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Contragredients and a multiplicity one theorem for general spin groups

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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.

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Notes

  1. According to [3, p.3] the term Pin was coined by J-P. Serre as a joke, though we do not know exactly what the joke was.

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Acknowledgements

Melissa Emory was partially supported by an AMS-Simons Travel Award, the NSF grant DMS-2002085, and FY2019 JSPS Postdoctoral Fellowship for Research in Japan (short term). Shuichiro Takeda was partially supported by the Simons Foundations Collaboration Grant #584704. This project was initiated while both of the authors were attending the conference “On the Langlands Program: Endoscopy and Beyond” from Dec 2018 to Jan 2019 at the Institute for Mathematical Sciences in Singapore, and part of the research was done while they were attending the Oberwolfach workshop “New developments in representation theory of p-adic groups” in October 2019. We would like to thank their hospitality. Melissa Emory would like to thank Hiraku Atobe for helpful conversations and his hospitality while hosting her for a week at Hokkaido University, and would like to thank Kyoto University for their hospitality while hosting her fellowship during the summer of 2019, and thank Atsushi Ichino for his interest in this project. Also part of the paper was completed while Shuichiro Takeda was visiting the National University of Singapore in spring 2020, and he would like to thank their hospitality, and would like to thank Wee Teck Gan for his interest in this project. Lastly, the authors would like thank the anonymous referee for his/her helpful comments.

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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

  1. (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,

  2. (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

  3. (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.

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Emory, M., Takeda, S. Contragredients and a multiplicity one theorem for general spin groups. Math. Z. 303, 70 (2023). https://doi.org/10.1007/s00209-023-03228-3

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