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Multiplicity One Theorem for General Spin Groups:
The Archimedean Case

Melissa Emory Department of Mathematics, Oklahoma State University; 401 Mathematical Sciences, Oklahoma State University Stillwater, OK 74078 melissa.emory@okstate.edu ,  Yeansu Kim Department of Mathematics Education, Chonnam National University;77 Yongbong-ro, Buk-gu, Gwangju, Korea ykim@jnu.ac.kr  and  Ayan Maiti Department of Mathematics, Purdue University; 150 North University Street, West Lafayette, Indiana 47907 maitia@purdue.edu
Abstract.

Let GSpin⁡(V)\operatorname{GSpin}(V) (resp. GPin⁡(V)\operatorname{GPin}(V)) be a general spin group (resp. a general Pin group) associated with a nondegenerate quadratic space VV of dimension nn over an Archimedean local field FF. For a nondegenerate quadratic space WW of dimension n−1n-1 over FF, we also consider GSpin⁡(W)\operatorname{GSpin}(W) and GPin⁡(W)\operatorname{GPin}(W). We prove the multiplicity-at-most-one theorem in the Archimedean case for a pair of groups (GSpin⁡(V),GSpin⁡(W)\operatorname{GSpin}(V),\operatorname{GSpin}(W)) and also for a pair of groups (GPin⁡(V),GPin⁡(W)\operatorname{GPin}(V),\operatorname{GPin}(W)); namely, we prove that the restriction to GSpin⁡(W)\operatorname{GSpin}(W) (resp. GPin⁡(W)\operatorname{GPin}(W)) of an irreducible Casselman-Wallach representation of GSpin⁡(V)\operatorname{GSpin}(V) (resp. GPin⁡(V)\operatorname{GPin}(V)) is multiplicity free.

1. Introduction

Restriction problems are one of the most natural problems regarding representations, and can be formulated as follows. Let G{\rm G} be a reductive group over a local field FF of characteristic zero. Let G′{\rm G}^{\prime} be a reductive subgroup of G{\rm G}, also defined over FF. Let also G′:=G′​(F)G^{\prime}:={\rm G}^{\prime}(F) and G:=G​(F)G:={\rm G}(F) be their FF-points. When π\pi and π′\pi^{\prime} are irreducible admissible representations of GG and G′G^{\prime}, respectively, the restriction problem asks how many times π′\pi^{\prime} appears as a quotient of π\pi when π\pi is restricted to G′G^{\prime}. Formally it asks about the dimension (over ℂ\mathbb{C}) of the following vector space:

HomG′​(π,π′),\text{Hom}_{G^{\prime}}(\pi,\pi^{\prime}),

where π\pi in the Hom\operatorname{Hom} space is thought of as the restriction to G′G^{\prime}. There are two flavors to this problem. One is about the non-vanishing of the above Hom space, which is codified as the Gan-Gross-Prasad conjecture [GGP12]. Another question concerns if the Hom space has dimension at most one, i.e. whether

dimℂ​(HomG′​(π,π′))≤1\text{dim}_{\mathbb{C}}(\text{Hom}_{G^{\prime}}(\pi,\pi^{\prime}))\leq 1

(which is known as a multiplicity-at-most-one or multiplicity-free theorem). For a reductive group GG and its reductive subgroup G′G^{\prime}, the multiplicity-at-most-one theorem in the Archimedean case (i.e., when FF is an Archimedean local field) explores how many times an irreducible Casselman-Wallach representation of G′G^{\prime} appears in the restriction to G′G^{\prime} of an irreducible Casselman-Wallach representation of GG. The Archimedean case of the multiplicity-at-most-one theorem is proven for classical groups by Sun and Zhu in [SZ12]. Using a different method of proof, Aizenbud and Gourevitch also handle the case for general linear groups in [AG09b]. The main purpose of this paper is to prove the multiplicity-at-most-one theorem in the Archimedean case for a pair of two reductive non-classical groups: the general Pin groups and the general Spin groups. More specifically, let FF be an Archimedean local field, i.e., either ℝ\mathbb{R} or ℂ\mathbb{C}, and let VV be a nondegenerate quadratic space over FF with dimension nn. Correspondingly, we let GG be the FF-points of either a general Pin group or a general Spin group defined over FF, which is denoted as

GPin⁡(V),GSpin⁡(V)\operatorname{GPin}(V),\ \ \operatorname{GSpin}(V)

(see Section 2.3.2 for exact definitions) and let G′G^{\prime} be respectively its subgroups

GPin⁡(W),GSpin⁡(W)\operatorname{GPin}(W),\ \ \operatorname{GSpin}(W)

where W⊂VW\subset V is a nondegenerate subspace of dimension n−1n-1. The multiplicity-at-most-one theorem that we prove in this paper is the following:

Theorem 1.1.

Let (G,G′)(G,G^{\prime}) be either (GPin⁡(V),GPin⁡(W))(\operatorname{GPin}(V),\operatorname{GPin}(W)) or (GSpin⁡(V),GSpin⁡(W))(\operatorname{GSpin}(V),\operatorname{GSpin}(W)) and let π\pi be an irreducible Cassleman-Wallach representation of GG and π′\pi^{\prime} be an irredcuible Casselman-Wallach representation of G′G^{\prime}. Then the space of G′G^{\prime}-invariant continuous bilinear functional on π×π′\pi\times\pi^{\prime} has dimension at most one, i.e.

dimℂHomG′⁡(π′,π)≤1.\dim_{\mathbb{C}}\operatorname{Hom}_{G^{\prime}}(\pi^{\prime},\pi)\leq 1.

The technical result that will imply the above theorem (through a version of Gelfand-Kazhdan criterion) is as follows:

Theorem 1.2.

Let (G,G′)(G,G^{\prime}) be either (GPin⁡(V),GPin⁡(W))(\operatorname{GPin}(V),\operatorname{GPin}(W)) or (GSpin⁡(V),GSpin⁡(W))(\operatorname{GSpin}(V),\operatorname{GSpin}(W)). Then there exists a real algebraic anti-automorphism τW\tau_{W} on G′G^{\prime} preserving GG with the following property: every generalized function on GG which is invariant under the adjoint action of G′G^{\prime} is automatically τW\tau_{W}-invariant.

The implication of Theorem 1.1 from Theorem 1.2 follows from [SZ11, Corollary 2.5]. To be clear, we have

Proposition 1.3.

Theorem 1.2 implies Theorem 1.1.

Remark 1.4.

A main ingredient in [ET23] is that they give an explicit description of the contragredient of an irreducible admissible representation of GSpin\operatorname{GSpin} and GPin\operatorname{GPin} which is necessary to prove the equivalent of Proposition 1.3. Thanks to [SZ11, Corollary 2.5], this is not required in the Archimedean case.

Let VV be a vector space defined over either ℝ\mathbb{R} or ℂ\mathbb{C} with dimension nn. We denote GPin~(V):=⟨g,β:g∈GPin(V)⟩\widetilde{{\rm GPin}}(V):=\langle g,\beta:g\in{\rm GPin}(V)\rangle such that g​β=β​gg\beta=\beta g for all g∈GPin⁡(V)g\in\operatorname{GPin}(V) and β2=1\beta^{2}=1. Let Span{e1,…,en}=V\{e_{1},\dots,e_{n}\}=V, where ei′e_{i}^{\prime}s constitute an orthogonal basis. Let

W=S​p​a​n​{e1,…,en−1}​ so that ​V=W⊕F​e​ with ​e:=en.W=Span\{e_{1},\dots,e_{n-1}\}\text{ so that }V=W\oplus Fe\textit{ with }e:=e_{n}.

Let GPin~​(W)≃GPin⁡(W)⋊{1,e​β}\widetilde{\operatorname{GPin}}(W)\simeq\operatorname{GPin}(W)\rtimes\{1,e\beta\} act on GPin⁡(V)\operatorname{GPin}(V) (viewed merely as a set) by letting GPin⁡(W)\operatorname{GPin}(W) act by conjugation and e​βe\beta by τW\tau_{W} (defined in eq 2.2). Let χ\chi be the surjection from GPin~​(W)\widetilde{\operatorname{GPin}}(W) to {±1}\{\pm 1\}, which sends e​βe\beta to −1-1 and let χ\chi be a continuous complex character of GPin⁡(W)\operatorname{GPin}(W). We denote 𝒮∗​(GPin⁡(V))GPin~​(W),χ\mathcal{S}^{*}(\operatorname{GPin}(V))^{\widetilde{\operatorname{GPin}}(W),\chi} to be the space of Schwartz distributions, in which GPin~​(W)\widetilde{\operatorname{GPin}}(W) acts via χ\chi.

Theorem 1.2 can be reduced to the following vanishing assertions:

(1.1) 𝒮∗​(GPin⁡(V))GPin~​(W),χ=0;\mathcal{S}^{*}(\operatorname{GPin}(V))^{\widetilde{\operatorname{GPin}}(W),\chi}=0;
(1.2) 𝒮∗​(GSpin⁡(V))GSpin~​(W),χ=0.\mathcal{S}^{*}(\operatorname{GSpin}(V))^{\widetilde{\operatorname{GSpin}}(W),\chi}=0.

Namely, every GPin⁡(W)\operatorname{GPin}(W)-invariant (respectively GSpin\operatorname{GSpin}-invariant) distribution on GPin⁡(V)\operatorname{GPin}(V) (resp. GSpin)\operatorname{GSpin}) is also invariant under the involution τW\tau_{W}. The involution τW\tau_{W} which satisfies (1.1) and (1.2) is defined in (2.2) and (4.2). Thanks to Proposition 1.3 we only need to show the vanishing assertion of (1.1) and we are done.

The aim of this article is to prove the above vanishing assertion in the Archimedean case. In the non-archimedean case, the multiplicity-at-most-one theorem is proven in [AGRS10], [Wal12] and [ET23] for the classical groups, the general Spin groups, and general Pin groups, respectively.

This work completes the multiplicity-at-most-one theorem for the general Spin groups and the general Pin groups and completes the first step in proving the Gas-Gross-Prasad conjecture for general Spin groups. These general Spin groups are of interest in part because they are a GL1\operatorname{GL}_{1} extension of the special orthogonal groups, and the representation theory of GSpin\operatorname{GSpin} completely subsumes that of SOn\operatorname{SO}_{n}. In addition, work on GSpin\operatorname{GSpin} groups is needed for arithmetic application purposes as mentioned in [MP16]; orthogonal Shimura varieties are Shimura varieties of abelian type, but are finite ´etale quotients of GSpin Shimura varieties. According to [MP16] one can easily deduce results for them from the corresponding ones for their GSpin counterparts.

The organization of the paper is as follows: in Section 2, we provide the background and preliminaries including definitions of the groups GPin\operatorname{GPin} and GSpin\operatorname{GSpin}. In Section 3, we prove our main theorem for GPin\operatorname{GPin} groups. First, we prove that our main theorem for GPin\operatorname{GPin} groups reduces to Theorem 3.3, which is the technical heart of our paper. The proof of Theorem 3.3 consists of four parts: elimination of VV, reduction to the semisimple orbits, reduction to classical groups case, and final step of proof. Subsequently, in Section 4 we prove the analogous results for GSpin\operatorname{GSpin} groups.

1.1. Notations and conventions

Throughout this article we assume our field FF to be either ℝ\mathbb{R} or ℂ\mathbb{C}. We assume VV to be a vector space defined over FF with dimension nn. We write {e1,…,en}\{e_{1},\dots,e_{n}\} for an orthogonal basis of VV and let

W=S​p​a​n​{e1,…,en−1}​ so that ​V=W⊕F​e​ with ​e:=en.W=Span\{e_{1},\dots,e_{n-1}\}\text{ so that }V=W\oplus Fe\textit{ with }e:=e_{n}.

We denote X=GPin⁡(V)×VX=\operatorname{GPin}(V)\times V, which will be a Nash Manifold for our purpose. Let 𝔤​𝔭​𝔦​𝔫:=L​i​e​(GPin⁡(V))\mathfrak{gpin}:=Lie(\operatorname{GPin}(V)) be the Lie group of GPin⁡(V)\operatorname{GPin}(V). We denote GPin~(V):=⟨g,β:g∈GPin(V)⟩\widetilde{\operatorname{GPin}}(V):=\langle g,\beta:g\in\operatorname{GPin}(V)\rangle, such that g​β=β​gg\beta=\beta g for all g∈GPin⁡(V)g\in\operatorname{GPin}(V) and β2=1\beta^{2}=1. Therefore we have the following short exact sequence:

1⟶GPin⁡(V)⟶GPin~​(V)→𝜒{±1}⟶1,1\longrightarrow\operatorname{GPin}(V)\longrightarrow\widetilde{\operatorname{GPin}}(V)\xrightarrow{\;\,\chi\;\,}\{\pm 1\}\longrightarrow 1,

where the surjection χ\chi sends β\beta to −1-1, and GPin~​(V)≃GPin⁡(V)×{1,β}.\widetilde{\operatorname{GPin}}(V)\simeq\operatorname{GPin}(V)\times\{1,\beta\}. For a Nash manifold YY, we let 𝒮​(Y)\mathcal{S}(Y) be the Fréchet space of Schwartz functions on YY and 𝒮∗​(Y)\mathcal{S}^{*}(Y) be the space of Schwartz distributions on YY. Denote by 𝒟​(Y):=Cc∞​(Y)∗\mathcal{D}(Y):=C_{c}^{\infty}(Y)^{*}. Hence we have the following inclusion between the spaces of distributions:

𝒮∗​(Y)⊆𝒟​(Y).\mathcal{S}^{*}(Y)\subseteq\mathcal{D}(Y).

We let PP be the natural projection map of GPin\operatorname{GPin} groups onto the orthogonal groups defined in Section 2.3.2. For a group HH, we let HsH_{s} be the set of semisimple elements in HH and we also let Hs/∼H_{s}/\!\sim be the set of conjugacy classes in HsH_{s}.

Acknowledgements This collaboration was initiated at the Midwest Representation Theory Conference at the University of Michigan in March 2022 and discussed further at the Texas-Oklahoma Representations and Automorphic Forms (TORA) Conference at University of Oklahoma in October 2023. We thank both conferences for providing a wonderful atmosphere to meet each other and collaborate. The first author would like to thank Shuichiro Takeda and Wee teck Gan for their continued interest in the project. The second author is grateful to Purdue University for providing excellent working conditions during a one-year research visit (July 2022 - July 2023). The second author has been supported by the National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIP and MSIT) (No. RS-2022-0016551 and No. RS-2024-00415601 (G-BRL)).

2. Background and preliminaries

2.1. Cassleman-Wallach Representation

Let GG be a real reductive group and 𝔤ℂ\mathfrak{g}_{\mathbb{C}} be its complexified Lie algebra. Let (π,V1)(\pi,V_{1}) be a representation of GG and let 𝒵​(𝔤ℂ)\mathcal{Z}(\mathfrak{g}_{\mathbb{C}}) be the center of the universal enveloping algebra of 𝔤ℂ\mathfrak{g}_{\mathbb{C}}. The representation (π,V1)(\pi,V_{1}) is called admissible if every irreducible representation of a maximal compact subgroup KK of GG has finite multiplicity in V1V_{1}. The representation (π,V1)(\pi,V_{1}) is called of Harish-chandra type if it is admissible and 𝒵​(𝔤ℂ)\mathcal{Z}(\mathfrak{g}_{\mathbb{C}}) finite.
The representation (π,V1)(\pi,V_{1}) is called of moderate growth if the following condition holds: for every continuous seminorm |⋅|μ|\cdot|_{\mu} on V1V_{1}, there exists a positive, moderate growth function ϕ\phi on GG, and a continuous seminorm |⋅|ν|\cdot|_{\nu} on V1V_{1} such that

|g​v|μ≤|ϕ​(g)|​|v|ν,∀g∈G,v∈V1.|gv|_{\mu}\leq|\phi(g)|\ |v|_{\nu},\quad\forall\ g\in G,\ v\in V_{1}.

The representation (π,V1)(\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 FF of characteristics zero as discussed in [AG08].

2.2.1. Semi algebraic sets

A subset A⊂FnA\subset F^{n} is called a semi-algebraic set if it satisfies the following: there exist finitely many polynomials fi​j,gi​k∈F​[x1,x2,⋯​xn]f_{ij},g_{ik}\in F[x_{1},x_{2},\cdots x_{n}] such that

A=⋃i=1r{x∈Fn:fi​1​(x)>0,fi​2​(x)>0,⋯,fi​si>0,gi​1​(x)=0,gi​2​(x)=0,⋯​gi​ti=0}.A=\bigcup_{i=1}^{r}\{x\in F^{n}:f_{i1}(x)>0,f_{i2}(x)>0,\cdots,f_{is_{i}}>0,g_{i1}(x)=0,g_{i2}(x)=0,\cdots g_{it_{i}}=0\}.

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 ν\nu between two semi-algebraic sets A⊂Fn,B⊂FmA\subset F^{n},B\subset F^{m} is called semi-algebraic if the graph of the map ν\nu is a semialgebraic subset of Fm+nF^{m+n}. The open semi-algebraic sets that define the topology can be realized in the following lemma:

Lemma 2.1.

Let X⊂FnX\subset F^{n} be a semi-algebraic set. Then every open semi-algebraic subset of XX can be presented as a finite union of sets of the form {x∈X|fi​(x)>0,i=1​…​n}\{x\in X|f_{i}(x)>0,i=1...n\}, where fif_{i} are polynomials in nn variables.

Let U,V⊂FnU,V\subset F^{n} be two open semi-algebraic sets. A smooth, semi-algebraic map between UU and VV 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 FnF^{n} is a semi-algebraic, smooth submanifold of FnF^{n}. A Nash group GG is a Nash manifold such that the following is a Nash map:

G×G⟶GG\times G\longrightarrow G
(g,h)↦g​h−1.(g,h)\mapsto gh^{-1}.

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

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

In this subsection we introduce the definitions of the groups GSpin⁡(V)\operatorname{GSpin}(V) and GPin⁡(V)\operatorname{GPin}(V) and their properties. A reference for the material can be found in [Sch85] and [Shi04]. In the literature, the group which we refer to as GPin⁡(V)\operatorname{GPin}(V) is sometimes called the Clifford group and GSpin⁡(V)\operatorname{GSpin}(V) is sometimes referred to as the special Clifford group, and are denoted by Γ​(V)\Gamma(V) and SΓ​(V)\Gamma(V), respectively.

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

2.3.1. Clifford Algebra

Let T​(V)=⨁l=0∞V⊗lT(V)=\bigoplus\limits_{l=0}^{\infty}V^{\otimes l} be the tensor algebra of VV and we define the Clifford algebra C​(V)C(V) by the following quotient:

C(V)=T(V)/⟨v⊗v−q(v)⋅1:v∈V⟩.C(V)=T(V)/\langle v\otimes v-q(v)\cdot 1:v\in V\rangle.

In C​(V)C(V) we have

v⋅v=q​(v)∀v∈V.v\cdot v=q(v)\quad\forall v\in V.

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

C+​(V)=∑l​ evenCl​(V)C−​(V)=∑l​ oddCl​(V)C^{+}(V)=\sum_{l\text{ even}}C^{l}(V)\qquad C^{-}(V)=\sum_{l\text{ odd}}C^{l}(V)

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

C​(V)=C+​(V)⊕C−​(V).C(V)=C^{+}(V)\oplus C^{-}(V).

The Clifford algebra is equipped with the natural involution ∗* by “reversing the indices” of v1​v2​⋯​vl∈Cl​(V)v_{1}v_{2}\cdots v_{l}\in C^{l}(V), namely

(v1​v2​⋯​vl)∗=vl​vl−1​⋯​v1.(v_{1}v_{2}\cdots v_{l})^{*}=v_{l}v_{l-1}\cdots v_{1}.

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

α:C​(V)⟶C​(V),α​(x++x−)=x+−x−,\alpha:C(V)\longrightarrow C(V),\quad\alpha(x^{+}+x^{-})=x^{+}-x^{-},

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

x¯=α​(x)∗=α​(x∗).\bar{x}=\alpha(x)^{*}=\alpha(x^{*}).

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

N:C​(V)⟶C​(V),N​(x)=x​x¯.N:C(V)\longrightarrow C(V),\quad N(x)=x\bar{x}.

2.3.2. GPin⁡(V)\operatorname{GPin}(V) and GSpin⁡(V)\operatorname{GSpin}(V)

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

GPin(V):={g∈C(V)×:α(g)Vg−1=V};\operatorname{GPin}(V):=\{g\in C(V)^{\times}:\quad\alpha(g)Vg^{-1}=V\};
GSpin(V):={g∈C(V)×:gVg−1=V},\operatorname{GSpin}(V):=\{g\in C(V)^{\times}:\quad gVg^{-1}=V\},

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

P​(g):V⟶V,P​(g)​v=α​(g)​v​g−1,P(g):V\longrightarrow V,\quad P(g)v=\alpha(g)vg^{-1},

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

1{1}GL⁡(1){\operatorname{GL}(1)}GPin⁡(V){\operatorname{GPin}(V)}O⁡(V){\operatorname{O}(V)}1{1}1{1}GL⁡(1){\operatorname{GL}(1)}GSpin⁡(V){\operatorname{GSpin}(V)}SO⁡(V){\operatorname{SO}(V)}1.{1.}P\scriptstyle{P}⊆{\subseteq}P\scriptstyle{P}⊆{\subseteq}

2.3.3. Involution

Let

sign:GPin⁡(V)⟶±1\text{sign}:\operatorname{GPin}(V)\longrightarrow{\pm{1}}

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

σV:GPin⁡(V)⟶GPin⁡(V),σV​(g)={g∗,if n=2​k;sign(g)k+1g∗,if n=2​k−1.\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}

Note that σV\sigma_{V} preserves the semisimple conjugacy classes of GPin⁡(V)\operatorname{GPin}(V). We denote GPin~(V):=⟨g,β:g∈GPin(V)⟩\widetilde{\operatorname{GPin}}(V):=\langle g,\beta:g\in\operatorname{GPin}(V)\rangle, such that g​β=β​gg\beta=\beta g for all g∈GPin⁡(V)g\in\operatorname{GPin}(V) and β2=1\beta^{2}=1. We now define the action of GPin~​(V)\widetilde{\operatorname{GPin}}(V) on GPin⁡(V)×V\operatorname{GPin}(V)\times V by

(2.1) g⋅(h,v)=(g​h​g−1,P​(g)​v)β⋅(h,v)=(σV​(h),−v)\displaystyle\begin{aligned} g\cdot(h,v)&=(ghg^{-1},P(g)v)\\ \beta\cdot(h,v)&=(\sigma_{V}(h),-v)\end{aligned}

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

We assume

W=S​p​a​n​{e1,…,en−1}​ so that ​V=W⊕F​e​ with ​e:=en.W=Span\{e_{1},\dots,e_{n-1}\}\text{ so that }V=W\oplus Fe\textit{ with }e:=e_{n}.

We then have GPin(V)e=GPin(W)\operatorname{GPin}(V)_{e}=\operatorname{GPin}(W) and GPin~(V)e=⟨g,eβ:g∈GPin(W)⟩.\widetilde{\operatorname{GPin}}(V)_{e}=\big{\langle}g,\,e\beta\;:\;g\in\operatorname{GPin}(W)\big{\rangle}. We define

GPin~​(W):=GPin~​(V)e.\widetilde{\operatorname{GPin}}(W):=\widetilde{\operatorname{GPin}}(V)_{e}.

Then we also have a short exact sequence as follows:

1⟶GPin⁡(W)⟶GPin~​(W)→𝜒{±1}⟶1,1\longrightarrow\operatorname{GPin}(W)\longrightarrow\widetilde{\operatorname{GPin}}(W)\xrightarrow{\;\,\chi\;\,}\{\pm 1\}\longrightarrow 1,

where the surjection χ\chi sends e​βe\beta to −1-1, and GPin~​(W)≃GPin⁡(W)⋊{1,e​β},\widetilde{\operatorname{GPin}}(W)\simeq\operatorname{GPin}(W)\rtimes\{1,e\beta\}, where e​βe\beta acts on GPin⁡(W)\operatorname{GPin}(W) by conjugation viewed inside GPin~​(V)\widetilde{\operatorname{GPin}}(V).

We define an involution

(2.2) τW:GPin⁡(V)⟶GPin⁡(V),τW​(g)=e​σV​(g)​e−1\tau_{W}:\operatorname{GPin}(V)\longrightarrow\operatorname{GPin}(V),\quad\tau_{W}(g)=e\sigma_{V}(g)e^{-1}

for g∈GPin⁡(V)g\in\operatorname{GPin}(V). This involution is precisely the action of the element e​β∈GPin~​(V)e\beta\in\widetilde{\operatorname{GPin}}(V). Note that it is direct to show that τW​(GPin⁡(W))=GPin⁡(W)\tau_{W}(\operatorname{GPin}(W))=\operatorname{GPin}(W) (See [ET23, Lemma 2.10] for more detail).

3. GPin\operatorname{GPin} case

In this section, we prove Theorem 1.2 for GPin\operatorname{GPin} groups. Recall that Lemma LABEL:mainthmtovanishing reduces our main theorem to the following vanishing assertion:

𝒮∗​(GPin⁡(V))GPin~​(W),χ=0.\mathcal{S}^{*}(\operatorname{GPin}(V))^{\widetilde{\operatorname{GPin}}(W),\chi}=0.

We first cite [AG09b, Theorem 2.2.5], which is true for all reductive groups as follows:

Proposition 3.1.

If 𝒮∗​(GPin⁡(V))GPin~​(W),χ=0\mathcal{S}^{*}(\operatorname{GPin}(V))^{\widetilde{\operatorname{GPin}}(W),\chi}=0, then 𝒟​(GPin⁡(V))GPin~​(W),χ=0.\mathcal{D}(\operatorname{GPin}(V))^{\widetilde{\operatorname{GPin}}(W),\chi}=0.

Furthermore, the following proposition is straightforward by definition:

Proposition 3.2.

If 𝒟​(GPin⁡(V))GPin~​(W),χ=0\mathcal{D}(\operatorname{GPin}(V))^{\widetilde{\operatorname{GPin}}(W),\chi}=0, then Theorem 1.2 holds. Therefore the multiplicity-at-most-one theorem for GPin\operatorname{GPin} groups also holds.

Therefore, our main theorem for GPin\operatorname{GPin} groups, i.e., Theorem 1.2 reduces to the following theorem, which is precisely the analogue of either [AG09b, Theorem A] or [ET23, Theorem 5.4]:

Theorem 3.3.

Let GPin~​(W)≃GPin⁡(W)⋊{1,e​β}\widetilde{\operatorname{GPin}}(W)\simeq\operatorname{GPin}(W)\rtimes\{1,e\beta\} act on GPin⁡(V)\operatorname{GPin}(V) (viewed merely as a set) by letting GPin⁡(W)\operatorname{GPin}(W) act by conjugation and e​βe\beta by τW\tau_{W}. Then we have

𝒮∗​(GPin⁡(V))GPin~​(W),χ=0.\mathcal{S}^{*}(\operatorname{GPin}(V))^{\widetilde{\operatorname{GPin}}(W),\chi}=0.

In other words, every GPin⁡(W)\operatorname{GPin}(W)-invariant distribution on GPin⁡(V)\operatorname{GPin}(V) is also invariant under the involution τW\tau_{W}.

The rest of this section is to prove Theorem 3.3, which is the technical heart of the paper. We adapt the arguments in [ET23, Section 7] to our case and it consists of three steps of reductions.

3.1. Reduction I: Elimination of WW

In this subsection, We reduce Theorem 3.3 to the follwoing vanishing assertion:

(3.1) 𝒮∗​(GPin⁡(V)×V)GPin~​(V),χ=0.\mathcal{S}^{*}(\operatorname{GPin}(V)\times V)^{\widetilde{\operatorname{GPin}}(V),\chi}=0.

The key ingredient is the following version of Frobenius descent [AG09a, Theorem 2.5.7].

Lemma 3.4 (Frobenius descent).

Let GG be a Nash group which is unimodular. Let XX and YY be Nash manifolds on which GG acts. Further assume that the action of GG on YY is transitive. Suppose we have a continuous GG-equivariant Nash map

ϕ:X→Y,\phi:X\rightarrow Y,

namely φ​(g⋅x)=g⋅φ​(x)\varphi(g\cdot x)=g\cdot\varphi(x) for all g∈Gg\in G and x∈Xx\in X. Fix y∈Yy\in Y. Assume the stabilizer Gy⊆GG_{y}\subseteq G of yy is unimodular which implies that there exists a GG-invariant measure on YY. Fix this measure. Let χ:G→ℂ1\chi:G\to\mathbb{C}^{1} be a character of GG. Then there is a canonical isomorphism

𝒮∗​(X)G,χ≃𝒮∗​(ϕ−1​(y))Gy,χ.\mathcal{S}^{*}(X)^{G,\chi}\simeq\mathcal{S}^{*}(\phi^{-1}(y))^{G_{y},\chi}.

We are now ready to prove the following:

Proposition 3.5.

If 𝒮∗​(GPin⁡(V)×V)GPin~​(V),χ=0\mathcal{S}^{*}(\operatorname{GPin}(V)\times V)^{\widetilde{\operatorname{GPin}}(V),\chi}=0 then 𝒮∗​(GPin⁡(V))GPin~(W)),χ=0\mathcal{S}^{*}(\operatorname{GPin}(V))^{\widetilde{\operatorname{GPin}}(W)),\chi}=0.

Proof.

We follow the main argument of the proof in [ET23, Proposition 7.2] and adapt those to 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

𝒮∗​(GPin⁡(V))GPin~​(W),χ⊆𝒮∗​(GPin⁡(V)×V)GPin~​(V),χ.\mathcal{S}^{*}(\operatorname{GPin}(V))^{\widetilde{\operatorname{GPin}}(W),\chi}\subseteq\mathcal{S}^{*}(\operatorname{GPin}(V)\times V)^{\widetilde{\operatorname{GPin}}(V),\chi}.

Let X:={(g,v)∈GPin⁡(V)×V:⟨v,v⟩=⟨e,e⟩}.X:=\{(g,v)\in\operatorname{GPin}(V)\times V\;:\;\langle v,v\rangle=\langle e,e\rangle\}. One can see that XX is invariant under GPin~​(V)\widetilde{\operatorname{GPin}}(V) and we have the following:

Lemma 3.6.

XX is closed in GPin⁡(V)×V\operatorname{GPin}(V)\times V.

Proof.

We have the quadratic form q:V→Fq:V\rightarrow F and we know that q​(v)=v2q(v)=v^{2} and v2∈Fv^{2}\in F. Hence ⟨v,v⟩=⟨e,e⟩\langle v,v\rangle=\langle e,e\rangle implies

q​(v+v)−q​(v)−q​(v)=q​(e+e)−q​(e)−q​(e),q(v+v)-q(v)-q(v)=q(e+e)-q(e)-q(e),
4​v2−2​v2=4​e2−2​e2,4v^{2}-2v^{2}=4e^{2}-2e^{2},
v2=e2.v^{2}=e^{2}.

Since v2,e2∈Fv^{2},e^{2}\in F, this is a discrete set. Therefore, XX is closed in GPin⁡(V)×V\operatorname{GPin}(V)\times V. ∎

Therefore, we have

𝒮∗​(X)GPin~​(V),χ⊆𝒮∗​(GPin⁡(V)×V)GPin~​(V),χ.\mathcal{S}^{*}(X)^{\widetilde{\operatorname{GPin}}(V),\chi}\subseteq\mathcal{S}^{*}(\operatorname{GPin}(V)\times V)^{\widetilde{\operatorname{GPin}}(V),\chi}.

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

Now, consider the projection

ϕ:X⟶Y,(g,v)↦v,\phi:X\longrightarrow Y,\quad(g,v)\mapsto v,

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

(3.2) 𝒮∗​(X)GPin~​(V),χ≃𝒮∗​(ϕ−1​(e))GPin~​(V)e,χ.\mathcal{S}^{*}(X)^{\widetilde{\operatorname{GPin}}(V),\chi}\simeq\mathcal{S}^{*}(\phi^{-1}(e))^{\widetilde{\operatorname{GPin}}(V)_{e},\chi}.

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

𝒮∗​(ϕ−1​(e))GPin~​(V)e,χ≃𝒮∗​(GPin⁡(V))GPin~​(W),χ.\mathcal{S}^{*}(\phi^{-1}(e))^{\widetilde{\operatorname{GPin}}(V)_{e},\chi}\simeq\mathcal{S}^{*}(\operatorname{GPin}(V))^{\widetilde{\operatorname{GPin}}(W),\chi}.

Hence we have

𝒮∗​(GPin⁡(V))GPin~​(W),χ≃𝒮∗​(X)GPin~​(V),χ⊆𝒮∗​(GPin⁡(V)×V)GPin~​(V),χ.\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}.

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 subsection we further reduce the vanishing assertion to the classical group scenarios as presented in [SZ12] or in [Wal12].

The main idea is that any distribution in 𝒮∗​(GPin⁡(V)×V)GPin⁡(V)~,χ\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:

Theorem 3.7.

(Localization principle). Let a real reductive group GG act on a smooth algebraic variety XX. Let YY be an algebraic variety and ϕ:X→Y\phi:X\rightarrow Y be an affine algebraic GG-invariant map. Let χ\chi be a character of GG. Suppose that for any y∈Y​(F)y\in Y(F) we have 𝒟X​(F)​(ϕ−1​(y)​(F))G,χ=0\mathcal{D}_{X(F)}\left(\phi^{-1}(y)(F)\right)^{G,\chi}=0. Then 𝒟​(X​(F))G,χ=0\mathcal{D}(X(F))^{G,\chi}=0.

Proof.

See [AG09b, Corollary A.0.1]. ∎

Remark 3.8.

We can swiftly interchange the notation 𝒟\mathcal{D} and 𝒮∗\mathcal{S}^{*}, due to the proof of [AG09b, Theorem 4.0.2].

Corollary 3.9.

Let a real reductive group GG act on a smooth algebraic variety XX. Let YY be an algebraic variety and ϕ:X→Y\phi:X\rightarrow Y be an affine algebraic GG-invariant submersion. Suppose that for any y∈Y​(F)y\in Y(F) we have 𝒮∗​(ϕ−1​(y))G,χ=0\mathcal{S}^{*}\left(\phi^{-1}(y)\right)^{G,\chi}=0. Then 𝒟​(X​(F))G,χ=0\mathcal{D}(X(F))^{G,\chi}=0.

Proof.

See [AG09b, Corollary A.0.3]. ∎

Proposition 3.10.

Define a map

θ:GPin(V)×V⟶O(V)s/∼\theta:\operatorname{GPin}(V)\times V\longrightarrow\operatorname{O}(V)_{s}/\!\sim

by

(g,v)↦P(g)s/∼,(g,v)\mapsto P(g)_{s}/\!\sim,

where P(g)s/∼P(g)_{s}/\!\sim is the conjugacy class of the semisimple part of P​(g)P(g) under the Jordan decomposition.

If

𝒮∗​(θ−1​(γ))GPin~​(V),χ=0\mathcal{S}^{*}(\theta^{-1}(\gamma))^{\widetilde{\operatorname{GPin}}(V),\chi}=0

for each semisimple conjugacy class γ∈O(V)s/∼\gamma\in\operatorname{O}(V)_{s}/\!\sim, then

𝒮∗​(GPin⁡(V)×V)GPin~​(V),χ=0.\mathcal{S}^{*}(\operatorname{GPin}(V)\times V)^{\widetilde{\operatorname{GPin}}(V),\chi}=0.
Proof.

We follow the main argument of the proof in [ET23, Proposition 7.4] and we briefly write the main ideas of the proof but in case the proof is different we write in the details. Let YY be the space of polynomials of degree at most n=dimVn=\dim V, which is a smooth algebraic variety and let ϕ:GPin⁡(V)×V⟶Y\phi:\operatorname{GPin}(V)\times V\longrightarrow Y be as in proof of [ET23, Proposition 7.4]. One can see that ϕ\phi is an affine algebraic GG-invariant submersion. Let f∈Yf\in Y be a polynomial. Since the fiber ϕ−1​(f)\phi^{-1}(f) is preserved by GPin~​(V)\widetilde{\operatorname{GPin}}(V), Theorem 3.7 implies that if 𝒮∗​(ϕ−1​(f))GPin~​(V),χ=0\mathcal{S}^{*}(\phi^{-1}(f))^{\widetilde{\operatorname{GPin}}(V),\chi}=0 for all f∈Yf\in Y, then 𝒮∗​(GPin⁡(V)×V)GPin~​(V),χ=0\mathcal{S}^{*}(\operatorname{GPin}(V)\times V)^{\widetilde{\operatorname{GPin}}(V),\chi}=0. Furthermore, we have ϕ−1​(f)=ℱf×V\phi^{-1}(f)=\mathcal{F}_{f}\times V where ℱf={g∈GPin⁡(V):the char. poly. of P​(g) is f},\mathcal{F}_{f}=\{g\in\operatorname{GPin}(V)\;:\;\text{the char.\ poly.\ of $P(g)$ is $f$}\}, which is a Nash manifold.

To use the Bernstein localization principle, we need to show that each element in P(ℱf)s/∼P(\mathcal{F}_{f})_{s}/\!\sim is closed and that P(ℱf)s/∼P(\mathcal{F}_{f})_{s}/\sim is an embedded submanifold of P​(ℱf)sP(\mathcal{F}_{f})_{s} (basically it means the topology is a subspace topology). The first claim follows from [BHC62, Proposition 10.1]. The second one is well-known and can be followed from [MZ40, Theorem 2.13].

We can now consider the map θ:ℱf×V⟶P(ℱf)⟶P(ℱf)s⟶P(ℱf)s/∼,\theta:\mathcal{F}_{f}\times V\longrightarrow P(\mathcal{F}_{f})\longrightarrow P(\mathcal{F}_{f})_{s}\longrightarrow P(\mathcal{F}_{f})_{s}/\!\sim, exactly as in the proof of [ET23, Proposition 7.4]. Then this map θ\theta is indeed an affine algebraic map. In the Archimedean case, the involution σV\sigma_{V} also preserves the semisimple conjugacy classes. Therefore, for each semisimple conjucagy class γ∈P(ℱf)s/∼\gamma\in P(\mathcal{F}_{f})_{s}/\!\sim the fiber θ−1​(γ)\theta^{-1}(\gamma) is invariant under GPin~​(V)\widetilde{\operatorname{GPin}}(V) and Theorem 3.7 implies that if 𝒮∗​(θ−1​(γ))GPin~​(V),χ=0\mathcal{S}^{*}(\theta^{-1}(\gamma))^{\widetilde{\operatorname{GPin}}(V),\chi}=0 for all semisimple conjugacy class γ\gamma of O⁡(V)\operatorname{O}(V), then 𝒮∗​(GPin⁡(V)×V)GPin~​(V),χ=0,\mathcal{S}^{*}(\operatorname{GPin}(V)\times V)^{\widetilde{\operatorname{GPin}}(V),\chi}=0, which completes the proof of the Proposition.

∎

3.3. Reduction III: O⁡(V)\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 [SZ12].

Remark 3.11.

Let 𝒰⊆GPin⁡(V)\mathcal{U}\subseteq\operatorname{GPin}(V) be the set of unipotent elements in GPin⁡(V)\operatorname{GPin}(V). Then, for each g∈GPin⁡(V)g\in\operatorname{GPin}(V) both 𝒰\mathcal{U} and 𝒰g\mathcal{U}_{g} are closed as smooth algebraic variety in GPin⁡(V)\operatorname{GPin}(V). It is also well known that the restriction to 𝒰\mathcal{U} of the canonical projection P:GPin⁡(V)→O⁡(V)P:\operatorname{GPin}(V)\to\operatorname{O}(V) is one-to-one, which allows us to identify the set of unipotent elements in O⁡(V)\operatorname{O}(V) with 𝒰\mathcal{U}.

For a semisimple conjugacy class γ⊆O(V)s\gamma\subseteq\operatorname{O}(V)_{s} of P​(g)P(g), we cosnider the following map exactly as in [ET23]:

θ:θ−1​(γ)⟶γ,(g,v)↦P​(g)s,\theta:\theta^{-1}(\gamma)\longrightarrow\gamma,\quad(g,v)\mapsto P(g)_{s},

Applying the Frobenius descent to the above map, we have

𝒮∗​(θ−1​(γ))GPin~​(V),χ≃𝒮∗​(Z∘​g​𝒰g×V)GPin~​(V)g,χ\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}

since θ−1​(P​(g))=Z∘​g​𝒰g×V\theta^{-1}(P(g))=Z^{\circ}g\,\mathcal{U}_{g}\times V.

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

Lemma 3.12.

If 𝒮∗​(g​𝒰g×V)GPin~​(V)g,χ=0\mathcal{S}^{*}(g\,\mathcal{U}_{g}\times V)^{\widetilde{\operatorname{GPin}}(V)_{g},\chi}=0 for all semisimple g∈GPin⁡(V)g\in\operatorname{GPin}(V), then 𝒮∗​(Z∘​g​𝒰g×V)GPin~​(V)g,χ=0\mathcal{S}^{*}(Z^{\circ}g\,\mathcal{U}_{g}\times V)^{\widetilde{\operatorname{GPin}}(V)_{g},\chi}=0.

Proof.

First, since z​gzg is semisimple for all z∈Z∘z\in Z^{\circ} and 𝒰z​g=𝒰g\mathcal{U}_{zg}=\mathcal{U}_{g}, we have

𝒮∗​(z​g​𝒰g×V)GPin~​(V)g,χ=0\mathcal{S}^{*}(zg\,\mathcal{U}_{g}\times V)^{\widetilde{\operatorname{GPin}}(V)_{g},\chi}=0

for all z∈Z∘z\in Z^{\circ}.

We then consider the following map:

Z∘​g​𝒰g×V⟶Z∘,z​g​𝒰g↦z.Z^{\circ}g\,\mathcal{U}_{g}\times V\longrightarrow Z^{\circ},\quad zg\,\mathcal{U}_{g}\mapsto z.

Now, as each fiber z​g​𝒰g×Vzg\,\mathcal{U}_{g}\times V is preserved by GPin~​(V)g\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 O⁡(V)\operatorname{O}(V) situation. Recall that GPin⁡(V)\operatorname{GPin}(V) and O⁡(V)\operatorname{O}(V) have the same set of unipotent elements. This implies that for each semisimple gg we have the bijection

g​𝒰g⟶P​(g​𝒰g)g\,\mathcal{U}_{g}\longrightarrow P(g\,\mathcal{U}_{g})

induced by the canonical projection PP. Furthermore, this map intertwines the actions of GPin~​(V)g\widetilde{\operatorname{GPin}}(V)_{g} and P​(GPin~​(V)g)P(\widetilde{\operatorname{GPin}}(V)_{g}). Note that the kernel Z0Z_{0} of PP act trivially on the space 𝒮∗​(g​𝒰g×V)\mathcal{S}^{*}(g\,\mathcal{U}_{g}\times V). Therefore, we have

𝒮∗​(g​𝒰g×V)GPin~​(V)g,χ≃𝒮∗​(P​(g​𝒰g)×V)P​(GPin~​(V)g),χ.\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}.

Hence to show our main theorem, it suffices to show the following O⁡(V)\operatorname{O}(V) situation [SZ12, Theorem A]:

(3.3) 𝒮∗(P(g𝒰g)×V)P​(GPin~​(V)g),χ⊆𝒮∗(P(GPin(V)g)×V)P​(GPin~​(V)g),χ=0\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

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

3.4. End of proof

In case P(GPin(V)g)=O(V)P​(g)P(\operatorname{GPin}(V)_{g})=\operatorname{O}(V)_{P(g)}, the vanishing assertion (3.3) is equivalent to show 𝒮∗(O(V)P​(g)×V)O(V)P​(g),χ=0\mathcal{S}^{*}(\operatorname{O}(V)_{P(g)}\times V)^{\operatorname{O}(V)_{P(g)},\chi}=0, which is precisely the assertion proven in [SZ12]. However, we do not always have P(GPin(V)g)=O(V)P​(g)P(\operatorname{GPin}(V)_{g})=\operatorname{O}(V)_{P(g)} as shown in [ET23, Lemma 3.11]. Accordingly, we need to modify [SZ12]. The difference is that P(GPin(V)g)P(\operatorname{GPin}(V)_{g}) might have a factor of SO\operatorname{SO} as in [ET23, Lemma 3.11], for which we need the result of Sun-Zhu [SZ12] for the SO\operatorname{SO} case. The complete argument in the case of locally compact totally disconnected space is present in [ET23]. 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 GPin⁡(V)\operatorname{GPin}(V) and GPin⁡(W)\operatorname{GPin}(W).

4. GSpin case

In this section, we prove Theorem 1.2 for GSpin⁡(V)\operatorname{GSpin}(V). The proof follows the same line as the GPin case but we need to make appropriate changes. Note that we follow the main arguments in [ET23, Proposition 9] and we adapt its arguments to Archimedean case and we briefly explain the main ideas. To point out the difference, you can see the proof of Proposition 4.2. In particular, we define a group GSpin~​(V)\widetilde{\operatorname{GSpin}}(V) which is the analogue of SO~​(V)\widetilde{\operatorname{SO}}(V) following exactly as the non-archimedean case. First, we recall our basic setup. As before, VV is a quadratic space with

dimFV=n={2​k;2​k−1.\dim_{F}V=n=\begin{cases}2k;\\ 2k-1.\end{cases}

We fix an orthogonal basis e1,…,en−1,ene_{1},\dots,e_{n-1},e_{n}, and assume W=Span⁡{e1,…,en−1}W=\operatorname{Span}\{e_{1},\dots,e_{n-1}\} so that V=W​⨁F​eV=W\bigoplus Fe with e:=ene:=e_{n}. The group GSpin~​(V)\widetilde{\operatorname{GSpin}}(V) we define as

GSpin~(V)=⟨g,ekβ:g∈GSpin(V)⟩⊆GPin~(V),\widetilde{\operatorname{GSpin}}(V)=\big{\langle}g,\,e^{k}\beta\;:\;g\in\operatorname{GSpin}(V)\big{\rangle}\subseteq\widetilde{\operatorname{GPin}}(V),

so that we have

1⟶GSpin⁡(V)⟶GSpin~​(V)→𝜒{±1}⟶1.1\longrightarrow\operatorname{GSpin}(V)\longrightarrow\widetilde{\operatorname{GSpin}}(V)\xrightarrow{\;\,\chi\;\,}\{\pm 1\}\longrightarrow 1.

The surjection χ\chi sends ek​βe^{k}\beta to −1-1, and

GSpin~​(V)≃GSpin⁡(V)×{1,ek​β},\widetilde{\operatorname{GSpin}}(V)\simeq\operatorname{GSpin}(V)\times\{1,e^{k}\beta\},

where ek​βe^{k}\beta acts on GSpin⁡(V)\operatorname{GSpin}(V) by conjugation viewed inside GPin~​(V)\widetilde{\operatorname{GPin}}(V). Since GSpin~​(V)\widetilde{\operatorname{GSpin}}(V) is a subgroup of GPin~​(V)\widetilde{\operatorname{GPin}}(V), it acts on GSpin⁡(V)×V\operatorname{GSpin}(V)\times V (viewed merely as a set) by restricting the action of GPin~​(V)\widetilde{\operatorname{GPin}}(V) as

(4.1) g⋅(h,v)=(g​h​g−1,P​(g)​v)ek​β⋅(h,v)=(ek​σV​(h)​e−k,−P​(e)k​v),\displaystyle\begin{aligned} g\cdot(h,v)&=(ghg^{-1},P(g)v)\\ e^{k}\beta\cdot(h,v)&=(e^{k}\sigma_{V}(h)e^{-k},-P(e)^{k}v),\end{aligned}

where (h,v)∈GSpin⁡(V)×V(h,v)\in\operatorname{GSpin}(V)\times V.

We let GSpin~​(V)e\widetilde{\operatorname{GSpin}}(V)_{e} be the stabilizer of e∈Ve\in V under the action of GSpin~​(V)\widetilde{\operatorname{GSpin}}(V) on VV as usual. Analogously to the SO⁡(V)\operatorname{SO}(V) case, one can then show

GSpin~(V)e=⟨g,en−1k−1eβ:g∈GSpin(W)⟩,\widetilde{\operatorname{GSpin}}(V)_{e}=\big{\langle}g,e_{n-1}^{k-1}e\beta\;:\;g\in\operatorname{GSpin}(W)\big{\rangle},

and we define

GSpin~​(W):=GSpin~​(V)e.\widetilde{\operatorname{GSpin}}(W):=\widetilde{\operatorname{GSpin}}(V)_{e}.

We have

1⟶GSpin⁡(W)⟶GSpin~​(W)→𝜒{±1}⟶1,1\longrightarrow\operatorname{GSpin}(W)\longrightarrow\widetilde{\operatorname{GSpin}}(W)\xrightarrow{\;\,\chi\;\,}\{\pm 1\}\longrightarrow 1,

where the surjection χ\chi sends en−1k−1​e​βe_{n-1}^{k-1}e\beta to −1-1, and

GSpin~​(W)≃GSpin⁡(W)⋊{1,en−1k−1​e​β},\widetilde{\operatorname{GSpin}}(W)\simeq\operatorname{GSpin}(W)\rtimes\{1,e_{n-1}^{k-1}e\beta\},

where the action of en−1k−1​e​βe_{n-1}^{k-1}e\beta is by conjugation viewed inside GPin~​(V)\widetilde{\operatorname{GPin}}(V).

We define an involution

(4.2) τW:GSpin⁡(V)⟶GSpin⁡(V),τW​(g)=(en−1k−1​e)​σV​(g)​(en−1k−1​e)−1,\tau_{W}:\operatorname{GSpin}(V)\longrightarrow\operatorname{GSpin}(V),\quad\tau_{W}(g)=(e_{n-1}^{k-1}e)\sigma_{V}(g)(e_{n-1}^{k-1}e)^{-1},

for g∈GSpin⁡(V)g\in\operatorname{GSpin}(V). This is the action of en−1k−1​e​β∈GSpin~​(V)ee_{n-1}^{k-1}e\beta\in\widetilde{\operatorname{GSpin}}(V)_{e} on GSpin⁡(V)\operatorname{GSpin}(V). Since ee commutes with all the elements in GSpin⁡(W)\operatorname{GSpin}(W), we have τW​(GSpin⁡(W))=GSpin⁡(W)\tau_{W}(\operatorname{GSpin}(W))=\operatorname{GSpin}(W).

We have the canonical projection

P:GSpin~​(V)⟶SO~​(V),g↦P​(g),ek​β↦rek​β,P:\widetilde{\operatorname{GSpin}}(V)\longrightarrow\widetilde{\operatorname{SO}}(V),\quad g\mapsto P(g),\;e^{k}\beta\mapsto r_{e}^{k}\beta,

which is nothing but the restriction of the canonical projection P:GPin~​(V)→O~​(V)P:\widetilde{\operatorname{GPin}}(V)\to\widetilde{\operatorname{O}}(V). We then have

P​(GSpin~​(V)e)=SO~​(V)e.P(\widetilde{\operatorname{GSpin}}(V)_{e})=\widetilde{\operatorname{SO}}(V)_{e}.

Let g∈GSpin⁡(V)g\in\operatorname{GSpin}(V) be semisimple, and set h:=P​(g)∈SO⁡(V)h:=P(g)\in\operatorname{SO}(V). If

O(V)h≃G1×⋯×Gm×O(V+)×O(V−)\operatorname{O}(V)_{h}\simeq G_{1}\times\cdots\times G_{m}\times\operatorname{O}(V_{+})\times\operatorname{O}(V_{-})

as before, then

SO(V)h≃G1×⋯×Gm×S(O(V+)×O(V−)),\operatorname{SO}(V)_{h}\simeq G_{1}\times\cdots\times G_{m}\times S(\operatorname{O}(V_{+})\times\operatorname{O}(V_{-})),

where

S​(O⁡(V+)×O⁡(V−))=(O⁡(V+)×O⁡(V−))∩SO⁡(V+⊕V−)S(\operatorname{O}(V_{+})\times\operatorname{O}(V_{-}))=(\operatorname{O}(V_{+})\times\operatorname{O}(V_{-}))\cap\operatorname{SO}(V_{+}\oplus V_{-})

by [ET23, Proposition A.4]. Following exactly as in the proof of [ET23, Lemma 9.1], we have the following Lemma:

Lemma 4.1.

Keeping the above notation, we have

P(GSpin(V)g)≃G1×⋯×Gm×SO(V+)×SO(V−)⊂SO(V)h.P(\operatorname{GSpin}(V)_{g})\simeq G_{1}\times\cdots\times G_{m}\times\operatorname{SO}(V_{+})\times\operatorname{SO}(V_{-})\subset\operatorname{SO}(V)_{h}.

4.1. Vanishing of distribution

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

(4.3) 𝒮∗​(GSpin⁡(V))GSpin~​(W),χ=0,\mathcal{S}^{*}(\operatorname{GSpin}(V))^{\widetilde{\operatorname{GSpin}}(W),\chi}=0,

where GSpin~​(W)≃GSpin⁡(W)×{1,en−1k−1​e​β}\widetilde{\operatorname{GSpin}}(W)\simeq\operatorname{GSpin}(W)\times\{1,e_{n-1}^{k-1}e\beta\} acts on GPin⁡(V)\operatorname{GPin}(V) by restricting the actions (4.1). In particular, the element en−1k−1​e​βe_{n-1}^{k-1}e\beta acts via the involution τW\tau_{W}, which preserves GSpin⁡(W)\operatorname{GSpin}(W) setwise. Recall the action of GSpin~​(V)\widetilde{\operatorname{GSpin}}(V) on GSpin⁡(V)×V\operatorname{GSpin}(V)\times V is defined in (4.1).

Proposition 4.2.

We have a natural inclusion

𝒮∗​(GSpin⁡(V))GSpin~​(W),χ⊆𝒮∗​(GSpin⁡(V)×V)GSpin~​(V),χ.\mathcal{S}^{*}(\operatorname{GSpin}(V))^{\widetilde{\operatorname{GSpin}}(W),\chi}\subseteq\mathcal{S}^{*}(\operatorname{GSpin}(V)\times V)^{\widetilde{\operatorname{GSpin}}(V),\chi}.

Hence if

𝒮∗​(GSpin⁡(V)×V)GSpin~​(V),χ=0,\mathcal{S}^{*}(\operatorname{GSpin}(V)\times V)^{\widetilde{\operatorname{GSpin}}(V),\chi}=0,

then 𝒮∗​(GSpin⁡(V))GSpin~​(W),χ=0\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

X\displaystyle X :={(g,v)∈GSpin⁡(V)×V:⟨v,v⟩=⟨e,e⟩}\displaystyle:=\{(g,v)\in\operatorname{GSpin}(V)\times V\;:\;\langle v,v\rangle=\langle e,e\rangle\}
Y\displaystyle Y :={v∈V:⟨v,v⟩=⟨e,e⟩},\displaystyle:=\{v\in V\;:\;\langle v,v\rangle=\langle e,e\rangle\},

and consider the projection

ϕ:X⟶Y.\phi:X\longrightarrow Y.

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

𝒮∗​(X)GSpin~​(V),χ≃𝒮∗​(GSpin⁡(V)×{e})GSpin~​(V)e,χ,\mathcal{S}^{*}(X)^{\widetilde{\operatorname{GSpin}}(V),\chi}\simeq\mathcal{S}^{*}(\operatorname{GSpin}(V)\times\{e\})^{\widetilde{\operatorname{GSpin}}(V)_{e},\chi},

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

𝒮∗​(GSpin⁡(V)×{e})GSpin~​(V)e,χ≃𝒮′​(GSpin~​(V))GSpin~​(W),χ.\mathcal{S}^{*}(\operatorname{GSpin}(V)\times\{e\})^{\widetilde{\operatorname{GSpin}}(V)_{e},\chi}\simeq\mathcal{S}^{\prime}(\widetilde{\operatorname{GSpin}}(V))^{\widetilde{\operatorname{GSpin}}(W),\chi}.

The proposition follows. ∎

4.2. Reducing to classical group situation

It is now enough to show

𝒮∗​(GSpin⁡(V)×V)GSpin~​(V),χ=0.\mathcal{S}^{*}(\operatorname{GSpin}(V)\times V)^{\widetilde{\operatorname{GSpin}}(V),\chi}=0.

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

𝒮∗​(g​𝒰g×V)GSpin~​(V)g,χ=0\mathcal{S}^{*}(g\,\mathcal{U}_{g}\times V)^{\widetilde{\operatorname{GSpin}}(V)_{g},\chi}=0

for all semisimple g∈GSpin⁡(V)g\in\operatorname{GSpin}(V). Since the canonical projection PP is bijective on g​𝒰gg\,\mathcal{U}_{g}, we have the natural isomorphism

𝒮∗​(g​𝒰g×V)GSpin~​(V)g,χ≃𝒮′​(P​(g​𝒰g)×V)P​(GSpin~​(V)g),χ.\mathcal{S}^{*}(g\,\mathcal{U}_{g}\times V)^{\widetilde{\operatorname{GSpin}}(V)_{g},\chi}\simeq\mathcal{S}^{\prime}(P(g\,\mathcal{U}_{g})\times V)^{P(\widetilde{\operatorname{GSpin}}(V)_{g}),\chi}.

Since we have

𝒮∗(P(g𝒰g)×V)P​(GSpin~​(V)g),χ⊆𝒮′(P(GSpin(V)g)×V)P​(GSpin~​(V)g),χ,\mathcal{S}^{*}(P(g\,\mathcal{U}_{g})\times V)^{P(\widetilde{\operatorname{GSpin}}(V)_{g}),\chi}\subseteq\mathcal{S}^{\prime}(P(\operatorname{GSpin}(V)_{g})\times V)^{P(\widetilde{\operatorname{GSpin}}(V)_{g}),\chi},

it is enough to show

𝒮′(P(GSpin(V)g)×V)P​(GSpin~​(V)g),χ=0.\mathcal{S}^{\prime}(P(\operatorname{GSpin}(V)_{g})\times V)^{P(\widetilde{\operatorname{GSpin}}(V)_{g}),\chi}=0.

We know P(GSpin(V)g)P(\operatorname{GSpin}(V)_{g}) is as in Lemma 4.1 and P​(GSpin~​(V)g)P(\widetilde{\operatorname{GSpin}}(V)_{g}) is generated by P(GSpin(V)g)P(\operatorname{GSpin}(V)_{g}) and the element γ​β\gamma\beta, where γ=(γ1,…,γm,γ+,γ−)∈O⁡(V1)×⋯×O⁡(Vm)×O⁡(V+)×O⁡(V−)\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 γ​h−1​γ−1=h\gamma h^{-1}\gamma^{-1}=h, where h:=P​(g)∈SO⁡(V)h:=P(g)\in\operatorname{SO}(V) as before (see also [ET23, Equation 8.1]. Note that since the orthogonal factor of P(GSpin(V)g)P(\operatorname{GSpin}(V)_{g}) is SO⁡(V+)×SO⁡(V−)\operatorname{SO}(V_{+})\times\operatorname{SO}(V_{-}), we always choose γ±=re±k±\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 GSpin⁡(V)\operatorname{GSpin}(V) and GSpin⁡(W)\operatorname{GSpin}(W).

References

  • [AG08] Avraham Aizenbud and Dmitry Gourevitch. Schwartz functions on Nash manifolds. Int. Math. Res. Not. IMRN, (5):Art. ID rnm 155, 37, 2008.
  • [AG09a] Avraham Aizenbud and Dmitry Gourevitch. Generalized harish-chandra descent and applications to gelfand pairs and an archimedean analog of jacquet-rallis’ theorem. Duke Math. J., 149(3), 2009.
  • [AG09b] Avraham Aizenbud and Dmitry Gourevitch. Multiplicity one theorem for (GLn+1​(ℝ),GLn​(ℝ))({\rm GL}_{n+1}(\mathbb{R}),{\rm GL}_{n}(\mathbb{R})). Selecta Math. (N.S.), 15(2):271–294, 2009.
  • [AGRS10] Avraham Aizenbud, Dmitry Gourevitch, Stephen Rallis, and Gérard Schiffmann. Multiplicity one theorems. Ann. of Math. (2), 172(2):1407–1434, 2010.
  • [BHC62] Armand Borel and Harish-Chandra. Arithmetic subgroups of algebraic groups. Ann. of Math. (2), 75(3):485–535, 1962.
  • [ET23] Melissa Emory and Shuichiro Takeda. Contragredients and a multiplicity one theorem for general spin groups. Math. Z., 303(3):Paper No. 70, 54, 2023.
  • [GGP12] Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad. Symplectic local root numbers, central critical LL values, and restriction problems in the representation theory of classical groups. Number 346, pages 1–109. 2012. Sur les conjectures de Gross et Prasad. I.
  • [MP16] Keerthi Madapusi Pera. Integral canonical models for spin Shimura varieties. Compos. Math., 152(4):769–824, 2016.
  • [MZ40] Deane Montgomery and Leo Zippin. Topological transformation groups. i. Ann. of Math. (2), 41(4):778–791, 1940.
  • [Sch85] Winfried Scharlau. Quadratic and Hermitian forms, volume 270 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1985.
  • [Shi04] Goro Shimura. Arithmetic and analytic theories of quadratic forms and Clifford groups, volume 109 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2004.
  • [SZ11] Binyong Sun and Chen-Bo Zhu. A general form of Gelfand-Kazhdan criterion. Manuscripta Math., 136(1-2):185–197, 2011.
  • [SZ12] Binyong Sun and Chen-Bo Zhu. Multiplicity one theorems: the Archimedean case. Ann. of Math. (2), 175(1):23–44, 2012.
  • [Wal12] Jean-Loup Waldspurger. Une variante d’un résultat de Aizenbud, Gourevitch, Rallis et Schiffmann. Number 346, pages 313–318. 2012. Sur les conjectures de Gross et Prasad. I.