, which gives rise to the involution

$$\begin{aligned} \sigma :A\longrightarrow A,\quad x\mapsto x^{-1}. \end{aligned}$$

We often use the exponential notation $f^\sigma$ instead of $\sigma (f)$ for $f\in A$.

We view the space V as an A-module in the obvious way, namely $f\cdot v=f(h)v$ for $f\in A$ and $v\in V$. Then for each $f\in A$

$$\begin{aligned} f^\sigma \cdot v=f(h^{-1})v \end{aligned}$$

and

$$\begin{aligned} \langle f\cdot v, v'\rangle =\langle v, f^\sigma \cdot v'\rangle \end{aligned}$$

for $v, v\in V$.

Since h is semisimple, we can write $p(x)=p_1(x)\cdots p_k(x)$, where $p_i(x)$’s are distinct irreducible polynomials, so that we have

$$\begin{aligned} F[x]/(p(x))=F[x]/(p_1(x))\times \cdots \times F[x]/(p_k(x)), \end{aligned}$$

where each

$$\begin{aligned} A_i:=F[x]/(p_i(x)) \end{aligned}$$

is a field because $p_i(x)$ is irreducible. Let $V_i=\ker p_i(h)$. Then we can write

$$\begin{aligned} V=V_1\oplus \cdots \oplus V_k. \end{aligned}$$

Since $hV_i=V_i$, we can view each $V_i$ as an $A_i$-module via $q(x)\cdot v_i=q(h)v_i$ for $q(x)\in A_i$, and hence as an A-module via the canonical surjection $A\rightarrow A_i$.

Since $(p(x)^\sigma )=(p(x))$ viewed in $F[x, x^{-1}]$, for each i we have $(p_i(x)^{\sigma })=(p_{\sigma (i)}(x))$ for some $\sigma (i)\in \{1,\dots , k\}$. There are two possibilities: either $\sigma (i)=i$ or $\sigma (i)\ne i$. Assume $\sigma (i)=i$. In this case, $\sigma$ restricts to an involution on the field $A_i$. Then $V_i$ is orthogonal to all $V_j$ with $i\ne j$, because for each $a_i\in A_i$ we have $\langle a_iv_i, v_j\rangle =\langle v_i, a_i^{\sigma }v_j\rangle =0$ for all $v_i\in V_i$ and $v_j\in V_j$ with $i\ne j$. On the other hand, assume $\sigma (i)\ne i$. One can then similarly see that $V_i\oplus V_{\sigma (i)}$ is orthogonal to all the other $V_j$’s and both $V_i$ and $V_{\sigma (i)}$ are isotropic. Let us set

$$\begin{aligned} B_i={\left\{ \begin{array}{ll} A_i\times A_{\sigma (i)},&{}\text {if}\, \sigma (i)\ne i;\\ A_i,&{}\text {if}\, \sigma (i)=i.\end{array}\right. } \end{aligned}$$

Let us first consider the case $\sigma (i)=i$, so that $B_i=A_i$ is a field with the involution $\sigma$.

Lemma A.1

Assume $B_i=A_i$. Then there is an $A_i$-Hermitian form

$$\begin{aligned} \langle \!\langle -,-\rangle \!\rangle _i:V_i\times V_i\longrightarrow A_i \end{aligned}$$

with respect to $\sigma$, namely

$$\begin{aligned} \langle \!\langle av, v'\rangle \!\rangle _i=a\langle \!\langle v, v'\rangle \!\rangle _i\quad \text {and}\quad \langle \!\langle v, v'\rangle \!\rangle _i^\sigma =\langle \!\langle v', v\rangle \!\rangle _i \end{aligned}$$

for all $v, v'\in V_i$ and $a\in A_i$, such that

$$\begin{aligned} \langle -, -\rangle ={\text {tr}}_{A_i/F}(\langle \!\langle -, -\rangle \!\rangle _i), \end{aligned}$$

where ${\text {tr}}_{A_i/F}:A_i\rightarrow F$ is the trace form.

Proof

We suppress the subscript i to ease the notation, so $A=A_i$, etc. It is elementary to show that any F-linear functional $\ell :A\rightarrow F$ is written as $\ell (a)={\text {tr}}_{A/F}(a\alpha )$ for some $\alpha \in A$. Now, for each fixed $v, v'\in V$ consider the F-linear functional $A\rightarrow F$ by $a\mapsto \langle av, v'\rangle$. Then there exists some $\langle \!\langle v, v'\rangle \!\rangle \in A$ such that

$$\begin{aligned} \langle av, v'\rangle ={\text {tr}}_{A/F}(a\langle \!\langle v, v'\rangle \!\rangle ) \end{aligned}$$

for all $a\in A$. One can readily see that the assignment $\langle \!\langle -,-\rangle \!\rangle :V\times V\rightarrow A$ is a nondegenerate Hermitian form on V over A with respect to the involution $\sigma$. $\square$

In the above lemma, it should be noted that if the involution $\sigma$ on $A_i$ is trivial then the polynomial $p_i(x)$ has to be either $p_i(x)=x-1$ or $p_i(x)=x+1$, in which case $A=F$ and the Hermitian form $\langle \!\langle -,-\rangle \!\rangle _i$ on $V_i$ is simply the restriction of our symmetric bilinear form $\langle -,-\rangle$. For $p_i(x)=x-1$ we set $V_+=V_i$ and $A_+=A_i$, and for $p_i(x)=x+1$ we set $V_-=V_i$ and $A_-=A_i$. (Of course $V_+$ or $V_-$ can be zero, depending on h.)

Next consider the case $\sigma (i)\ne i$. Let us set $j=\sigma (i)$, so that

$$\begin{aligned} B_i=A_i\times A_j. \end{aligned}$$

We then have the field isomorphism

$$\begin{aligned} \sigma :A_i=F[x]/(p_i(x))\xrightarrow {\;\sim \;} F[x]/(p_j(x))=A_j,\quad f(x)\mapsto f(x)^\sigma . \end{aligned}$$

Note that under this isomorphism we have $x\mapsto x^{-1}$. By identifying $A_j$ with $B_i$ under this isomorphism, we can write

$$\begin{aligned} B_i=A_i\times A_i. \end{aligned}$$

Since the identification of $A_j$ with $A_i$ is made via $\sigma$, the involution $\sigma$ acts on $B_i=A_i\times A_i$ as switching the two factors.

Let $(h_i, h_j)\in A_i\times A_j$ be the image of h in $B_i$. Since the isomorphism $A_i\rightarrow A_j$ maps x to $x^{-1}$, under the identification $B_i=A_i\times A_i$ we have

$$\begin{aligned} (h_i, h_j)=(h_i, h_i^{-1}). \end{aligned}$$

We often write

$$\begin{aligned} h_i=(h_i, h_i^{-1}) \end{aligned}$$

by slight abuse of notation. With this said, we have the following.

Lemma A.2

Assume $B_i=A_i\times A_i$, so that $V_i$ and $V_{\sigma (i)}$ are $A_i$-vector spaces. Recall both $V_i$ and $V_{\sigma (i)}$ are totally isotropic such that the restriction of our symmetric form $\langle -,-\rangle$ on the sum $V_i\oplus V_{\sigma (i)}$ is nondegenerate. Then there exists a nondegenerate $A_i$-bilinear pairing

$$\begin{aligned} \langle \!\langle -,-\rangle \!\rangle _i:V_i\times V_{\sigma (i)}\longrightarrow A_i \end{aligned}$$

such that

$$\begin{aligned} \langle -,-\rangle ={\text {tr}}_{A_i/F}(\langle \!\langle -,-\rangle \!\rangle _i). \end{aligned}$$

Via this bilinear pairing, we have the identification

$$\begin{aligned} V_{\sigma (i)}=V_i^*={{\,\textrm{Hom}\,}}_F(V_i, F). \end{aligned}$$

Proof

The proof is essentially the same as the other case. Again let us suppress the subscript i, and write $V_{\sigma (i)}=V_\sigma$. For each fixed $v\in V$ and $v'\in V_\sigma$, define the F-linear form on A by

$$\begin{aligned} a\mapsto \langle av, v'\rangle . \end{aligned}$$

Then there exists a unique element $\langle \!\langle v,v'\rangle \!\rangle \in A$ such that

$$\begin{aligned} \langle av, v'\rangle ={\text {tr}}_{A/F}(a\langle \!\langle v,v'\rangle \!\rangle ). \end{aligned}$$

The assignment $\langle \!\langle -,-\rangle \!\rangle :V\times V_\sigma \rightarrow A$ is indeed a nondegenerate A-bilinear pairing. $\square$

In the above case, let us write

$$\begin{aligned} X_i=V_i\quad \text {and}\quad X_i^*=V_{\sigma (i)}. \end{aligned}$$

It should be noted that we have the natural isomorphism

$$\begin{aligned} {{\,\textrm{Hom}\,}}_{A_i}(X_i, A_i)\xrightarrow {\;\sim \;}{{\,\textrm{Hom}\,}}_F(X_i, F),\quad \ell \mapsto {\text {tr}}_{A_i/F}\circ \ell , \end{aligned}$$

of F-vector spaces. Hence the dual $X_i^*$ can be interpreted either over F or over $A_i$.

Now, by re-choosing the indices we can write

$$\begin{aligned} V=(X_1\oplus X_1^*)\oplus \cdots \oplus (X_\ell \oplus X_\ell ^*)\oplus V_{\ell +1}\oplus \cdots \oplus V_m\oplus V_+\oplus V_-, \end{aligned}$$

and

$$\begin{aligned} A=B_1\times \cdots \times B_\ell \times A_{\ell +1}\times \cdots \times A_m\times A_+\times A_-, \end{aligned}$$

where

• (a) for $i=1,\dots , \ell$, we have $B_i=A_i\times A_i$, and $X_i$ is an $A_i$-vector space and $X_i^*$ its dual,

• (b) for $i=\ell +1,\dots , m$, we have that $A_i$ is a field and $V_i$ is equipped with a Hermitian form over $A_i$, and

• (c) $A_{\pm }=F$ and $V_{\pm }$ is a nondegenerate quadratic subspace of V.

Our involution $\sigma$ on A restricts an involution on each $B_i$, and we write

$$\begin{aligned} \sigma =\sigma _1\otimes \cdots \otimes \sigma _m\otimes \sigma _+\otimes \sigma _-, \end{aligned}$$

where on $B_i=A_i\times A_i$ the involution $\sigma _i$ switches the two factors, on $B_i=A_i$ the involution $\sigma _i$ is of the second kind and on $A_{\pm }$ the involution $\sigma _{\pm }$ is trivial.

If we view our h as an element of A, we can write

$$\begin{aligned} h=(h_1,\ldots ,h_m, h_+, h_-), \end{aligned}$$

where $h_i\in B_i$ and $h_{\pm }=1_{V_{\pm }}$. Recall by our convention that if $B_i=A_i\times A_i$ then

$$\begin{aligned} h_i=(h_i, h_i^{-1}). \end{aligned}$$

Then

$$\begin{aligned} \sigma (h)=h^{-1}&=(h_1^{-1},\ldots , h_m^{-1}, h_+, h_-)\\&=(\sigma _1(h_1),\ldots ,\sigma _m(h_m),\sigma _+(h_+), \sigma _-(h_-)), \end{aligned}$$

where if $B_i=A_i\times A_i$ then $\sigma _i(h_i)$ is actually

$$\begin{aligned} \sigma _i(h_i, h_i^{-1})=(h_i^{-1}, h_i), \end{aligned}$$

because $\sigma _i$ switches the two factors of $A_i\times A_i$. Also note that if $B_i=A_i\ne A_{\pm }$ then $\sigma _i$ is a Galois conjugation, and hence $h_i\in A_i$ is such that

$$\begin{aligned} \sigma _i(h_i)=h_i^{-1}, \end{aligned}$$

namely $h_i$ is a norm one element in $A_i$.

We then have the following.

Proposition A.3

The centralizer ${\text {O}}(V)_h$ is of the form

$$\begin{aligned} {\text {O}}(V)_h\simeq G_1\times \cdots \times G_m\times {\text {O}}(V_+)\times {\text {O}}(V_-), \end{aligned}$$

where

$$\begin{aligned} G_i={\left\{ \begin{array}{ll}{\text {GL}}_{A_i}(X_i), &{}\text {if}\, B_i=A_i\times A_i;\\ U_{A_i}(V_i),&{}\text {if}\, B_i=A_i.\end{array}\right. } \end{aligned}$$

Here by ${\text {GL}}_{A_i}(V_i)$ we actually mean the “diagonal”

$$\begin{aligned} {\text {GL}}_{A_i}(X_i)\simeq \{(g_i, {g_i^*}^{-1})\;:\;g_i\in {\text {GL}}_{A_i}(X_i)\}\subseteq {\text {GL}}_{A_i}(X_i)\times {\text {GL}}_{A_i}(X_i^*), \end{aligned}$$

where $g_i^*$ is the adjoint of $g_i$ with resect to the canonical pairing $X_i\times X_i^*\rightarrow A_i$, and by $U_{A_i}(V_i)$ we mean the unitary group for the Hermitian space $V_i$ over $A_i$.

Further, if $B_i=A_i\times A_i$ then each $h_i=(h_i, h_i^{-1})$ is viewed as the central element $h_iI_{X_i}$ of ${\text {GL}}_{A_i}(X_i)$, and if $B_i=A_i$ (including $A_{\pm }$) then each $h_i$ is the central element $h_iI_{V_i}$ of $U_{A_i}(V_i)$.

Proof

Let $g\in {\text {O}}(V)_h$. Assume $B_i=A_i\times A_i$. Since $V_i=\ker p_i(h)$, one can readily see that g preserves each of the spaces $X_i$ and $X_i^*$. Let $g_i$ be the restriction of g on $X_i$ and $g_i'$ that on $X_i^*$. Note that at this point, $g_i$ and $g_i'$ are only F-linear.

Then $(g_i, g_i')\in {\text {GL}}_F(X_i)\times {\text {GL}}_F(X_i^*)$ commutes with h if and only if $g_i$ and $g_i'$ are $A_i$-linear because $A_i$ is the field $F[x]/(p_i(x))$ which acts via the evaluation at $x=h$. Further $(g_i, g_i')$ preserves the original form $\langle -,-\rangle$ if and only if

$$\begin{aligned} \langle \!\langle g_i v_i, g_i'v_i\rangle \!\rangle _i=\langle \!\langle v_i, v_i'\rangle \!\rangle \end{aligned}$$

for all $v_i\in V_i$ and $v_i^*\in V_i^*$, where $\langle \!\langle -,-\rangle \!\rangle _i$ is the canonical pairing. Hence we must have $g_i'={g_i^*}^{-1}$, where $g_i^*$ is the adjoint of $g_i$ with respect to $\langle \!\langle -,-\rangle \!\rangle _i$. This shows that the set of all $(g_i, g_i')$ commuting with h is of the form

$$\begin{aligned} \{(g_i, {g_i^*}^{-1})\;:\;g_i\in {\text {GL}}_{A_i}(X_i)\}, \end{aligned}$$

which is isomorphic to ${\text {GL}}_{A_i}(X_i)$.

Assume $B_i=A_i$ (including $A_{\pm }$). Then one can see that g preserves the space $V_i=\ker p_i(h)$. Let $g_i\in {\text {GL}}_F(V_i)$ be the restriction of g to $V_i$. Then $g_i$ commutes with h if and only if $g_i$ is $A_i$-linear. Also $g_i$ preserves the original form $\langle -,-\rangle$ if and only if it preserves the form $\langle \!\langle -,-\rangle \!\rangle _i$. This shows that $g_i\in U_{A_i}(V_i)$. $\square$

One can see that this is precisely Proposition 3.2.

Let us mention that if $B_i=A_i\times A_i$ then by Lemma A.2 we know that ${\text {GL}}_{A_i}(X_i)$ is in the Siegel Levi of the special orthogonal group ${\text {SO}}(X_i\oplus X_i^*)$, and in particular

$$\begin{aligned} {\text {GL}}_{A_i}(X_i)\subseteq {\text {SO}}(X_i\oplus X_i^*). \end{aligned}$$

If $B_i=A_i$ but not equal to $A_{\pm }$, then by Lemma A.1 we have

$$\begin{aligned} U_{A_i}(V_i)\subseteq {\text {SO}}(V_i). \end{aligned}$$

Note that $U_{A_i}(V_i)$ is in the special orthogonal group ${\text {SO}}(V_i)$ instead of just the orthogonal group ${\text {O}}(V_i)$ because the unitary group $U_{A_i}(V_i)$ is connected.

Finally, let us mention the ${\text {SO}}(V)$-analogue of the above proposition, whose proof is left to the reader.

Proposition A.4

Keep the above notation. Let $h\in {\text {SO}}(V)$ be semisimple. The centralizer ${\text {SO}}(V)_h$ is of the form

$$\begin{aligned} {\text {SO}}(V)_h\simeq G_1\times \cdots \times G_m\times S\big ({\text {O}}(V_+)\times {\text {O}}(V_-)\big ), \end{aligned}$$

where

$$\begin{aligned} G_i={\left\{ \begin{array}{ll}{\text {GL}}_{A_i}(X_i), &{}\text {if}\, B_i=A_i\times A_i;\\ U_{A_i}(V_i),&{}\text {if}\, B_i=A_i,\end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} S\big ({\text {O}}(V_+)\times {\text {O}}(V_-)\big )=\big ({\text {O}}(V_+)\times {\text {O}}(V_-)\big )\cap {\text {SO}}(V_+\oplus V_-), \end{aligned}$$

and further $\dim _FV_-$ is always even.

Appendix B: Summary of involutions

In this appendix, we summarize the involutions we use in this paper.

Canonical involution $g^*$: For $g\in {\text {GPin}}(V)$, the canonical involution $g^*$ is defined by reversing the order of the vectors that appear in g viewed in the Clifford algebra C(V); namely if we write $g=v_1v_2\cdots v_\ell$, where $v_i\in V$, then

$$\begin{aligned} g^*=(v_1v_2\cdots v_\ell )^*=v_\ell v_{\ell -1}\cdots v_1. \end{aligned}$$

Clifford involution $\overline{g}$: For $g\in {\text {GPin}}(V)$, the Clifford involution $\overline{g}$ is defined as the “signed canonical involution”; namely for $g=v_1v_2\cdots v_\ell$,

$$\begin{aligned} \overline{g}=(-1)^\ell (v_1v_2\cdots v_\ell )^*=(-1)^\ell v_\ell v_{\ell -1}\cdots v_1. \end{aligned}$$

In other words

$$\begin{aligned} \overline{g}={\text {sign}}(g)g^*, \end{aligned}$$

where ${\text {sign}}:{\text {GPin}}(V)\rightarrow \{\pm 1\}$ is the sign map that sends the nonidentity component to $-1$. Note that the Clifford norm $N:{\text {GPin}}(V)\rightarrow F^\times$ is defined by $N(g)=g\,\overline{g}$, so that

$$\begin{aligned} g^{-1}=\frac{1}{N(g)}\overline{g}. \end{aligned}$$

Involution $\sigma _V$: The involution $\sigma _V$ is the involution on ${\text {GPin}}(V)$ defined by

$$\begin{aligned} \sigma _V(g)={\left\{ \begin{array}{ll}g^*&{}\text {if}\, n=2k;\\ {\text {sign}}(g)^{k+1}g^*&{}\text {if}\, n=2k-1.\end{array}\right. } \end{aligned}$$

The important property of $\sigma _V$ is that it preserves the semisimple conjugacy classes of ${\text {GPin}}(V)$. This, in particular, implies $\pi ^\vee \simeq \pi ^\sigma$ for all $\pi \in {{\,\textrm{Irr}\,}}({\text {GPin}}(V))$, where $\pi ^{\sigma }(g):=\pi (\sigma _V(g)^{-1})$. Also, this property allows us to reduce the vanishing of invariant distributions to semisimple orbits by using Bernstein’s localization principle.

Remark B.1

All the three involutions (canonical, Clifford and $\sigma _V$) are equal on ${\text {GSpin}}(V)$.

Involution $e^{k}\sigma _V(g)e^{-k}$: The involution $g\mapsto e^k\sigma _V(g) e^{-k}$ on ${\text {GPin}}(V)$ is also defined. This involution preserves the semisimple conjugacy classes of ${\text {GSpin}}(V)$, and hence plays the same role as $\sigma _V$ of the ${\text {GPin}}(V)$ case.

Group $\widetilde{{\text {GPin}}}(V)$ and Involution $\tau _W$: The group $\widetilde{{\text {GPin}}}(V)$ is defined as

$$\begin{aligned} \widetilde{{\text {GPin}}}(V)=\big \langle g,\,\beta \;:\;g\in {\text {GPin}}(V)\big \rangle \end{aligned}$$

with the relations $g\beta =\beta g$ and $\beta ^2=1$, namely

$$\begin{aligned} \widetilde{{\text {GPin}}}(V)={\text {GPin}}(V)\times \{1,\beta \}. \end{aligned}$$

The action of $\widetilde{{\text {GPin}}}(V)$ on the set ${\text {GPin}}(V)\times V$ is defined as in (5.10). In particular, $\beta$ acts on ${\text {GPin}}(V)$ via the involution $\sigma _V$.

Assume $V=W\oplus Fe$, where e is anisotropic. We set

$$\begin{aligned} \widetilde{{\text {GPin}}}(W):=\widetilde{{\text {GPin}}}(V)_e=\big \langle g, e\beta \;:\;g\in {\text {GPin}}(W)\big \rangle , \end{aligned}$$

so that

$$\begin{aligned} \widetilde{{\text {GPin}}}(W)\simeq {\text {GPin}}(W)\rtimes \{1, e\beta \}. \end{aligned}$$

The involution $\tau _W$ on ${\text {GPin}}(V)$ is defined by $\tau _W(g)=e\sigma _V(g) e^{-1}$, and the element $e\beta$ acts on ${\text {GPin}}(V)$ via this involution.

Since $\tau _W({\text {GPin}}(W))={\text {GPin}}(W)$, the involution $\tau _W$ acts on the space

$$\begin{aligned} \mathcal {S}'({\text {GPin}}(V))^{{\text {GPin}}(W)} \end{aligned}$$

of the ${\text {GPin}}(V)$ invariant distributions. We showed that the $-1$-eigenspace of the involution $\tau _W$ vanishes, which is equivalent to the assertion

$$\begin{aligned} \mathcal {S}'({\text {GPin}}(V))^{\widetilde{{\text {GPin}}}(W),\chi }=0. \end{aligned}$$

However, we reduce this vanishing assertion to

$$\begin{aligned} \mathcal {S}'({\text {GPin}}(V)\times V)^{\widetilde{{\text {GPin}}}(V),\chi }=0, \end{aligned}$$

where the space W no longer appears. Hence the involution $\tau _W$ does not play any direct role in our proof.

Group $\widetilde{{\text {GSpin}}}(V)$ and Involution $\tau _W$: The group $\widetilde{{\text {GSpin}}}(V)$ is defined as

$$\begin{aligned} \widetilde{{\text {GSpin}}}(V)=\big \langle g,\,e^k\beta \;:\;g\in {\text {GPin}}(V)\big \rangle \subseteq \widetilde{{\text {GPin}}}(V), \end{aligned}$$

so that

$$\begin{aligned} \widetilde{{\text {GSpin}}}(V)\simeq {\text {GSpin}}(V)\rtimes \{1, e^k\beta \}. \end{aligned}$$

The action of $\widetilde{{\text {GSpin}}}(V)$ on the set ${\text {GSpin}}(V)\times V$ is simply the restriction of the action of $\widetilde{{\text {GPin}}}(V)$ as in (9.1). In particular, $e^k\beta$ acts on ${\text {GSpin}}(V)$ via the involution $g\mapsto e^k\sigma _V(g)e^{-k}$, which preserves the semisimple conjugacy classes of ${\text {GSpin}}(V)$.

Assume $V=W\oplus Fe$, where e is anisotropic and fix an orthogonal basis $\{e_1,\dots , e_{n-1}\}$ of W. We set

$$\begin{aligned} \widetilde{{\text {GSpin}}}(W)\simeq \widetilde{{\text {GSpin}}}(V)_e=\big \langle g, e_{n-1}^{k-1}e\beta \;:\;g\in {\text {GSpin}}(W)\big \rangle , \end{aligned}$$

so that

$$\begin{aligned} \widetilde{{\text {GSpin}}}(W)={\text {GSpin}}(W)\rtimes \{1, e_{n-1}^{k-1}e\beta \}. \end{aligned}$$

The involution $\tau _W$ on ${\text {GSpin}}(V)$ is defined by

$$\begin{aligned} \tau _W(g)=(e_{n-1}^{k-1}e)\sigma _V(g) (e_{n-1}^{k-1}e)^{-1}, \end{aligned}$$

and the element $e_{n-1}^{k-1}e\beta$ acts on ${\text {GPin}}(V)$ via this involution. This involution plays the same role as the $\tau _W$ of the ${\text {GPin}}$ case.

Our main theorem can be shown by showing the vanishing assertion

$$\begin{aligned} \mathcal {S}'({\text {GSpin}}(V))^{\widetilde{{\text {GSpin}}}(W),\chi }=0 \end{aligned}$$

just as the ${\text {GPin}}$ case.

Involution $\tau _V$ on classical groups: Let $G(V)={\text {GL}}(V), {\text {U}}(V), {\text {O}}(V)$ or ${\text {SO}}(V)$. The involution $\tau _V$ on G(V) is defined as follows:

$$\begin{aligned} \tau _V(g)= {\left\{ \begin{array}{ll} g^t&{}\text {for}\, {\text {GL}}(V);\\ \beta g^{-1}\beta &{}\text {for}\, {\text {U}}(V);\\ g^{-1}&{}\text {for}\, {\text {O}}(V);\\ r_e^{k}g^{-1}r_e^{-k}&{}\text {for}\, {\text {SO}}(V), \end{array}\right. } \end{aligned}$$

where $\beta :V\rightarrow V$ for ${\text {U}}(V)$ is Galois conjugation and $r_e\in {\text {O}}(V)$ is the reflection in the hyperplane orthogonal to e. This involution preserves the (semisimple) conjugacy classes of G(V), and hence plays the same role as our $\sigma _V$ for the ${\text {GPin}}(V)$ case.

The groups $\widetilde{G}(V)$ and $\widetilde{G}(W)$ and the involution $\tau _W$ are defined similarly to the ${\text {GPin}}(V)$ case.

Remark B.2

It should be pointed out here that one can show $\tau _V(g)$ and g are conjugate in G(V) not just for semisimple $g\in G(V)$ but for all $g\in G(V)$. (See [8, I.2. Proposition, p.79].) This allows one to prove the assertion on contragredient without using Harish-Chandra’s regularity theorem. Thus the existence of MVW-involution can be shown for G(V) even when the characteristic of F is not zero.