2.1. The $\operatorname {\mathrm {GSpin}}$ group

In this section, we recall how the $\operatorname {\mathrm {GSpin}}$ group is realized via the Clifford algebra. Our main reference is [Reference Shimura18], which introduces the Clifford algebras associated with quadratic spaces over arbitrary field of characteristic different from 2.

Let F be a local field of characteristic zero. Let V be a finite-dimensional quadratic space over F equipped with a non-degenerate quadratic form $q_V$ , giving rise to the groups $\operatorname {\mathrm {O}}(V)$ and $\operatorname {\mathrm {SO}}(V)$ :

$$ \begin{align*} \begin{aligned} \operatorname{\mathrm{O}}(V)&=\{g\in \operatorname{\mathrm{GL}}_F(V): q_V(gv)=q_V(v) \text{ for every } v\in V\},\\ \operatorname{\mathrm{SO}}(V)&=\{g\in \operatorname{\mathrm{O}}(V):\det(g)=1\}. \end{aligned} \end{align*} $$

The Clifford algebra $C(V)$ associated with $(V, q_V)$ is the quotient of the tensor algebra ${T(V)=\bigoplus _{d=0}^{\infty } V^{\otimes d}}$ by the two-sided ideal generated by all $v\otimes v - q_V(v)$ , for $v\in V$ . To simplify notation, we also write $v_1v_2\dots v_k$ for $v_1\otimes v_2\otimes \cdots \otimes v_k$ .

The inclusion map $V\to T(V)$ induces a canonical injection $\iota : V\to C(V)$ . Identifying v with $\iota (v)$ for every $v\in V$ , we view V as a subspace of $C(V)$ , so that $v^2=q_V(v)$ . For $v_1, v_2\in V$ , we set $\langle v_1, v_2\rangle : = q_V(v_1+v_2)-q_V(v_1)-q_V(v_2)$ the induced symmetric bilinear form, so that $\langle v, v \rangle =2q_V(v)$ . Similarly, for $x, y\in C(V)$ , we denote $\langle x, y\rangle =(x+y)^2-x^2-y^2$ . In particular, $\langle x, y\rangle $ measures if x and y anti-commute in $C(V)$ since

(1) $$ \begin{align} \langle x, y\rangle= (x+y)^2-x^2-y^2 = xy+yx \in C(V). \end{align} $$

Denote by $ C^\pm (V) $ the even and odd parts of $ C(V) $ , so that $C(V)=C^+(V)\oplus C^-(V)$ . The Clifford algebra $C(V)$ has a canonical involution $*:C(V)\to C(V)$ , induced by reversing the order of products in the tensor algebra, that is, $(v_1v_2\dots v_k)^*=v_k\dots v_2v_1$ for $v_1, \ldots , v_k\in V$ . The canonical involution preserves both $C^+(V)$ and $C^-(V)$ . We also define

$$ \begin{align*} \alpha:C(V)\to C(V), \quad \alpha(v_{+}+v_{-})=v_{+}-v_{-}, \end{align*} $$

where $v_{+}\in C^{+}(V)$ and $v_{-}\in C^{-}(V)$ . So $\alpha $ acts as the identity on $C^{+}(V)$ and as multiplication by $-1$ on $C^{-}(V)$ , and $\alpha (v_1v_2\dots v_k)=(-1)^k v_1 v_2\dots v_k$ for $v_1, \ldots , v_k\in V$ . Moreover, for all $x\in C(V),$ we define the Clifford involution

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

and the Clifford norm

$$ \begin{align*} N: C(V)\to C(V), \quad x\mapsto x \overline{x}. \end{align*} $$

It is clear that $N(\lambda x)=\lambda ^2 N(x)$ for all $\lambda \in F$ .

The $\operatorname {\mathrm {GPin}}$ group and $\operatorname {\mathrm {GSpin}}$ group of $(V, q_V)$ are defined by

$$ \begin{align*} \begin{aligned} \operatorname{\mathrm{GPin}}(V) &=\{g\in C(V)^\times: \alpha(g) V g^{-1}=V\}, \\ \operatorname{\mathrm{GSpin}}(V) &=\operatorname{\mathrm{GPin}}(V)\cap C^{+}(V). \end{aligned} \end{align*} $$

These are also called the Clifford group and the even Clifford group in the literature. Since $\alpha $ acts as identity on $C^+(V)$ , we have the inclusion $\operatorname {\mathrm {GSpin}}(V)\subset \operatorname {\mathrm {GPin}}(V)$ . In fact, we have $[\operatorname {\mathrm {GPin}}(V):\operatorname {\mathrm {GSpin}}(V)]=2$ , and $\operatorname {\mathrm {GSpin}}(V)$ is the identity component of $\operatorname {\mathrm {GPin}}(V)$ .

There is a natural projection map

(2) $$ \begin{align} \mathrm{pr}:\operatorname{\mathrm{GPin}}(V)\to \operatorname{\mathrm{O}}(V) \end{align} $$

sending $g\in \operatorname {\mathrm {GPin}}(V)$ to the map $v\mapsto \alpha (g)vg^{-1}$ . The kernel of this homomorphism consists of scalars, thus we get an exact sequence

$$ \begin{align*} 1\to F^\times \to \operatorname{\mathrm{GPin}}(V)\to \operatorname{\mathrm{O}}(V)\to 1. \end{align*} $$

Moreover, $\mathrm {pr}^{-1}(\operatorname {\mathrm {SO}}(V))=\operatorname {\mathrm {GSpin}}(V)$ , and we have a commutative diagram

where the rows are exact. It is known (see, e.g., [Reference Meinrenken14, Theorem 3.1]) that the elements of $\operatorname {\mathrm {GPin}}(V)$ are all products

(3) $$ \begin{align} g=v_1\dots v_k, \end{align} $$

where $v_1, \ldots , v_k\in V$ are non-isotropic. Given $g=v_1\dots v_k\in \operatorname {\mathrm {GPin}}(V)$ , the corresponding element $\mathrm {pr}(g)\in \operatorname {\mathrm {O}}(V)$ is a product

$$ \begin{align*} \mathrm{pr}(g)=r_{v_1}\dots r_{v_k}, \end{align*} $$

where each $r_{v_i}\in \operatorname {\mathrm {O}}(V)$ is the reflection in the hyperplane orthogonal to $Fv_i$ , that is, $r_{v_i} v_i=-v_i$ and $r_{v_i}x=x$ for every $x\in (Fv_i)^\perp $ . In particular, $\operatorname {\mathrm {GSpin}}(V)$ is given by products in (3) with k even.

Since $\operatorname {\mathrm {GPin}}(V)$ consists of products in (3), the restriction of the Clifford norm N to $\operatorname {\mathrm {GPin}}(V)$ has its image in $F^\times $ , thus we have a homomorphism

$$ \begin{align*} N:\operatorname{\mathrm{GPin}}(V)\to F^\times , \quad g\mapsto g\overline{g}. \end{align*} $$

For any $z\in \ker (\mathrm {pr})=F^\times $ , we have $N(z)=z^2$ . Also, for any $g\in \operatorname {\mathrm {GPin}}(V)$ , we have $g^{-1}=\frac {1}{N(g)}\overline {g}$ . In particular, for any $g\in \operatorname {\mathrm {GSpin}}(V)$ , we have $g^{-1}=\frac {1}{N(g)}\overline {g}=\frac {1}{g g^*}g^*$ .

We now describe the centers of $\operatorname {\mathrm {GPin}}(V)$ and $\operatorname {\mathrm {GSpin}}(V)$ . For this purpose, take an orthogonal basis $\{b_1, \ldots , b_n\}$ of V and denote $\zeta =b_1\dots b_n$ . It is clear that $\zeta \in \operatorname {\mathrm {GSpin}}(V)$ if $\dim (V)$ is even, and $\zeta \in \operatorname {\mathrm {GPin}}(V)\backslash \operatorname {\mathrm {GSpin}}(V)$ if $\dim (V)$ is odd.

Note that if $W\subset V$ is a non-degenerate subspace, then we have the inclusions of Clifford algebra $C(W)\subset C(V)$ and of even Clifford algebra $C^+(W)\subset C^+(V)$ , which map $F^\times $ onto itself. These induce the inclusions $\operatorname {\mathrm {GPin}}(W)\subset \operatorname {\mathrm {GPin}}(V)$ and $\operatorname {\mathrm {GSpin}}(W)\subset \operatorname {\mathrm {GSpin}}(V)$ .

Now let $V=V_1\oplus V_2$ be an orthogonal sum decomposition, where both $V_1, V_2$ are non-degenerate. If $g_1=v_1\dots v_k\in \operatorname {\mathrm {GPin}}(V_1)$ and $g_2=w_1\dots w_m\in \operatorname {\mathrm {GPin}}(V_2)$ , by using (1), we see that

$$ \begin{align*} g_1 g_2 g_1^{-1}=(-1)^{km} g_2. \end{align*} $$

In particular, if $g_1\in \operatorname {\mathrm {GSpin}}(V_1)$ and $g_2\in \operatorname {\mathrm {GSpin}}(V_2)$ , then $g_1g_2=g_2g_1$ . Thus, we can form the direct product $\operatorname {\mathrm {GSpin}}(V_1)\times \operatorname {\mathrm {GSpin}}(V_2)$ . We have a homomorphism $\operatorname {\mathrm {GSpin}}(V_1)\times \operatorname {\mathrm {GSpin}}(V_2)\to \operatorname {\mathrm {GSpin}}(V)$ given by $(g_1, g_2)\mapsto g_1g_2$ for $g_1\in \operatorname {\mathrm {GSpin}}(V_1), g_2\in \operatorname {\mathrm {GSpin}}(V_2)$ , which induces an injection

$$ \begin{align*} (\operatorname{\mathrm{GSpin}}(V_1)\times \operatorname{\mathrm{GSpin}}(V_2))/ \{(z, z^{-1}): z\in \operatorname{\mathrm{GL}}_1\} \to \operatorname{\mathrm{GSpin}}(V). \end{align*} $$

Moreover, the diagram

is commutative, where $\iota _{V_1,V_2}$ is the block diagonal embedding.

2.2. Parabolic subgroups

In this section, we recall an explicit realization of parabolic subgroups of $\operatorname {\mathrm {GSpin}}(V)$ in terms of Clifford algebra from [Reference Pollack15]. Let $X\subset V$ be an isotropic subspace, and we consider the parabolic subgroup $P_X$ of $\operatorname {\mathrm {GSpin}}(V)$ that stabilizes X, namely,

$$ \begin{align*} P_X=\{ g\in \operatorname{\mathrm{GSpin}}(V): g X g^{-1}=X\}. \end{align*} $$

The unipotent radical $N_X$ of $P_X$ has a two-step filtration $N_X\supset N_X^\prime \supset 1$ . Inside the Clifford algebra, its Lie algebra is given by $X^\perp \cdot X$ , and the exponential map identifies this nilpotent Lie algebra with $N_X$ . Moreover, this filtration identifies

$$\begin{align*}N_X'\cong \wedge^2 X, \qquad N_X/N_X'\cong (X^\perp/X)\otimes X. \end{align*}$$

If X is one-dimensional, then $N_X^\prime =1$ , while if $\dim (V)$ is even and $\dim (X)=\dim (V)/2$ , then $X^\perp =X$ and hence $N_X=N_X^\prime $ . In all other cases, we have both $N_X/N_X^\prime $ and $N_X^\prime $ are nontrivial. The image of $P_X$ under $\mathrm {pr}$ , $\mathrm {pr}(P_X)$ , is the corresponding parabolic subgroup of $\operatorname {\mathrm {SO}}(V)$ .

Associated with X, one can define an increasing filtration $\mathcal {W}^{X}_{\bullet }$ on $C(V)$ as follows. First, we define an increasing filtration on V by

$$ \begin{align*} \mathcal{W}_{k}^{X}(V):=\begin{cases} 0 & \text{ if }k<-1,\\ X & \text{ if }k=-1,\\ X^\perp & \text{ if }k=0,\\ V &\text{ if }k\ge 1. \end{cases} \end{align*} $$

Next, we define

$$ \begin{align*} \mathcal{W}_{k}^{X}(C(V)):=\{x\in C(V): x=\sum x_1 x_2 \dots x_m, x_i\in \mathcal{W}_{\alpha_i}^{X}(V), \alpha_1+\alpha_2+\dots \alpha_m\le k \}. \end{align*} $$

Lemma 2.3 [Reference Pollack15, Lemma 2.3].

The parabolic subgroup $P_X$ and its unipotent radical $N_X$ are given by

$$ \begin{align*} P_X = \mathcal{W}_{0}^{X}(C(V))\cap \operatorname{\mathrm{GSpin}}(V), \quad N_X=(1+ \mathcal{W}_{-1}^{X}(C(V))) \cap \operatorname{\mathrm{GSpin}}(V). \end{align*} $$

Now suppose Y is an isotropic subspace of V dual to X, and we take an orthogonal sum decomposition $V=X\oplus W\oplus Y$ . Let $M_X$ be the Levi subgroup of the parabolic subgroup $P_X$ of $\operatorname {\mathrm {GSpin}}(V)$ . Then the image in $\operatorname {\mathrm {SO}}(V)$ of $M_X$ is identified with $\operatorname {\mathrm {GL}}_F(X)\times \operatorname {\mathrm {SO}}(W)$ . To describe the lifting of the $\operatorname {\mathrm {GL}}_F(X)$ factor to $\operatorname {\mathrm {GSpin}}(V)$ explicitly, we take $X=\mathrm {Span}\{e_1, \ldots , e_m\}$ and $Y=\mathrm {Span}\{e_{-m}, \ldots , e_{-1}\}$ such that $\langle e_i, e_{-j}\rangle =\delta _{i,j}$ for $1\le i, j\le m$ . Let $\{w_1, \ldots , w_{\dim W}\}$ be a basis of W. Then we have the following basis of V:

(4) $$ \begin{align} \{e_1, \ldots, e_m, w_1, \ldots, w_{\dim W}, e_{-m}, \ldots, e_{-1}\}. \end{align} $$

Let $J_m$ denote the $m\times m$ anti-diagonal matrix with entries 1 on the anti-diagonal and 0 everywhere else.

Lemma 2.4 [Reference Shimura18, Proposition 4.8].

There exists an injective algebraic group homomorphism

$$ \begin{align*} \ell_m:\operatorname{\mathrm{GL}}_F(X)=\operatorname{\mathrm{GL}}_m(F)\to P_X \end{align*} $$

with the following properties:

1. 1. For any $g\in \operatorname {\mathrm {GL}}_m(F)$ , we have $\mathrm {pr}(\ell _m(g))=\begin {pmatrix} g & & \\ &I_W &\\ & & J_m {}^tg^{-1} J_m \end {pmatrix}\in \operatorname {\mathrm {SO}}(V)$ , where the matrix is written with respect to the ordered basis (4).

2. 2. For any $g\in \operatorname {\mathrm {GL}}_m(F)$ , we have $N(\ell _m(g))=\det (g)$ .

3. 3. For any diagonal element $\operatorname {\mathrm {diag}}(a_1, \ldots , a_m)\in \operatorname {\mathrm {GL}}_m(F)$ , we have $\ell _m(\operatorname {\mathrm {diag}}(a_1, \ldots , a_m))=\prod _{i=1}^m (a_i e_ie_{-i}+e_{-i} e_i)$ .

4. 4. If $1\le m^\prime <m$ and $g_0\in \operatorname {\mathrm {GL}}_F(\mathrm {Span}\{e_1, \ldots , e_{m^\prime } \} )=\operatorname {\mathrm {GL}}_{m^\prime }(F)$ , let $g\in \operatorname {\mathrm {GL}}_m(F)$ be the element such that g acts as $g_0$ on $\mathrm {Span}\{e_1, \ldots , e_{m^\prime } \}$ and acts trivially on $\mathrm {Span}\{e_{m^\prime +1}, \ldots , e_{m} \}$ . Then $\ell _m(g)=\ell _{m^\prime }(g_0)$ .

5. 5. The image $\ell _m(\operatorname {\mathrm {GL}}_m(F))$ is an algebraic subgroup of $P_X$ .

Moreover, $\ell _m$ is uniquely determined, as an algebraic group homomorphism, by properties (1)–(3).

Let $Z_m\subset \operatorname {\mathrm {GL}}_m(F)$ be the unipotent radical of the Borel subgroup $B_m=T_m Z_m$ of $\operatorname {\mathrm {GL}}_m(F)$ stabilizing the flag

$$ \begin{align*} \mathrm{Span}\{e_1\} \subset \mathrm{Span}\{e_1, e_2\} \subset \cdots \subset \mathrm{Span}\{e_1, e_2, \ldots, e_m\}. \end{align*} $$

In the next lemma, we describe the lifting of the unipotent group $Z_m$ .

Proof. Set $x=\gamma e_i e_{-j}$ . Since $i\neq j$ , we have $\langle e_i,e_{-j}\rangle =e_i e_{-j}+e_{-j} e_i=0$ , and hence $(e_i e_{-j})^2=e_i (-e_i e_{-j})e_{-j}=0$ . Therefore, $(1+x)^{-1}=1-x.$ By the uniqueness of $\ell _m$ in Lemma 2.4, it suffices to check that $1+x$ agrees with $u_{i,j}(\gamma )$ on properties (1)–(3) of Lemma 2.4.

We first compute the action of $1+x$ on V by conjugation. For $e_j$ , using $e_{-j} e_j=1-e_je_{-j}$ , we get

$$ \begin{align*} \begin{aligned} (1+x)e_j(1-x) =e_j+x e_j-e_jx-xe_jx = e_j+\gamma e_i e_{-j} e_j-\gamma e_j e_i e_{-j}-\gamma^2 e_i e_{-j} e_j e_i e_{-j}. \end{aligned} \end{align*} $$

Now $e_i e_{-j} e_j=e_i(1-e_je_{-j})=e_i-e_i e_j e_{-j}$ and $e_j e_i e_{-j}=-e_i e_j e_{-j}.$ Moreover, the last term is zero because $e_i e_{-j} e_je_i=-e_{-j} e_ie_j e_i=e_{-j} e_j e_i^2=0$ . Hence, $(1+x)e_j(1-x)=e_j+\gamma e_i.$ For $k\neq j$ , a direct calculation gives $(1+x)e_k(1-x)=e_k.$ Thus, the action of $1+x$ on $X=\mathrm {Span}\{e_1,\dots ,e_m\}$ is given by

$$\begin{align*}e_j\mapsto e_j+\gamma e_i,\quad e_k\mapsto e_k\quad \text{for}\ k\neq j. \end{align*}$$

Next, since every $w\in W$ is orthogonal to both $e_i$ and $e_{-j}$ , it commutes with the product $e_ie_{-j}$ . Hence,

$$\begin{align*}(1+x)w(1-x)=w. \end{align*}$$

On the Y-part, one checks similarly that the action of $1+x$ is given by

$$ \begin{align*} e_{-i}\mapsto e_{-i}-\gamma e_{-j}, \quad e_{-k}\mapsto e_{-k} \quad \text{ for } k\neq i. \end{align*} $$

We conclude that the action of $1+x$ on V is given by

$$\begin{align*}\begin{pmatrix} u_{i,j}(\gamma) & & \\ & I_W & \\ & & J_m{}^tu_{i,j}(\gamma)^{-1}J_m \end{pmatrix}. \end{align*}$$

We now compute the Clifford norm of $1+x$ . Since $(e_ie_{-j})^*=e_{-j} e_i=-e_i e_{-j},$ we have ${(1+\gamma e_ie_{-j})^*=1-\gamma e_ie_{-j}.}$ Thus,

$$\begin{align*}N(1+\gamma e_ie_{-j}) = (1+\gamma e_ie_{-j})(1-\gamma e_ie_{-j}) = 1-\gamma^2(e_ie_{-j})^2 = 1. \end{align*}$$

This agrees with $\det (u_{i,j}(\gamma ))=1.$

Finally, the unipotent matrix $u_{i,j}(\gamma )$ is diagonal if and only if $\gamma =0$ , reducing it to the identity matrix $I_m=\operatorname {\mathrm {diag}}(1, \ldots , 1)$ , which agrees with $\prod _{i=1}^m (e_i e_{-i}+e_{-i} e_i)=1$ .

By the uniqueness of the lift $\ell _m$ in Lemma 2.4, we obtain $\ell _m(u_{i,j}(\gamma ))=1+\gamma e_i e_{-j}$ .

3.1. The Bessel subgroup H and the character $\xi $

Let V be a non-degenerate quadratic space over F, and assume that $\dim (V)>3$ (see Remark 1.2). Let $V_0\subset V$ be a non-degenerate subspace such that $V_0^\perp $ is split of dimension $2r+1$ . Write ${V_0^\perp =X_r\oplus Y_r\oplus Fe_0,}$ where $X_r$ and $Y_r$ are maximal isotropic subspaces of $V_0^\perp $ in duality, and $e_0$ is non-isotropic. Set ${V_0'=V_0\oplus Fe_0}$ . Thus, $V=X_r\oplus V_0'\oplus Y_r.$

Let $P_r=N_{P_r}\rtimes M_{P_r}$ be the parabolic subgroup of $\operatorname {\mathrm {GSpin}}(V)$ stabilizing $X_r$ , namely,

$$\begin{align*}P_r=\{g\in \operatorname{\mathrm{GSpin}}(V):gX_rg^{-1}=X_r\}. \end{align*}$$

Then the Levi subgroup $M_{P_r}=\ell _r(\operatorname {\mathrm {GL}}_F(X_r))\times \operatorname {\mathrm {GSpin}}(V_0^\prime )\cong \operatorname {\mathrm {GL}}_F(X_r) \times \operatorname {\mathrm {GSpin}}(V_0^\prime )$ , where we recall $\ell _r$ is the map from Lemma 2.4. The unipotent radical $N_{P_r}$ has a two-step filtration $N_{P_r}\supset N_{P_r}'\supset 1,$ where $N_{P_r}'\cong \wedge ^2 X_r$ and $N_{P_r}/N_{P_r}'\cong X_r\otimes V_0'.$ We have a short exact sequence

Let $l_{X_r}:X_r\to F$ be a nonzero F-linear homomorphism, and let $l_{V_0}:V_0'\to F$ be a nonzero F-linear homomorphism which vanishes on $V_0$ . Then $l_{V_0}$ is supported on the line $Fe_0$ . We obtain a linear map $l_{X_r}\otimes l_{V_0}:X_r\otimes V_0'\longrightarrow F.$ Explicitly, we have

$$ \begin{align*} (l_{X_r}\otimes l_{V_0})(x\otimes (v+c e_0))=c\cdot l_{X_r}(x) l_{V_0}(e_0)\quad \text{ for }x\in X_r, v\in V_0, c\in F. \end{align*} $$

By composing with the natural quotient map, we define a map $l_{N_{P_r}}:N_{P_r}\longrightarrow F$ by

Let $Z_{X_r}$ be a maximal unipotent subgroup of $\operatorname {\mathrm {GL}}_F(X_r)$ stabilizing $l_{X_r}$ . Then the subgroup $Z_{X_r}\times \operatorname {\mathrm {GSpin}}(V_0)$ of the standard Levi subgroup $M_{P_r}$ fixes $l_{N_{P_r}}$ . Indeed, $Z_{X_r}$ fixes $l_{X_r}$ by definition, and $\operatorname {\mathrm {GSpin}}(V_0)$ acts trivially on $Fe_0$ .

Set

$$\begin{align*}N=N_{P_r}\rtimes Z_{X_r}\ \subset \operatorname{\mathrm{GSpin}}(V). \end{align*}$$

We define the rth Bessel subgroup H of $\operatorname {\mathrm {GSpin}}(V)$ by

$$\begin{align*}H= N\rtimes \operatorname{\mathrm{GSpin}}(V_0). \end{align*}$$

We extend $l_{N_{P_r}}$ to $N=N_{P_r}\rtimes Z_{X_r}$ by making it trivial on $Z_{X_r}$ . Let $\psi $ be a nontrivial additive unitary character of F, and let

$$\begin{align*}\lambda_{X_r}:Z_{X_r}\to \mathbb{S}^1 \end{align*}$$

be a generic unitary character. We define a character $\xi $ of H by

$$\begin{align*}\xi(nzh)=\lambda_{X_r}(z)\psi(l_{N_{P_r}}(n)), \end{align*}$$

for $z\in Z_{X_r}$ , $n\in N_{P_r}$ , and $h\in \operatorname {\mathrm {GSpin}}(V_0)$ . This is well-defined because $Z_{X_r}$ and $\operatorname {\mathrm {GSpin}}(V_0)$ fix $l_{N_{P_r}}$ . The pair $(H,\xi )$ is uniquely determined, up to conjugacy in $G=\operatorname {\mathrm {GSpin}}(V)\times \operatorname {\mathrm {GSpin}}(V_0)$ , by the pair $V_0\subset V$ .

We now give a more explicit description of $(H,\xi )$ . Let $\{e_1,\dots ,e_r\}$ be a basis of $X_r$ , and let $\{e_{-r},\dots ,e_{-1}\}$ be the dual basis of $Y_r$ , with $\langle e_i,e_{-j}\rangle =\delta _{ij}$ for $1\le i,j\le r.$ Thus, in the Clifford algebra, we have

(5) $$ \begin{align} e_i^2=e_{-i}^2=0,\quad e_i e_{-i}+e_{-i}e_i=1,\quad e_i e_{-j}=-e_{-j}e_i \end{align} $$

for $1\le i,j\le r$ with $i\neq j$ .

Let $Q_r=N_{Q_r}\rtimes M_{Q_r}$ denote the standard parabolic subgroup of $\operatorname {\mathrm {GSpin}}(V)$ stabilizing the flag

(6) $$ \begin{align} 0=X_0\subset X_1=Fe_1\subset X_2=Fe_1\oplus Fe_2 \subset\cdots\subset X_r=Fe_1\oplus\cdots\oplus Fe_r. \end{align} $$

Then the Levi subgroup $M_{Q_r} =\ell _r((\operatorname {\mathrm {GL}}_1)^r)\times \operatorname {\mathrm {GSpin}}(V_0^\prime ) \cong (\operatorname {\mathrm {GL}}_1)^r\times \operatorname {\mathrm {GSpin}}(V_0^\prime )$ , where $(\operatorname {\mathrm {GL}}_1)^r\subset \operatorname {\mathrm {GL}}_F(X_r)$ is the diagonal torus with respect to the basis $\{e_1, \ldots , e_r\}$ . The $(\operatorname {\mathrm {GL}}_1)^r$ factor is represented explicitly by $\prod _{i=1}^r(a_i e_i e_{-i}+e_{-i}e_i)\in \operatorname {\mathrm {GSpin}}(V),$ where $a_i\in F^\times $ . It acts by

(7) $$ \begin{align} \prod_{i=1}^r(a_i e_i e_{-i}+e_{-i}e_i) e_k \prod_{i=1}^r(a_i e_i e_{-i}+e_{-i}e_i) ^{-1}=a_k e_k, \quad 1\le k\le r, \end{align} $$

and

(8) $$ \begin{align} \prod_{i=1}^r(a_i e_i e_{-i}+e_{-i}e_i) e_{-k} \prod_{i=1}^r(a_i e_i e_{-i}+e_{-i}e_i) ^{-1}=a_k^{-1}e_{-k}, \quad 1\le k\le r. \end{align} $$

The torus element $\prod _{i=1}^r(a_i e_i e_{-i}+e_{-i}e_i)$ fixes the line $Fe_0$ . Moreover, $\operatorname {\mathrm {GSpin}}(V_0')$ acts trivially on $X_r\oplus Y_r$ . The unipotent radical of $Q_r$ is $N_{Q_r}=Z_r\ltimes N_{P_r},$ where $Z_r=Z_{X_r}$ is the unipotent radical of the Borel subgroup $B_r=T_rZ_r$ of $\operatorname {\mathrm {GL}}_r=\operatorname {\mathrm {GL}}_F(X_r)$ stabilizing the flag (6). Thus, with this choice of $Z_r$ , we have $N_{Q_r}=N.$

Define a map

$$\begin{align*}l_r:N_{Q_r}\longrightarrow F^r \end{align*}$$

by

$$\begin{align*}l_r(u)=(x_1,\dots,x_{r-1},z), \end{align*}$$

where

(9) $$ \begin{align} x_i=x_i(u)=\langle \mathrm{pr}(u)e_{i+1},e_{-i}\rangle, \quad 1\le i\le r-1, \end{align} $$

and

(10) $$ \begin{align} z=z(u)=\langle \mathrm{pr}(u)e_0,e_{-r}\rangle. \end{align} $$

We first observe that $l_r$ is fixed by $\operatorname {\mathrm {GSpin}}(V_0)$ -conjugation. Indeed, for $h\in \operatorname {\mathrm {GSpin}}(V_0)$ and $u\in N_{Q_r}$ , since $\mathrm {pr}(h)$ fixes each $e_i$ , $e_{-i}$ , and $e_0$ , we have

$$ \begin{align*} \langle \mathrm{pr}(huh^{-1})e_{i+1},e_{-i}\rangle &= \langle \mathrm{pr}(h)\mathrm{pr}(u)\mathrm{pr}(h^{-1})e_{i+1},e_{-i}\rangle \\ &= \langle \mathrm{pr}(h)\mathrm{pr}(u)e_{i+1},e_{-i}\rangle \\ &= \langle \mathrm{pr}(u)e_{i+1},\mathrm{pr}(h^{-1})e_{-i}\rangle \\ &= \langle \mathrm{pr}(u)e_{i+1},e_{-i}\rangle. \end{align*} $$

Similarly,

$$\begin{align*}\langle \mathrm{pr}(huh^{-1})e_0,e_{-r}\rangle = \langle \mathrm{pr}(u)e_0,e_{-r}\rangle. \end{align*}$$

Hence, $l_r$ is invariant under $\operatorname {\mathrm {GSpin}}(V_0)$ -conjugation. The conjugation action of the torus $t=\prod _{i=1}^r(a_i e_i e_{-i}+e_{-i}e_i)$ , where $a_i\in F^\times $ , is given by

$$\begin{align*}\begin{aligned} \langle \mathrm{pr}(tut^{-1})e_{i+1},e_{-i}\rangle &= \langle \mathrm{pr}(t)\mathrm{pr}(u)\mathrm{pr}(t^{-1})e_{i+1},e_{-i}\rangle \\ &= a_{i+1}^{-1} \langle \mathrm{pr}(t)\mathrm{pr}(u)e_{i+1},e_{-i}\rangle \quad (\text{by}\ (7)) \\ &= a_{i+1}^{-1} \langle \mathrm{pr}(u)e_{i+1},\mathrm{pr}(t^{-1})e_{-i}\rangle \\ &= a_i a_{i+1}^{-1} \langle \mathrm{pr}(u)e_{i+1},e_{-i}\rangle \quad (\text{by}\ (8)). \end{aligned} \end{align*}$$

Similarly, since t fixes $e_0$ and $t^{-1}e_{-r}t=a_r e_{-r}$ , we have $\langle \mathrm {pr}(tut^{-1})e_0,e_{-r}\rangle =a_r \langle \mathrm {pr}(u) e_0,e_{-r}\rangle $ . The subgroup of $M_{Q_r}$ fixing $l_r$ is precisely $\operatorname {\mathrm {GSpin}}(V_0)$ .

For a nontrivial additive unitary character $\psi $ of F, define a character $\xi $ of $N_{Q_r}$ by

(11) $$ \begin{align} \xi(u)=\psi\left(\sum_{i=1}^{r-1}x_i+z\right), \end{align} $$

where $x_i$ and z are given by (9) and (10). Since $l_r$ is invariant under $\operatorname {\mathrm {GSpin}}(V_0)$ -conjugation, $\xi $ is invariant under $\operatorname {\mathrm {GSpin}}(V_0)$ -conjugation. Hence, $\xi $ extends to a character

$$\begin{align*}\xi:N_{Q_r}\rtimes \operatorname{\mathrm{GSpin}}(V_0)\longrightarrow \mathbb{C}^\times \end{align*}$$

by $\xi (nh)=\xi (n)$ for $n\in N_{Q_r},\ h\in \operatorname {\mathrm {GSpin}}(V_0).$ As noted above, up to conjugacy in $G=\operatorname {\mathrm {GSpin}}(V)\times \operatorname {\mathrm {GSpin}}(V_0)$ , the pair $(H, \xi )$ depends only on the pair $V_0\subset V$ , and not on the choices of $\psi $ , or the basis vectors $e_i$ .

We now prove that the Bessel subgroup H is unimodular.

3.2. The group $\operatorname {\mathrm {GSpin}}(V^\prime )$

Consider $V^\prime =V\oplus F f_0$ . We equip $V^\prime $ with a symmetric bilinear form $\langle \cdot , \cdot \rangle _{V^\prime }$ so that

$$ \begin{align*} \langle v_1, v_2 \rangle_{V^\prime}=\langle v_1, v_2\rangle, \quad \langle v, f_0\rangle_{V^\prime} =0, \quad \langle f_0, f_0\rangle_{V^\prime}=-\langle e_0, e_0\rangle, \quad \text{ for all }v, v_1, v_2\in V. \end{align*} $$

Denote $e_{r+1}=e_0+f_0$ , $e_{-(r+1)}=\frac {1}{2\langle e_0, e_0\rangle }(e_0-f_0)$ . Then $V^\prime $ is the orthogonal sum

$$ \begin{align*} V^\prime= (X^\prime_{r+1}\oplus Y^\prime_{r+1}) \oplus V_0, \end{align*} $$

where

$$ \begin{align*} X_{r+1}^\prime= X_r\oplus F e_{r+1}, \quad Y^\prime_{r+1}= Y_r \oplus F e_{-(r+1)} \end{align*} $$

are totally isotropic subspaces.

Recall $V_0^\prime =V_0\oplus F e_0$ , and $P_r=N_{P_r} \rtimes M_{P_r}$ is the parabolic subgroup of $\operatorname {\mathrm {GSpin}}(V)$ stabilizing $X_r$ , where $M_{P_r}\cong \operatorname {\mathrm {GSpin}}(V_0^\prime )\times \operatorname {\mathrm {GL}}_r$ , $\operatorname {\mathrm {GL}}_r=\operatorname {\mathrm {GL}}_F(X_r)$ . Denote by $P_{r+1}^\prime = N_{P_{r+1}^\prime } \rtimes M_{P_{r+1}^\prime }$ the parabolic subgroup of $\operatorname {\mathrm {GSpin}}(V^\prime )$ stabilizing $X_{r+1}^\prime $ . Then we have

$$ \begin{align*} \begin{aligned} M_{P_{r+1}^\prime} \cong \operatorname{\mathrm{GSpin}}(V_0)\times \operatorname{\mathrm{GL}}_{r+1}, \quad \operatorname{\mathrm{GL}}_{r+1}:=\operatorname{\mathrm{GL}}_F(X_{r+1}^\prime) \supset \operatorname{\mathrm{GL}}_r. \end{aligned} \end{align*} $$

We view $\operatorname {\mathrm {GL}}_r$ as a subgroup of $\operatorname {\mathrm {GL}}_{r+1}$ by acting trivially on $e_{r+1}$ . Recall that $Z_r$ is the unipotent radical of the Borel subgroup $B_{r}=T_{r}Z_{r}$ of $\operatorname {\mathrm {GL}}_F(X_{r} )$ stabilizing the flag (6). Let $Z_{r+1}$ be the unipotent radical of the Borel subgroup $B_{r+1}=T_{r+1}Z_{r+1}$ of $\operatorname {\mathrm {GL}}(X_{r+1}^\prime )$ stabilizing the flag

(12) $$ \begin{align} X_1 \subset X_2 \subset\cdots \subset X_r \subset X_{r+1}^\prime. \end{align} $$

We observe that

(13) $$ \begin{align} H=N_{P_r}\rtimes (\operatorname{\mathrm{GSpin}}(V_0)\times Z_r)\subset P_r= N_{P_r} \rtimes (\operatorname{\mathrm{GSpin}}(V_0^\prime)\times \operatorname{\mathrm{GL}}_r) \end{align} $$

and

$$ \begin{align*} H \subset P_{r+1}^\prime\cap \operatorname{\mathrm{GSpin}}(V). \end{align*} $$

Now we analyze the structure of the group $P_{r+1}^\prime \cap \operatorname {\mathrm {GSpin}}(V)$ . Let $P^\prime _r=N_{P_{r}^\prime } \rtimes M_{P_{r}^\prime }$ be the parabolic subgroup of $\operatorname {\mathrm {GSpin}}(V^\prime )$ stabilizing $X_r$ .

Proof. We first show that $P_{r+1}^\prime \cap \operatorname {\mathrm {GSpin}}(V) \subset P_r^\prime \cap \operatorname {\mathrm {GSpin}}(V)$ . Take any $g\in P_{r+1}^\prime \cap \operatorname {\mathrm {GSpin}}(V)$ . It suffices to show that $\mathrm {pr}(g)$ also stabilizes $X_r$ , that is, for any $x\in X_r$ , we need to show that ${\langle \mathrm {pr}(g)x, e_0-f_0 \rangle _{V^\prime }=0}$ . We compute that

$$ \begin{align*} \begin{aligned} \langle \mathrm{pr}(g)x, e_0-f_0 \rangle_{V^\prime} &= \langle x, \mathrm{pr}(g)^{-1}(e_0-f_0) \rangle_{V^\prime} \\ &= \langle x, \mathrm{pr}(g)^{-1}(e_{r+1}-2f_0) \rangle_{V^\prime} \\ &= \langle x, \mathrm{pr}(g)^{-1}e_{r+1} \rangle_{V^\prime} -2\langle x, \mathrm{pr}(g)^{-1} f_0 \rangle_{V^\prime}. \end{aligned} \end{align*} $$

Since $g^{-1}\in \operatorname {\mathrm {GSpin}}(V)$ , $g^{-1}$ acts trivially on $f_0$ , that is, $\mathrm {pr}(g^{-1})f_0=f_0$ . Hence,

$$ \begin{align*} \langle x, \mathrm{pr}(g)^{-1} f_0 \rangle_{V^\prime}=\langle x, f_0 \rangle_{V^\prime}=0. \end{align*} $$

Since $g^{-1}\in P_{r+1}^\prime $ , it stabilizes $X_{r+1}^\prime $ and $\mathrm {pr}(g^{-1})e_{r+1}\in X_{r+1}^\prime $ . Since $X_{r+1}^\prime $ is totally isotropic, we have

$$ \begin{align*} \langle x, \mathrm{pr}(g)^{-1}e_{r+1}\rangle_{V^\prime}=0. \end{align*} $$

Thus,

$$ \begin{align*} \langle \mathrm{pr}(g)x, e_0-f_0 \rangle_{V^\prime}=0. \end{align*} $$

This proves $P_{r+1}^\prime \cap \operatorname {\mathrm {GSpin}}(V) \subset P_r^\prime \cap \operatorname {\mathrm {GSpin}}(V)$ .

Next, we show that as a subgroup of $P_r^\prime \cap \operatorname {\mathrm {GSpin}}(V)$ ,

$$ \begin{align*}P_{r+1}^\prime\cap \operatorname{\mathrm{GSpin}}(V) =(N_{P_{r}^\prime}\cap \operatorname{\mathrm{GSpin}}(V)) \rtimes (\operatorname{\mathrm{GL}}(X_r)\times \operatorname{\mathrm{GSpin}}(V_0)). \end{align*} $$

Let h be an element of $P_r^\prime \cap \operatorname {\mathrm {GSpin}}(V)$ , so that $\mathrm {pr}(h)x\in X_{r}$ for all $x\in X_r$ and $\mathrm {pr}(h)f_0=f_0$ . Note that

$$ \begin{align*} \begin{aligned} h\in P_{r+1}^\prime\cap \operatorname{\mathrm{GSpin}}(V) &\iff \mathrm{pr}(h) e_{r+1}\in X_{r+1}^\prime \\ &\iff \mathrm{pr}(h)e_0+f_0\in X_{r+1}^\prime\\ &\iff \mathrm{pr}(h)e_0-e_0\in X_{r+1}^\prime. \end{aligned} \end{align*} $$

Also, if $x\in X_r$ , then $\mathrm {pr}(h)^{-1}x\in X_r$ and hence

$$ \begin{align*} \langle \mathrm{pr}(h)e_0, x\rangle_{V^\prime}= \langle e_0, \mathrm{pr}(h)^{-1}x \rangle_{V^\prime}=0. \end{align*} $$

Thus, we may write $\mathrm {pr}(h)e_0=\lambda e_0+w+x$ for some $\lambda \in F$ , $w\in V_0$ , $x\in X_r$ . The condition $\mathrm {pr}(h)e_0-e_0\in X_{r+1}^\prime $ if and only if $\lambda =1$ and $w=0$ , implying that $\mathrm {pr}(h)$ fixes $e_0$ modulo $X_r$ and hence stabilizes $V_0$ modulo $X_r$ , consequently, $\mathrm {pr}(h)\in (N_{P_{r}^\prime }\cap \operatorname {\mathrm {GSpin}}(V)) \rtimes (\operatorname {\mathrm {GL}}(X_r)\times \operatorname {\mathrm {GSpin}}(V_0))$ , as desired. This completes the proof.

Let $\alpha :P_{r+1}^\prime \twoheadrightarrow \ell _{r+1}(\operatorname {\mathrm {GL}}_{r+1})\times \operatorname {\mathrm {GSpin}}(V_0)\cong \operatorname {\mathrm {GL}}_{r+1}\times \operatorname {\mathrm {GSpin}}(V_0)$ be the natural projection, whose kernel is the unipotent radical $N_{P_{r+1}^\prime }$ .

Lemma 3.3. The natural projection map $\alpha :P_{r+1}^\prime \twoheadrightarrow \operatorname {\mathrm {GSpin}}(V_0)\times \operatorname {\mathrm {GL}}_{r+1}$ induces the following commutative diagram with exact rows, where the vertical arrows are inclusions:

Here, $R\subset \operatorname {\mathrm {GL}}(X_{r+1}^\prime )$ is the mirabolic subgroup which stabilizes the co-dimension one subspace ${X_r\subset X_{r+1}^\prime }$ and fixes $e_{r+1}$ modulo $X_r$ , and $\mathrm {Hom}(F e_{r+1}, X_r)$ is the unipotent radical of R.

Proof. We consider the restriction of $\alpha $ to the subgroup $P_{r+1}^\prime \cap \operatorname {\mathrm {GSpin}}(V) =N_{P_{r}} \rtimes (\operatorname {\mathrm {GL}}(X_r)\times \operatorname {\mathrm {GSpin}}(V_0))$ . Observe that the subgroup $\operatorname {\mathrm {GL}}_r\times \operatorname {\mathrm {GSpin}}(V_0)$ is mapped isomorphically to its image in $\operatorname {\mathrm {GL}}_{r+1} \times \operatorname {\mathrm {GSpin}}(V_0)$ , and its image is precisely $\operatorname {\mathrm {GL}}_r\times \operatorname {\mathrm {GSpin}}(V_0)$ , which is the Levi subgroup of $R\times \operatorname {\mathrm {GSpin}}(V_0)$ .

Next, we analyze the kernel of $\alpha $ restricted to $P_{r+1}^\prime \cap \operatorname {\mathrm {GSpin}}(V)$ , which is given by

$$ \begin{align*}\ker(\alpha|_{P_{r+1}^\prime\cap \operatorname{\mathrm{GSpin}}(V)}) =N_{P_{r+1}^\prime}\cap \operatorname{\mathrm{GSpin}}(V). \end{align*} $$

We claim that $\ker (\alpha |_{P_{r+1}^\prime \cap \operatorname {\mathrm {GSpin}}(V)})$ is contained in $N_{P_r^\prime }\cap \operatorname {\mathrm {GSpin}}(V)$ . To prove this, take $h\in N_{P_{r+1}^\prime }\cap \operatorname {\mathrm {GSpin}}(V)$ , so that $\mathrm {pr}(h)f_0=f_0$ . Our goal is to show that $\mathrm {pr}(h)$ acts trivially on $X_r$ and acts trivially on $V_0^\prime $ modulo $X_r$ . Since $h\in N_{P_{r+1}^\prime }$ , $\mathrm {pr}(h)$ acts trivially on $X_{r+1}^\prime $ and acts trivially on $V_0$ modulo $X_{r+1}^\prime $ . Since $X_r\subset X_{r+1}^\prime $ , $\mathrm {pr}(h)$ acts trivially on $X_r$ . Also, for $w\in V_0$ , we have $\mathrm {pr}(h)w-w\in X_{r+1}^\prime \cap V=X_r$ . It remains to show that $\mathrm {pr}(h)e_0-e_0\in X_r=X_{r+1}^\prime \cap V$ . Since $\mathrm {pr}(h)e_0-e_0\in V$ , it suffices to show that $\mathrm {pr}(h)e_0-e_0\in X_{r+1}^\prime $ . We have

$$ \begin{align*} \mathrm{pr}(h)e_0-e_0=\mathrm{pr}(h)(e_0+f_0)-(e_0+f_0)=\mathrm{pr}(h)e_{r+1}-e_{r+1} \in X_{r+1}^\prime, \end{align*} $$

as desired. This proves the claim.

Now we show that the map $\alpha $ induces an isomorphism

$$ \begin{align*} (N_{P_r^\prime}\cap\operatorname{\mathrm{GSpin}}(V)) / (N_{P_{r+1}^\prime}\cap \operatorname{\mathrm{GSpin}}(V)) \cong \mathrm{Hom}(F e_{r+1}, X_r). \end{align*} $$

If $h\in N_{P_r^\prime }\cap \operatorname {\mathrm {GSpin}}(V)$ , then $\mathrm {pr}(h)$ acts trivially on $X_r$ , and $\mathrm {pr}(h)$ acts trivially on $V_0^\prime $ modulo $X_r$ , and $\mathrm {pr}(h)f_0=f_0$ . Then

$$ \begin{align*} \mathrm{pr}(h)e_{r+1}-e_{r+1}=\mathrm{pr}(h)(e_0+f_0)-(e_0+f_0)=\mathrm{pr}(h)e_0-e_0\in X_r. \end{align*} $$

It follows that $\alpha (h)$ lies in the unipotent radical $\mathrm {Hom}(Fe_{r+1}, X_r)$ of R. It remains to prove that $\alpha |_{P_{r+1}^\prime \cap \operatorname {\mathrm {GSpin}}(V)}$ is surjective onto $\mathrm {Hom}(Fe_{r+1}, X_r)$ . For any $x\in X_r$ , set $h_x=1+\frac {1}{\langle e_0,e_0\rangle }xe_0.$ Then $h_x\in N_{P_r^\prime }\cap \operatorname {\mathrm {GSpin}}(V)$ , fixes $X_r$ and $V_0$ , and satisfies $\mathrm {pr}(h_x)e_0=e_0+x$ . Then $\mathrm {pr}(\alpha (h_x))(e_{r+1})=x$ , as desired. Thus, we have the commutative diagram.

By Lemma 3.3, we see that the preimage of $Z_{r+1}\times \operatorname {\mathrm {GSpin}}(V_0)$ in $P_{r+1}^\prime \cap \operatorname {\mathrm {GSpin}}(V)$ under the restriction of $\alpha $ is precisely the Bessel subgroup

$$ \begin{align*} H=N_{P_r}\rtimes (\operatorname{\mathrm{GSpin}}(V_0)\times Z_r). \end{align*} $$

Thus, the restriction of $\alpha $ gives a surjective homomorphism

$$ \begin{align*} \alpha|_H:H \twoheadrightarrow Z_{r+1}\times\operatorname{\mathrm{GSpin}}(V_0). \end{align*} $$

Moreover, the character $\xi :H\to \mathbb {C}^\times $ is trivial on the kernel of $\alpha |_H$ . Indeed, the kernel of $\alpha |_H$ is $N_{P_{r+1}'}\cap \operatorname {\mathrm {GSpin}}(V)$ , and this subgroup is contained in the kernel of $\xi $ . Hence, $\xi $ descends to a character on $Z_{r+1}\times \operatorname {\mathrm {GSpin}}(V_0)$ . Equivalently, there exists a character $\chi $ of $Z_{r+1}\times \operatorname {\mathrm {GSpin}}(V_0)$ such that

(14) $$ \begin{align} \xi=\chi\circ \alpha|_{H}. \end{align} $$

Here, $\chi $ is a unitary generic character on $Z_{r+1}$ , extended to a character on $Z_{r+1}\times \operatorname {\mathrm {GSpin}}(V_0)$ by ${\chi (uh)=\chi (u)}$ for $u\in Z_{r+1}, h\in \operatorname {\mathrm {GSpin}}(V_0)$ .

3.3. Induced representations of $\operatorname {\mathrm {GSpin}}(V')$

Let $\pi _0$ be an irreducible admissible representation of $\operatorname {\mathrm {GSpin}}(V_0)$ and let $\sigma $ be an irreducible admissible $\chi ^{-1}$ -generic representation of $\operatorname {\mathrm {GL}}_{r+1}$ . The existence of such $\sigma $ is well known (see, e.g., [Reference Zelevinsky24, Theorem 9.7] when F is non-archimedean and [Reference Casselman, Hecht and Miličić5, Theorem 9.1] when F is archimedean). Put

$$ \begin{align*} \rho:= \pi_0\otimes \sigma, \end{align*} $$

which is an admissible representation of the Levi subgroup $M_{P_{r+1}'}\cong \operatorname {\mathrm {GSpin}}(V_0)\times \operatorname {\mathrm {GL}}_{r+1}.$ For $s\in \mathbb C$ , define $\rho _s(h m)=|\det (m)|^s\rho (h m),$ for $h\in \operatorname {\mathrm {GSpin}}(V_0),\ m\in \operatorname {\mathrm {GL}}_{r+1}.$ Throughout, we identify $\operatorname {\mathrm {GL}}_{r+1}$ with its lift $\ell _{r+1}(\operatorname {\mathrm {GL}}_{r+1})$ in $\operatorname {\mathrm {GSpin}}(V^\prime )$ , where $\ell _{r+1}$ is given in Lemma 2.4.

Let $\pi _s'=\operatorname {\mathrm {Ind}}_{P_{r+1}'}^{\operatorname {\mathrm {GSpin}}(V')}(\rho _s)$ denote the (unnormalized) induced representation of $\operatorname {\mathrm {GSpin}}(V^\prime )$ , consisting of all smooth functions $f:\operatorname {\mathrm {GSpin}}(V^\prime )\to \rho $ satisfying

(15) $$ \begin{align} f( ugm x) =|\det(m)|^{s} \rho(g m)(f(x)) \end{align} $$

for all $u\in N_{P_{r+1}^\prime }$ , $g\in \operatorname {\mathrm {GSpin}}(V_0)$ , $m\in \operatorname {\mathrm {GL}}_{r+1}$ , and $x\in \operatorname {\mathrm {GSpin}}(V^\prime )$ . Note that for $z\in F^\times \subset \operatorname {\mathrm {GSpin}}(V_0)$ , we have

(16) $$ \begin{align} f(zx)=\rho(z)f(x)=\omega_{\pi_0}(z)f(x), \quad \forall x\in \operatorname{\mathrm{GSpin}}(V^\prime). \end{align} $$

We have the following lemma, which will be used in Section 4.3.

4.1. A local integral

Recall that $\sigma $ is $\chi ^{-1}$ -generic. We fix a nonzero $\chi ^{-1}$ -Whittaker functional $\lambda $ of $\sigma $ , satisfying

(17) $$ \begin{align} \lambda(\sigma(u) v_\sigma)=\chi^{-1}(u)\lambda(v_\sigma), \quad \text{ for all }u\in Z_{r+1}, v_\sigma\in \sigma. \end{align} $$

Then we define a linear map $\Lambda :\rho \to \pi _0$ by

$$ \begin{align*} \begin{aligned} \Lambda: \rho=\pi_0\otimes \sigma &\to \pi_0 \\ v_0 \otimes v_\sigma &\mapsto \lambda(v_\sigma)v_0. \end{aligned} \end{align*} $$

Let $\pi $ , $\pi _0$ be irreducible admissible representations of $\operatorname {\mathrm {GSpin}}(V)$ and $\operatorname {\mathrm {GSpin}}(V_0),$ respectively, and let

$$ \begin{align*} \mu: \pi\otimes \pi_0\to \mathbb{C} \end{align*} $$

be a Bessel functional, that is, a map in the space $\mathrm {Hom}_{H}(\pi \otimes \pi _0, \xi )$ .

We have the following result.

Lemma 4.2. For every $s\in \mathbb {C}$ , $v\in \pi ,$ and $f\in \pi _s^\prime $ , the smooth function on $\operatorname {\mathrm {GSpin}}(V)$ defined by

$$ \begin{align*} g\mapsto \mu(\pi(g)v, \Lambda(f(g))) \end{align*} $$

is left invariant under H.

Proof. Let $g\in \operatorname {\mathrm {GSpin}}(V)$ and $b\in H \subset P_{r+1}^\prime $ . We may write $b=nhm$ , where $n\in N_{P_{r+1}^\prime }$ , $h\in \operatorname {\mathrm {GSpin}}(V_0)$ , and $m\in Z_{r+1}$ . Recall that we have identified $Z_{r+1}$ with its lift $\ell _{r+1}(Z_{r+1})\subset \operatorname {\mathrm {GSpin}}(V^\prime )$ . Then

$$ \begin{align*} \Lambda(f(bg))= \Lambda(\rho(hm) f(g)) = \chi^{-1}(m)\pi_0(h)(\Lambda(f(g))) =\xi^{-1}(b) \pi_0(h)(\Lambda(f(g))) \end{align*} $$

and hence

$$ \begin{align*} \begin{aligned} \mu(\pi(bg)v, \Lambda(f(bg))) &=\xi^{-1}(b) \mu(\pi(bg)v, \pi_0(h)(\Lambda(f(g)))) \\ &= \mu( \pi(b^{-1}) \pi(bg)v, \pi_0(b^{-1})\pi_0(h)(\Lambda(f(g)))) \\ &=\mu(\pi(g)v, \Lambda(f(g))), \end{aligned} \end{align*} $$

where the last equality holds because b maps to h under the quotient map $H\to \operatorname {\mathrm {GSpin}}(V_0)$ , and $\pi _0$ is viewed as a representation of H via inflation.

Since $\operatorname {\mathrm {GSpin}}(V)$ and H are unimodular, there exists a right $\operatorname {\mathrm {GSpin}}(V)$ -invariant positive measure on $H\backslash \operatorname {\mathrm {GSpin}}(V)$ unique up to scalar, and we write $dg$ for such a measure. The following local zeta integral plays an important role in the proof of Theorem 1.1:

(18) $$ \begin{align} \mathcal{Z}_{\mu}(f,v):=\int_{H\backslash \operatorname{\mathrm{GSpin}}(V)} \mu(\pi(g)v, \Lambda(f(g))) dg, \quad f\in \pi_s^\prime, v\in \pi. \end{align} $$

Whenever the integral converges absolutely, the right $\operatorname {\mathrm {GSpin}}(V)$ -invariance of $dg$ implies that, for every $g_0\in \operatorname {\mathrm {GSpin}}(V)$ , we have

(19) $$ \begin{align} \mathcal{Z}_{\mu}(\pi_s^\prime(g_0)f,\pi(g_0)v)= \mathcal{Z}_{\mu}(f,v). \end{align} $$

Indeed, this follows from the change of variable $gg_0\mapsto g$ .

We postpone the proof of the following proposition to Section 4.2.

We also have the following result.

Proof. We prove this result by reducing from $\operatorname {\mathrm {GSpin}}(V)$ to $\operatorname {\mathrm {SO}}(V)$ . A similar approach has been used in [Reference Asgari, Cogdell and Shahidi2, Theorem 7.1(i)] and [Reference Kaplan, Lau and Liu13, Proposition 3.1].

Recall that the kernel of the projection map $\mathrm {pr}:\operatorname {\mathrm {GSpin}}(V)\to \operatorname {\mathrm {SO}}(V)$ is $\ker (\mathrm {pr})=F^\times $ . Since $F^\times \subset H$ , the projection induces a homeomorphism

(20) $$ \begin{align} H\backslash \operatorname{\mathrm{GSpin}}(V) \cong \mathrm{pr}(H)\backslash \operatorname{\mathrm{SO}}(V). \end{align} $$

Let $v\in \pi $ and $f\in \pi _s^\prime $ , and denote

$$ \begin{align*} \mathcal{G}_{\mu, f, v}(g):=\mu(\pi(g)v, \Lambda(f(g))), \qquad g\in \operatorname{\mathrm{GSpin}}(V). \end{align*} $$

By Lemma 4.2, $\mathcal {G}_{\mu , f, v}$ is left invariant under H. Note that $F^\times $ lies in the centers of both $\operatorname {\mathrm {GSpin}}(V)$ and $\operatorname {\mathrm {GSpin}}(V_0)$ . It follows that $\mathcal {G}_{\mu , f, v}$ is invariant under multiplication by $F^\times $ . Indeed, for any $z\in F^\times $ , we have

(21) $$ \begin{align} \begin{aligned} \mathcal{G}_{\mu, f, v}(gz)&= \mathcal{G}_{\mu, f, v}(zg) \\ &=\mu( \pi(zg)v, \Lambda(f(zg)))\\ &=\mu(\omega_{\pi}(z) \pi(g)v, \omega_{\pi_0}(z) \Lambda(f(g))) \quad \text{(by}\ (16))\\ &=\omega_\pi(z)\omega_{\pi_0}(z) \mu(\pi(g)v, \Lambda(f(g))) \\ &=\mu(\pi(g)v, \Lambda(f(g))) \\ &= \mathcal{G}_{\mu, f, v}(g), \end{aligned} \end{align} $$

where the second to the last equality follows from the assumption that $\omega _{\pi }|_{F^\times } \cdot \omega _{\pi _0}|_{F^\times }$ is the trivial character of $F^\times $ (see Remark 4.1). Thus, the function $\mathcal {G}_{\mu , f, v}$ descends to a function on $\operatorname {\mathrm {SO}}(V)$ as follows. For any $\overline {g}\in \operatorname {\mathrm {SO}}(V)$ , we set

$$ \begin{align*} \overline{\mathcal{G}}_{\mu, f, v}(\overline{g}):=\mathcal{G}_{\mu, f, v}(g), \end{align*} $$

where $g\in \operatorname {\mathrm {GSpin}}(V)$ is any lift of $\overline {g}$ . This is well-defined by (21). Moreover, $\overline {\mathcal {G}}_{\mu , f, v}$ is left invariant under $\mathrm {pr}(H)$ . Indeed, let $\overline {h}\in \mathrm {pr}(H)$ , and choose any lift $h\in H$ . Then we have

$$ \begin{align*} \overline{\mathcal{G}}_{\mu, f, v}( \overline{h} \overline{g}) = \overline{\mathcal{G}}_{\mu, f, v}( \overline{hg} ) = \mathcal{G}_{\mu, f, v}(hg) = \mathcal{G}_{\mu, f, v}(g)=\overline{\mathcal{G}}_{\mu, f, v}( \overline{g}). \end{align*} $$

Hence, $\overline {\mathcal {G}}_{\mu , f, v}$ is left invariant under $\mathrm {pr}(H)$ .

Using the homeomorphism (20), we may rewrite the local zeta integral $\mathcal {Z}_{\mu }(f,v)$ as

$$ \begin{align*} \begin{aligned} \mathcal{Z}_{\mu}(f,v) &= \int_{H\backslash \operatorname{\mathrm{GSpin}}(V)} \mathcal{G}_{\mu, f, v}(g) dg\\ &= \int_{\mathrm{pr}(H)\backslash \operatorname{\mathrm{SO}}(V)} \overline{\mathcal{G}}_{\mu, f, v}(\overline{g}) d\overline{g}, \end{aligned} \end{align*} $$

where $d\overline {g}$ denotes the quotient measure on $\mathrm {pr}(H)\backslash \operatorname {\mathrm {SO}}(V)$ transported from the quotient measure $dg$ on $H\backslash \operatorname {\mathrm {GSpin}}(V)$ through the homeomorphism (20). The absolute convergence of the integral $\int _{\mathrm {pr}(H)\backslash \operatorname {\mathrm {SO}}(V)} \overline {\mathcal {G}}_{\mu , f, v}(\overline {g}) d\overline {g}$ and hence of $\mathcal {Z}_\mu (f,v)$ now follows from the absolute convergence of the analogous local integral for special orthogonal groups in [Reference Jiang and Zhang12, Lemma 4.1]. Finally, in the archimedean case, the same reduction together with [Reference Jiang, Sun and Zhu11, Proposition 3.4] shows that $\mathcal {Z}_{\mu }(f,v)$ is continuous on the input data in the domain of absolute convergence.

4.2. Proof of Proposition 4.3

The goal of this section is to prove Proposition 4.3, and hence we assume that $\mu \in \mathrm {Hom}_{H}(\pi \otimes \pi _0, \xi )$ is nonzero. By twisting the representation $\sigma $ by a character if necessary, we may assume $s=0$ .

Denote $\overline {P}_r=\{g\in \operatorname {\mathrm {GSpin}}(V): gY_r g^{-1}=Y_r\}$ . This is the parabolic subgroup of $\operatorname {\mathrm {GSpin}}(V)$ opposite to $P_r$ with respect to the decomposition $V=X_r\oplus V_0^\prime \oplus Y_r$ . It has a Levi decomposition $\overline {P}_r= (\operatorname {\mathrm {GL}}_r \times \operatorname {\mathrm {GSpin}}(V_0^\prime ))\ltimes \overline {N}_{P_r}$ , where the $\operatorname {\mathrm {GL}}_r$ -factor is embedded in $\operatorname {\mathrm {GSpin}}(V)$ through the lift $\ell _r$ described in Lemma 2.4, and $\overline {N}_{P_r}$ is the unipotent radical, which is given by

$$ \begin{align*} \overline{N}_{P_r}=\{g\in \operatorname{\mathrm{GSpin}}(V): \mathrm{pr}(g)y=y \text{ for all }y\in Y_r, \mathrm{pr}(g) w \equiv w \pmod{Y_r} \text{ for all }w\in V_0^\prime \}. \end{align*} $$

By [Reference Borel4, Theorem 14.20], the multiplication map $P_r \times \overline {N}_{P_r}\to \operatorname {\mathrm {GSpin}}(V)$ given by $(p, \overline {n})\mapsto p\overline {n}$ is an F-morphic open immersion, and hence the product

(22) $$ \begin{align} P_r \overline{N}_{P_r}= \left(N_{P_r} \rtimes (\operatorname{\mathrm{GSpin}}(V_0^\prime)\times\operatorname{\mathrm{GL}}_r)\right) \overline{N}_{P_r} \end{align} $$

is the open Bruhat cell of $\operatorname {\mathrm {GSpin}}(V)$ . In particular, it is an open dense subset of $\operatorname {\mathrm {GSpin}}(V)$ , and its image in $H\backslash \operatorname {\mathrm {GSpin}}(V)$ has full measure in $H\backslash \operatorname {\mathrm {GSpin}}(V)$ . Therefore, the integral defining $\mathcal Z_\mu (f,v)$ may be computed over $H\backslash P_r \overline {N}_{P_r}$ .

Recall that

$$ \begin{align*} H=N_{P_r}\rtimes (\operatorname{\mathrm{GSpin}}(V_0)\times Z_r)\subset P_r= N_{P_r} \rtimes (\operatorname{\mathrm{GSpin}}(V_0^\prime)\times \operatorname{\mathrm{GL}}_r). \end{align*} $$

It follows that multiplication induces an identification

$$ \begin{align*} H\backslash P_r \overline{N}_{P_r} \cong (Z_r\backslash \operatorname{\mathrm{GL}}_r)\times (\operatorname{\mathrm{GSpin}}(V_0)\backslash \operatorname{\mathrm{GSpin}}(V_0^\prime))\times \overline N_{P_r}, \end{align*} $$

given by $(Z_r m, \operatorname {\mathrm {GSpin}}(V_0)g_0^\prime , \overline {n}) \mapsto H \ell _r(m) g_0^\prime \overline {n}$ . Thus, we have

(23) $$ \begin{align} \mathcal{Z}_{\mu}(f,v)= \int_{(Z_r\backslash \operatorname{\mathrm{GL}}_r)\times (\operatorname{\mathrm{GSpin}}(V_0)\backslash \operatorname{\mathrm{GSpin}}(V_0^\prime))\times \overline{N}_{P_r}} \mu(\pi(m g_0^\prime \overline{n} )v, \Lambda(f(m g_0^\prime \overline{n}))) \delta_{P_r}^{-1}(m)dm dg_0^\prime d\overline{n}, \end{align} $$

where $\delta _{P_r}$ is the modular character of $P_r$ defined by $p\mapsto |\det (\mathrm {Ad}_{p})|$ . We now briefly explain the Jacobian factor in (23). Since $\overline {N}_{P_r}$ is unipotent, the $\overline {N}_{P_r}$ part does not contribute to the Jacobian. Moreover, $\delta _{P_r}$ is trivial on the $\operatorname {\mathrm {GSpin}}(V_0^\prime )$ part, since $\operatorname {\mathrm {GSpin}}(V_0^\prime )$ acts on $\mathrm {Lie}(N_{P_r})$ through the projection to the orthogonal action which has determinant 1. Therefore, the factor $\delta _{P_r}^{-1}(m)$ in the right-hand side of (23) accounts for the full Jacobian. Here and in what follows, we identify $\operatorname {\mathrm {GL}}_r$ with its image under $\ell _r$ when no confusion can arise.

We have the following non-vanishing lemma on $\operatorname {\mathrm {GSpin}}(V_0)\backslash \operatorname {\mathrm {GSpin}}(V_0^\prime )$ .

Lemma 4.5. There exist a vector $v\in \pi $ and a smooth function $f_{\pi _0}:\operatorname {\mathrm {GSpin}}(V_0^\prime )\to \pi _0$ , compactly supported modulo $\operatorname {\mathrm {GSpin}}(V_0)$ , such that

$$ \begin{align*} f_{\pi_0}(g g_0^\prime)=\pi_0(g)f_{\pi_0}(g_0^\prime), \quad \text{ for all }g\in \operatorname{\mathrm{GSpin}}(V_0), \ g_0^\prime\in \operatorname{\mathrm{GSpin}}(V^\prime_0), \end{align*} $$

and

(24) $$ \begin{align} \int_{\operatorname{\mathrm{GSpin}}(V_0)\backslash \operatorname{\mathrm{GSpin}}(V_0^\prime)} \mu(\pi(g_0^\prime) v, f_{\pi_0}(g_0^\prime))dg_0^\prime\not=0. \end{align} $$

Proof. Since $\mu \not =0$ , we can find $v\in \pi $ and $v_0\in \pi _0$ such that $\mu (v, v_0)=1$ . Consider the function $\phi (g_0^\prime ):=\mu (\pi (g_0^\prime )v,v_0)$ , $g_0^\prime \in \operatorname {\mathrm {GSpin}}(V_0^\prime )$ . This is continuous, and $\phi (1)=1$ . Hence, there exists a compact neighborhood $K^\prime $ of the identity in $\operatorname {\mathrm {GSpin}}(V_0^\prime )$ such that $\mathrm {Re}(\phi (h))>0$ for all $h\in K^\prime $ . Let $\beta :\operatorname {\mathrm {GSpin}}(V_0^\prime )\to \operatorname {\mathrm {GSpin}}(V_0)\backslash \operatorname {\mathrm {GSpin}}(V_0^\prime )$ be the quotient map. Shrinking $K^\prime $ if necessary, we may assume that $\beta |_{K^\prime }$ is injective, and that there exists an open neighborhood $\Omega $ of the base point $\beta (1)$ in $\operatorname {\mathrm {GSpin}}(V_0)\backslash \operatorname {\mathrm {GSpin}}(V_0^\prime )$ together with a smooth section $\widetilde {\beta }:\Omega \to \operatorname {\mathrm {GSpin}}(V_0^\prime )$ such that $\widetilde {\beta }(\Omega )\subset K^\prime $ .

Let $\phi _1\in \mathcal {C}_c^\infty (\Omega )$ be a non-negative nonzero smooth function with compact support contained in $\Omega $ . Define

$$ \begin{align*} f_{\pi_0}(g_0^\prime)= \begin{cases} \phi_1(x)\,\pi_0(g)v_0, & \text{if } g_0^\prime=g \widetilde{\beta}(x) \text{ with } g\in \operatorname{\mathrm{GSpin}}(V_0),\ x\in \Omega,\\ 0, & \text{if } g_0^\prime\notin \operatorname{\mathrm{GSpin}}(V_0) \widetilde{\beta}(\Omega). \end{cases} \end{align*} $$

Since $\widetilde {\beta }$ is a section of the quotient map, the decomposition $g_0^\prime =g \widetilde {\beta }(x)$ is unique whenever $g_0^\prime \in \operatorname {\mathrm {GSpin}}(V_0)\widetilde {\beta }(\Omega )$ , so $f_{\pi _0}$ is well-defined. By construction, we have

$$ \begin{align*} f_{\pi_0}(g g_0^\prime)=\pi_0(g)f_{\pi_0}(g_0^\prime), \quad \text{ for all }g\in \operatorname{\mathrm{GSpin}}(V_0), \ g_0^\prime\in \operatorname{\mathrm{GSpin}}(V^\prime_0). \end{align*} $$

Moreover, $f_{\pi _0}$ is smooth. Indeed, on the open subset $\operatorname {\mathrm {GSpin}}(V_0)\widetilde {\beta }(\Omega )\subset \operatorname {\mathrm {GSpin}}(V_0^\prime )$ , the map

$$ \begin{align*} \operatorname{\mathrm{GSpin}}(V_0)\times \Omega \longrightarrow \operatorname{\mathrm{GSpin}}(V_0)\widetilde{\beta}(\Omega), \qquad (g,x)\longmapsto g\widetilde{\beta}(x) \end{align*} $$

is a diffeomorphism in the archimedean case and a homeomorphism onto an open subset in the non-archimedean case. Since $\phi _1$ is smooth and $g\mapsto \pi _0(g)v_0$ is smooth, the function $f_{\pi _0}$ is smooth on the open subset $\operatorname {\mathrm {GSpin}}(V_0) \widetilde {\beta }(\Omega )$ . Outside $\operatorname {\mathrm {GSpin}}(V_0) \widetilde {\beta }(\Omega )$ , the function $f_{\pi _0}$ is identically zero. Since $\mathrm {supp}(\phi _1)\subset \Omega $ is compact, the support of $f_{\pi _0}$ is contained in $\operatorname {\mathrm {GSpin}}(V_0)\widetilde {\beta }(\mathrm {supp}(\phi _1))$ , which is compact modulo $\operatorname {\mathrm {GSpin}}(V_0)$ , and $f_{\pi _0}$ vanishes near the boundary of $\operatorname {\mathrm {GSpin}}(V_0)\widetilde {\beta }(\Omega )$ . Hence, the extension by zero is smooth on all of $\operatorname {\mathrm {GSpin}}(V_0^\prime )$ .

Finally, we have

$$ \begin{align*} \int_{\operatorname{\mathrm{GSpin}}(V_0)\backslash \operatorname{\mathrm{GSpin}}(V_0^\prime)} \mu(\pi(g_0^\prime) v, f_{\pi_0}(g_0^\prime))dg_0^\prime = \int_{\Omega} \phi_1(x)\,\mu(\pi(\widetilde{\beta}(x))v,v_0)dx. \end{align*} $$

By taking the real parts, we obtain

$$ \begin{align*} \mathrm{Re}\left( \int_{\operatorname{\mathrm{GSpin}}(V_0)\backslash \operatorname{\mathrm{GSpin}}(V_0^\prime)} \mu(\pi(g_0^\prime) v, f_{\pi_0}(g_0^\prime))dg_0^\prime\right) = \int_{\Omega} \phi_1(x) \mathrm{Re} (\phi( \widetilde{\beta}(x) ))dx>0. \end{align*} $$

Thus, we have (24). This finishes the proof of the lemma.

Take $v\in \pi $ and $f_{\pi _0}$ as in Lemma 4.5. For $m\in \operatorname {\mathrm {GL}}_{r}$ and $\overline {n}\in \overline {N}_{P_r}$ , put

$$ \begin{align*} \Phi(m,\overline{n}):=\int_{\operatorname{\mathrm{GSpin}}(V_0)\backslash \operatorname{\mathrm{GSpin}}(V_0^\prime)} \mu(\pi(m g_0^\prime \overline{n} )v, f_{\pi_0}( g_0^\prime ) )dg_0^\prime, \end{align*} $$

which is a smooth function on $\operatorname {\mathrm {GL}}_r\times \overline {N}_{P_r}$ . By Lemma 4.5, $\Phi (1,1)\not =0$ , and hence $\Phi \not =0$ . Note that for any $z\in Z_r$ , $m\in \operatorname {\mathrm {GL}}_{r}$ , and $\overline {n}\in \overline {N}_{P_r}$ , we have

$$ \begin{align*} \begin{aligned} \Phi(zm, \overline{n}) &= \int_{\operatorname{\mathrm{GSpin}}(V_0)\backslash \operatorname{\mathrm{GSpin}}(V_0^\prime)} \mu(\pi( z m g_0^\prime \overline{n} )v, f_{\pi_0}( g_0^\prime ))dg_0^\prime \\ &= \int_{\operatorname{\mathrm{GSpin}}(V_0)\backslash \operatorname{\mathrm{GSpin}}(V_0^\prime)} \mu(\pi( z m g_0^\prime \overline{n} )v, \pi_0(z)f_{\pi_0}( g_0^\prime ))dg_0^\prime \\ &=\chi(z)\Phi(m, \overline{n}), \end{aligned} \end{align*} $$

where the second equality follows from the fact that the representation $\pi _0$ of H has trivial restriction to $Z_r$ , and the third equality follows from the fact that $\chi |_{Z_r}=\xi |_{Z_r}$ (see (14) and Remark 3.4).

Let $W_r$ be a smooth function on $\operatorname {\mathrm {GL}}_r$ with compact support modulo $Z_r$ such that

$$ \begin{align*} W_r(zm)=\chi^{-1}(z)W_r(m), \quad \text{ for all }z\in Z_r, m\in \operatorname{\mathrm{GL}}_r. \end{align*} $$

Then for each $\overline {n}\in \overline {N}_{P_r}$ , the function on $\operatorname {\mathrm {GL}}_r$ given by $m\mapsto \Phi (m, \overline {n})W_r(m)\delta _{P_r}^{-1}(m)$ is left-invariant under $Z_r$ . We remind the reader that $\operatorname {\mathrm {GL}}_r=\operatorname {\mathrm {GL}}_F(X_r)$ is viewed as a subgroup of $\operatorname {\mathrm {GL}}_{r+1}=\operatorname {\mathrm {GL}}_F(X_{r+1}^{\prime })$ by acting trivially on $e_{r+1}$ , and that $Z_r$ (resp. $Z_{r+1}$ ) is the unipotent radical of the Borel subgroup of $\operatorname {\mathrm {GL}}_r$ (resp. $\operatorname {\mathrm {GL}}_{r+1}$ ) stabilizing the flag (6) (resp. the flag (12)), as in Section 3.2.

Lemma 4.6. For every $W_r$ as above, there exists a vector $v_\sigma \in \sigma $ such that

$$ \begin{align*} W_r(m)=\lambda(\sigma(m)v_\sigma), \quad m\in \operatorname{\mathrm{GL}}_r. \end{align*} $$

Let $\phi _2$ be a smooth and compactly supported function on $\overline {N}_{P_r}$ . Since $\Phi \not =0$ , we can pick $W_r$ and $\phi _2$ such that

(25) $$ \begin{align} \int_{(Z_r\backslash \operatorname{\mathrm{GL}}_r) \times \overline{N}_{P_r}} \Phi(m, \overline{n}) W_r(m) \phi_2(\overline{n}) \delta_{P_r}^{-1}(m) dm d\overline{n} \not=0. \end{align} $$

We now define a section $f_\rho \in \pi _s'$ . Note that

$$ \begin{align*} P_{r+1}^\prime\cap \operatorname{\mathrm{GSpin}}(V) =N_{P_{r}} \rtimes (\operatorname{\mathrm{GL}}_r\times \operatorname{\mathrm{GSpin}}(V_0)), \end{align*} $$

and the multiplication map

$$ \begin{align*} \begin{aligned} \iota: (N_{P_{r+1}^\prime} \rtimes (\operatorname{\mathrm{GL}}_{r+1} \times \operatorname{\mathrm{GSpin}}(V_0^\prime))\ltimes \overline{N}_{P_r} ) &\to \operatorname{\mathrm{GSpin}}(V^\prime) \\ (u,m, g_0^\prime, \overline{n} ) &\mapsto umg_0^\prime \overline{n} \end{aligned} \end{align*} $$

is an open embedding. Put

$$ \begin{align*} f_{\rho}(x):= \begin{cases} \phi_2(\overline{n})f_{\pi_0}(g_0^\prime)\otimes (\sigma(m) v_\sigma) &\text{ if }x=\iota(u, m, g_0^\prime, \overline{n}),\\ 0 &\text{ if } x \text{ is not in the image of } \iota, \end{cases} \end{align*} $$

where $v_\sigma \in \sigma $ is as in Lemma 4.6. By construction, $f_\rho $ is smooth and compactly supported modulo $P_{r+1}^\prime $ . Moreover, it satisfies the equivariance property (15) as we explain below. Let ${x=\iota (u,m,g_0^\prime , \overline {n})}$ be in the image of $\iota $ . For any $u_0\in N_{P_{r+1}^\prime }$ , we have

$$ \begin{align*} f_{\rho}( u_0 x)= f_{\rho}(\iota( u_0 u, m, g_0^\prime, \overline{n}))= \phi_2(\overline{n})f_{\pi_0}(g_0^\prime)\otimes (\sigma(m) v_\sigma) =f_{\rho}(x). \end{align*} $$

For any $g_0\in \operatorname {\mathrm {GSpin}}(V_0)$ , we have

$$ \begin{align*} \begin{aligned} f_{\rho}( g_0 x) &= f_{\rho}( g_0 u m g_0^\prime \overline{n}) \\ &=f_{\rho} ( (g_0u g_0^{-1})g_0 m g_0^\prime \overline{n})\\ &=f_{\rho} ( (g_0u g_0^{-1}) m g_0 g_0^\prime \overline{n}) \quad (\text{by Lemma 2.7}) \\ &=f_{\rho}(\iota( g_0 u g_0^{-1}, m, g_0g_0^\prime, \overline{n}))\\ &=\rho(g_0) f_{\rho}(x). \end{aligned} \end{align*} $$

For any $m_0\in \operatorname {\mathrm {GL}}_{r+1}$ , we have

$$ \begin{align*} \begin{aligned} f_{\rho}(m_0 x) & = f_{\rho} ( m_0 u m g_0^\prime \overline{n}) = f_{\rho}( \iota( m_0 u m_0^{-1}, m_0 m, g_0^\prime, \overline{n}))=\rho(m_0)f_{\rho}(x). \end{aligned} \end{align*} $$

On the other hand, if x is not in the image of $\iota $ , then for any $u_0\in N_{P_{r+1}^\prime }$ , $g_0\in \operatorname {\mathrm {GSpin}}(V_0)$ , and ${m_0\in \operatorname {\mathrm {GL}}_{r+1}}$ , we have that $u_0 g_0 m_0 x$ is not in the image of $\iota $ and hence

$$ \begin{align*} f_{\rho}( u_0 g_0 m_0 x)=f_{\rho}(x)=0. \end{align*} $$

Thus, $f_\rho \in \pi _s^\prime $ (recall that s is assumed to be zero in this section). For any $m\in \operatorname {\mathrm {GL}}_r$ , $g_0^\prime \in \operatorname {\mathrm {GSpin}}(V_0^\prime ), \overline {n}\in \overline {N}_{P_r}$ , we have

$$ \begin{align*} \Lambda( f_{\rho}(m g_0^\prime \overline{n} ))=\Lambda( f_{\rho}( \iota(1, m, g_0^\prime, \overline{n}) )) = \Lambda( \phi_2(\overline{n})f_{\pi_0}(g_0^\prime)\otimes (\sigma(m) v_\sigma)) = \phi_2(\overline{n}) \lambda( \sigma(m) v_\sigma) f_{\pi_0}(g_0^\prime). \end{align*} $$

We are ready to finish the proof of Proposition 4.3. Starting from (23), we compute that

$$ \begin{align*} \begin{aligned} \mathcal{Z}_{\mu}(f_\rho,v) &= \int_{(Z_r\backslash \operatorname{\mathrm{GL}}_r)\times (\operatorname{\mathrm{GSpin}}(V_0)\backslash \operatorname{\mathrm{GSpin}}(V_0^\prime))\times \overline{N}_{P_r}} \mu(\pi(m g_0^\prime \overline{n} )v, \Lambda(f_\rho(m g_0^\prime \overline{n}))) \delta_{P_r}^{-1}(m)dm dg_0^\prime d\overline{n} \\ &= \int_{(Z_r\backslash \operatorname{\mathrm{GL}}_r)\times (\operatorname{\mathrm{GSpin}}(V_0)\backslash \operatorname{\mathrm{GSpin}}(V_0^\prime))\times \overline{N}_{P_r}} \mu(\pi(m g_0^\prime \overline{n} )v, \phi_2(\overline{n}) \lambda(\sigma(m)v_\sigma) f_{\pi_0}( g_0^\prime ) ) \delta_{P_r}^{-1}(m)dm dg_0^\prime d\overline{n} \\ &= \int_{(Z_r\backslash \operatorname{\mathrm{GL}}_r) \times \overline{N}_{P_r}} \Phi(m,\overline{n}) \phi_2(\overline{n}) W_r(m) \delta_{P_r}^{-1}(m)dm d\overline{n,} \end{aligned} \end{align*} $$

which converges to a nonzero number by (25). This concludes the proof of Proposition 4.3.

4.3. Proof of Theorem 1.1

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1.

We are given $\pi , \pi _0$ and a generic character $\xi $ of H as in (14) (and hence a unitary generic character $\chi $ of $Z_{r+1}$ ). We take $\sigma $ to be an irreducible $\chi ^{-1}$ -generic representation of $\operatorname {\mathrm {GL}}_{r+1}$ . For each $\mu \in \mathrm {Hom}_{H}(\pi \otimes \pi _0, \xi )$ , we may define the integral $\mathcal {Z}_{\mu }$ as in (18).

Let $\mathcal {F}$ be a finite-dimensional subspace of $\mathrm {Hom}_{H}(\pi \otimes \pi _0, \xi )$ . By Proposition 4.4, there exists a number $c_{\mathcal {F}}\in \mathbb {R}$ such that for all $\mu \in \mathcal {F}$ and all $s\in \mathbb {C}$ with $\mathrm {Re}(s)>c_{\mathcal {F}}$ , the integral $\mathcal {Z}_{\mu }(f,v)$ converges absolutely for all $f\in \pi _s^\prime $ and all $v\in \pi $ , and defines a linear functional on $\pi _s^\prime \otimes \pi $ .

By Lemma 3.5, we may choose one s with $\mathrm {Re}(s)>c_{\mathcal {F}}$ such that $\pi _s^\prime $ is irreducible. By (19) and Proposition 4.3, we obtain a linear embedding

$$ \begin{align*} \mathcal{F}\hookrightarrow \mathrm{Hom}_{\operatorname{\mathrm{GSpin}}(V)}(\pi_s^\prime\otimes \pi, \mathbb{C}), \quad \mu\mapsto \mathcal{Z}_{\mu}. \end{align*} $$

The injectivity follows from Proposition 4.3. The $\operatorname {\mathrm {GSpin}}(V)$ -equivariance of $\mathcal {Z}_\mu $ follows from (19). The linearity of the map $\mu \mapsto \mathcal {Z}_\mu $ follows from the definition of the integral $\mathcal {Z}_\mu $ and Proposition 4.4. The space $\mathrm {Hom}_{\operatorname {\mathrm {GSpin}}(V)}(\pi _s^\prime \otimes \pi , \mathbb {C})$ is at most one-dimensional (see [Reference Emory and Takeda8] when F is non-archimedean and [Reference Emory, Kim and Maiti7] when F is archimedean). Therefore, $\mathcal {F}$ is at most one-dimensional. Since $\mathcal {F}$ was arbitrary, we conclude that $\mathrm {Hom}_{H}(\pi \otimes \pi _0, \xi )$ is at most one-dimensional.