2.1. Some structural facts

Let $F_s$ denote a separable closure of F and $\Gamma =\operatorname {\mathrm {Gal}}(F_s/F)$ . Let $\mathbf {T}$ be a maximal F-torus of $\mathbf {G}$ . Let

$$ \begin{align*} \{ X, \ \Phi,; \ Y, \ \Phi^\vee \} \end{align*} $$

be the resulting (absolute) root system. Here X (respectively, Y) is the character lattice (respectively, cocharacter lattice) for $(\mathbf {G},\mathbf {T})$ over $F_s$ . Both X and Y are naturally $\mathbb {Z}[\Gamma ]$ -modules. The group $\Gamma $ acts on the root system $\Phi $ as well. Choose a set $\Delta \subset \Phi $ of simple roots from the set of roots $\Phi $ and $\Delta ^\vee $ the corresponding simple coroots from $\Phi ^\vee $ . Write $Y^{sc}\subset Y$ for the sublattice generated by $\Phi ^\vee $ . Let $\mathbf {B}\subset \mathbf {G}_{F_s}$ be the Borel ( $F_s$ -)subgroup determined by $\Delta $ .

For each $\alpha \in \Phi $ , one has the associated root subgroup $\mathbf {U}_\alpha \subset \mathbf {G}_{F_s}$ which is normalised by $\mathbf {T}_{F_s}$ . We shall fix a Chevalley system of épinglage for $(\mathbf {G}_{F_s},\mathbf {T}_{F_s})$ , so that for each $\alpha \in \Phi $ , one has an $F_s$ -isomorphism $x_\alpha :\mathbf {G}_a\simeq \mathbf {U}_\alpha $ .

Denote by $W=W(\mathbf {G}):=N(\mathbf {T}_{F_s})/\mathbf {T}_{F_s}$ the (absolute) Weyl group of $(\mathbf {G},\mathbf {T})$ , which we identify with the Weyl group of the coroot system.

2.2. Multiplicative $\mathbf {K}_2$ -torsors

The reductive group $\mathbf {G}$ defines a sheaf of groups on the big Zariski site on $\mathrm {Spec}(F)$ . Let $\mathbf {K}_2$ denote the sheaf of groups on $\mathrm {Spec}(F)$ associated to the $\mathbf {K}_2$ -group in Quillen’s K-theory. Then a multiplicative $\mathbf {K}_2$ -torsor is an extension

$$ \begin{align*} 1\to \mathbf{K}_2 \to \overline{\mathbf{G}} \to \mathbf{G} \to 1 \end{align*} $$

of sheaves of groups on $\mathrm {Spec}(F)$ . We consider the category $\mathbf {CExt}(\mathbf {G},\mathbf {K}_2)$ of such extensions where the morphisms between objects are given by morphisms of extensions. Given two such central extensions, one can form the Baer sum: this equips $\mathbf {CExt}(\mathbf {G},\mathbf {K}_2)$ with the structure of a commutative Picard category.

In [Reference Brylinski and Deligne1], Brylinski and Deligne made a deep study of $\mathbf {CExt}(\mathbf {G},\mathbf {K}_2)$ and obtained an elegant classification of this category when $\mathbf {G}$ is a connected reductive group. We first recall their results.

2.3. Tori

Suppose that $\mathbf {T}$ is an F-torus, with cocharacter lattice $Y=\operatorname {\mathrm {Hom}}(\mathbf {G}_m,\mathbf {T}_{F_s})$ and character lattice $X=\operatorname {\mathrm {Hom}}(\mathbf {T}_{F_s},\mathbf {G}_m)$ .

The isomorphism class of the extension $\mathcal {E}$ is completely determined by the commutator map and hence by the quadratic form Q. The extension $\mathcal {E}$ is obtained from $\overline {\mathbf {T}}$ as follows. Let $F_s(\!(\tau )\!)$ denote the field of Laurent series in the variable $\tau $ over $F_s$ . Then one has

$$ \begin{align*} 1\to \mathbf{K}_2(F_s(\!(\tau)\!)) \to \overline{\mathbf{T}}(F_s(\!(\tau)\!)) \to \mathbf{T}(F_s(\!(\tau)\!)) = Y\otimes_{\mathbb{Z}} F_s(\!(\tau)\!)^{\times}\to 1. \end{align*} $$

The map $y\mapsto y(\tau )$ defines a group homomorphism $Y\to \mathbf {T}(F_s(\!(\tau )\!))$ . Pulling back by this morphism and pushing out by the residue map

$$ \begin{align*} \mathrm{Res}:\mathbf{K}_2(F_s(\!(\tau)\!)) \to \mathbf{K}_1(F_s) = F_s^\times \end{align*} $$

defined by

$$ \begin{align*} \mathrm{Res}(f,g)=(-1)^{v(f)\cdot v(g)} \cdot \left(\dfrac{f^{v(g)}}{g^{v(f)}}(0)\right), \end{align*} $$

one obtains the desired extension $\mathcal {E}$ .

2.4. Simply connected groups

Suppose now that $\mathbf {G}$ is a simply connected semisimple group over F. Since $\mathbf {G}$ is simply connected, we have $Y^{sc}=Y$ and $\Delta ^\vee $ is a basis for Y.

We have the following ([Reference Weissman27] Proposition 3.15).

As a result, whenever we are given such a quadratic form Q on Y, Q gives rise to a multiplicative $\mathbf {K}_2$ -torsor $\overline {\mathbf {G}}_Q$ on $\mathbf {G}$ , unique up to unique isomorphism, which may be pulled back to a multiplicative $\mathbf {K}_2$ -torsor $\overline {\mathbf {T}}_Q$ on $\mathbf {T}$ and hence gives rise to an extension $\mathcal {E}_Q$ of Y by $F_s^\times $ . The automorphism group of the extension $\mathcal {E}_Q$ is $\operatorname {\mathrm {Hom}}_{\Gamma }(Y,F_s^\times )$ . Following [Reference Brylinski and Deligne1] Section 11 and [Reference Gan and Gao11] Subsection 2.4, one can rigidify $\mathcal {E}_Q$ by giving it an extra structure.

2.5. The Brylinski–Deligne liftings

We assume that $\mathbf {G}$ is simply connected. We also have a fixed Chevalley system of épinglage for $(\mathbf {G}_{F_s},\mathbf {T}_{F_s})$ . In particular, for each $\alpha \in \Phi $ with associated root subgroup $\mathbf {U}_\alpha $ , there is a fixed isomorphism over $F_s$ :

$$ \begin{align*} x_\alpha: \mathbf{G}_a \to \mathbf{U}_\alpha \subset \mathbf{G}. \end{align*} $$

Indeed, one has an embedding over $F_s$

$$ \begin{align*} i_\alpha: \mathbf{SL}_2 \hookrightarrow \mathbf{G} \end{align*} $$

which restricts to $x_{\pm \alpha }$ on the upper and lower triangular subgroups of unipotent matrices. By [Reference Brylinski and Deligne1], one has a canonical lifting

$$ \begin{align*} \tilde x_\alpha : \mathbf{G}_a \to \overline{\mathbf{U}}_\alpha \subset \overline{\mathbf{G}}. \end{align*} $$

For $t\in \mathbf {G}_m$ , we set

$$ \begin{align*} n_\alpha(t)=x_\alpha(t)\cdot x_{-\alpha}(-t^{-1}) \cdot x_\alpha(t) =i_\alpha \begin{pmatrix} 0 & t \\ -t^{-1} & 0 \end{pmatrix} \in N(\mathbf{T}_Q) \end{align*} $$

and

$$ \begin{align*} \tilde n_\alpha(t)=\tilde x_\alpha(t)\cdot \tilde x_{-\alpha}(-t^{-1}) \cdot \tilde x_\alpha(t)\in \overline{\mathbf{G}}_Q. \end{align*} $$

Then one has a map

$$ \begin{align*} s_\alpha: \mathbf{T}_\alpha := \alpha^\vee(\mathbf{G}_m) \to \overline{\mathbf{T}}_Q \end{align*} $$

given by

$$ \begin{align*} \alpha^\vee(t) \mapsto \tilde n_\alpha(t) \cdot \tilde n_\alpha(-1). \end{align*} $$

This is a section of $\overline {\mathbf {G}}_Q$ over $\overline {\mathbf {T}}_\alpha $ which is trivial at the identity element. This section is useful in describing the natural conjugation action of $N(\mathbf {T}_Q)$ on $\overline {\mathbf {T}}_Q$ . By [Reference Brylinski and Deligne1] Proposition 11.3, one has the nice formula

(1) $$ \begin{align} \tilde n_\alpha(1) \cdot \tilde t \cdot \tilde n_\alpha(1)^{-1}=\tilde t\cdot s_\alpha(\alpha^\vee(\alpha(t)^{-1})). \end{align} $$

We also use the following formula frequently ([Reference Gao10] (2)): for $t_1,t_2 \in \mathbf {G}_m$ ,

(2) $$ \begin{align} s_\alpha(\alpha^\vee(t_1)) \cdot s_\alpha(\alpha^\vee(t_2)) = s_\alpha(\alpha^\vee(t_1 t_2)) \cdot \{t_1,t_2\}^{Q(\alpha^\vee)}. \end{align} $$

The collection of sections $\{s_\alpha : \alpha \in \Delta \}$ provides a collection of elements $s_\alpha (\alpha ^\vee (a))\in \mathbf {T}_Q$ with $a\in \mathbf {G}_m$ and $\overline {\mathbf {T}}_Q$ is generated by $\mathbf {K}_2$ and the collection of $s_\alpha (\alpha ^\vee (a))$ .

Taking points in $F(\!(\tau )\!)$ , we have the element

$$ \begin{align*} s_\alpha (\alpha^\vee(\tau)) \in \overline{\mathbf{T}}_Q(F(\!(\tau)\!)), \end{align*} $$

which gives rise (via the construction of $\mathcal {E}_Q$ ) to an element

$$ \begin{align*} s_Q(\alpha^\vee) \in \mathcal{E}_Q. \end{align*} $$

Then we rigidify $\mathcal {E}_Q$ by equipping it with the set $\{s_Q(\alpha ^\vee )\mid \alpha ^\vee \in \Delta ^\vee \}$ : there is a unique automorphism of $\mathcal {E}_Q$ which fixes all of these elements.

In the following, we shall fix a choice of the data $(\overline {\mathbf {G}}_Q,\overline {\mathbf {T}}_Q,\mathcal {E}_Q)$ for each $\Gamma $ -invariant, W-invariant quadratic form Q on Y when $\mathbf {G}$ is simply connected.

2.6. Weyl group action on Brylinski–Deligne liftings

Observe that $s_Q(\alpha ^\vee )$ can be defined for every coroot (not necessarily simple coroot). We would like know how $s_Q(\alpha ^\vee )$ behaves under the action of the Weyl group. Recall that for every $\alpha \in \Phi $ , one can choose $w_\alpha :=n_\alpha (1) \in \mathbf {G}(F_s)$ as a representative of the simple reflection $\mathbf {w}_\alpha $ corresponding to $\alpha $ . Let $\mathbf {w}=\mathbf {w}_{\alpha _1} \cdots \mathbf {w}_{\alpha _{\ell }}$ be a minimum decomposition of $\mathbf {w}$ ; we choose the following representative of $\mathbf {w}$ :

$$ \begin{align*} w=w_{\alpha_1}\cdots w_{\alpha_{\ell}} \in \mathbf{G}(F_s). \end{align*} $$

This is independent of the minimum decomposition of $\mathbf {w}$ . We choose $\tilde w_\alpha = \tilde n_\alpha (1)$ as a lift of $w_\alpha $ in $\mathbf {G}(F_s)$ . In any case, the conjugation action of $\overline {\mathbf {G}}(F_s)$ on $\overline {\mathbf {G}}(F_s)$ descends to an action of $\mathbf {G}(F_s)$ .

Fix a pair of roots $\alpha $ and $\beta $ . Then we have a homomorphism

$$ \begin{align*} \mathrm{Int}(w_\alpha)\circ x_{\beta}=w_\alpha \cdot x_\beta \cdot w_\alpha^{-1}: \mathbf{G}_a \to \mathbf{U}_{\mathbf{w}_\alpha(\beta)}. \end{align*} $$

From [Reference Bruhat and Tits3] Subsection 3.2.2, there is a sign $\epsilon _{\alpha ,\beta }\in \{\pm 1\}$ associated to the Chevalley system of épinglage such that

$$ \begin{align*} w_\alpha \cdot x_\beta(t) \cdot w_\alpha^{-1}=x_{\mathbf{w}_\alpha(\beta)}(\epsilon_{\alpha,\beta}t),\qquad t \in \mathbf{G}_m. \end{align*} $$

This implies that for $t\in \mathbf {G}_m$ ,

(3) $$ \begin{align} \tilde w_\alpha \cdot \tilde x_{\beta}(t) \cdot \tilde w_\alpha^{-1}= \tilde x_{\mathbf{w}_\alpha(\beta)}(\epsilon_{\alpha,\beta} t). \end{align} $$

Lemma 2.4. For $t\in \mathbf {G}_m$ ,

$$ \begin{align*} \tilde w_\alpha \cdot s_{\beta}(\beta^{\vee}(t)) \cdot \tilde w_\alpha^{-1}= \{\epsilon_{\alpha,\beta},t\}^{Q(\beta^\vee)} \cdot s_{\mathbf{w}_\alpha(\beta)}(\mathbf{w}_\alpha(\beta)^\vee (t)). \end{align*} $$

Proof. The calculation here is identical to [Reference Gao10] Corollary 7.4. For $t\in \mathbf {G}_m$ , we have

$$ \begin{align*} \tilde w_\alpha\cdot \tilde n_\beta(t)\cdot \tilde w_\alpha^{-1}=(\tilde w_\alpha\cdot \tilde x_\beta(t)\cdot \tilde w_\alpha^{-1}) (\tilde w_\alpha\cdot \tilde x_{-\beta}(-t^{-1})\tilde w_\alpha^{-1}) (\tilde w_\alpha \cdot \tilde x_\beta(t)\cdot \tilde w_\alpha^{-1}). \end{align*} $$

By (3), this is

$$ \begin{align*} \tilde x_{\mathbf{w}_{\alpha}(\beta)}(\epsilon_{\alpha,\beta}t) \cdot \tilde x_{-\mathbf{w}_{\alpha}(\beta)}(-\epsilon_{\alpha,\beta}t^{-1}) \cdot \tilde x_{\mathbf{w}_{\alpha}(\beta)}(\epsilon_{\alpha,\beta}t) = \tilde n_{\mathbf{w}_\alpha(\beta)}(\epsilon_{\alpha,\beta}t). \end{align*} $$

From this we deduce that for any $t\in \mathbf {G}_m$ ,

$$ \begin{align*} \tilde w_\alpha \cdot \tilde n_\beta(t) \cdot \tilde n_\beta(-1) \cdot \tilde w_{\alpha}^{-1} &= \tilde n_{\mathbf{w}_\alpha(\beta)}(\epsilon_{\alpha,\beta} t) \cdot \tilde n_{\mathbf{w}_\alpha(\beta)}(-\epsilon_{\alpha,\beta}) \\ &= \{\epsilon_{\alpha,\beta}, t\}^{Q(\beta^\vee)} s_{\mathbf{w}_\alpha(\beta)}(\mathbf{w}_\alpha(\beta)^\vee(t)). \end{align*} $$

The last equality follows from [Reference Gao10] (3).

Proof. On the one hand, $\tilde w_{\alpha } \cdot s_{\beta }(\beta ^{\vee }(\tau )) \cdot \tilde w_{\alpha }^{-1} =s_{\mathbf {w}_\alpha (\beta )}(\mathbf {w}_\alpha (\beta )^\vee (\tau )) \cdot \{\epsilon _{\alpha ,\beta },\tau \}^{Q(\beta ^\vee )}$ . On the other hand, from (1), we obtain

$$ \begin{align*} \begin{aligned} & \tilde w_{\alpha} \cdot s_{\beta}(\beta^{\vee}(\tau)) \cdot \tilde w_{\alpha}^{-1} =s_{\beta}(\beta^{\vee}(\tau)) \cdot s_{\alpha}(\alpha^{\vee}(\alpha(\beta^\vee(\tau))^{-1})) \\ =&s_{\beta}(\beta^{\vee}(\tau))\cdot s_{\alpha}(\alpha^{\vee}(\tau^{-\langle\alpha,\beta^\vee\rangle}))=s_{\beta}(\beta^{\vee}(\tau))\cdot s_{\alpha}(\alpha^{\vee}(\tau)). \end{aligned} \end{align*} $$

Observe that under the map $\mathbf {K}_2(F) \to \mathbf {K}_1(F)=F^\times $ , $\mathrm {Res}\{\epsilon _{\alpha ,\beta },\tau \}=\epsilon _{\alpha ,\beta }$ . Now the first statement follows.

The argument for the second statement is similar. We also need the following consequences of (2):

$$ \begin{align*} \begin{aligned} s_\alpha(\alpha^{\vee}(\tau)) \cdot s_\alpha(\alpha^{\vee}(\tau^{-1})) = & \{\tau,\tau^{-1}\}^{Q(\alpha^\vee)}. \\ s_\alpha(\alpha^\vee(\tau))\cdot s_\alpha(\alpha^\vee(\tau^{-2})) = & s_\alpha(\alpha^\vee(\tau^{-1})) \{\tau, \tau^{-2}\}^{Q(\alpha^\vee)}. \end{aligned} \end{align*} $$

Note that

$$ \begin{align*} \tilde w_{\alpha} \cdot s_{\alpha}(\alpha^{\vee}(\tau)) \cdot \tilde w_{\alpha}^{-1} =s_{-\alpha}(-\alpha^\vee(\tau)) \cdot \epsilon_{\alpha,-\alpha}^{Q(\alpha^\vee)} =s_{-\alpha}(-\alpha^\vee(\tau)) \cdot (-1)^{Q(\alpha^\vee)}. \end{align*} $$

Again, by (1),

$$ \begin{align*} \tilde w_{\alpha} \cdot s_{\alpha}(\alpha^{\vee}(\tau)) \cdot \tilde w_{\alpha}^{-1} = s_{\alpha}(\alpha^{\vee}(\tau)) \cdot s_\alpha (\alpha^\vee(\tau^{-2})) = s_\alpha (\alpha^\vee(\tau^{-1}))\cdot \{\tau, \tau^{-2}\}^{Q(\alpha^\vee)}. \end{align*} $$

From these two equations, we deduce that

$$ \begin{align*} s_{-\alpha}(-\alpha^\vee(\tau)) \cdot (-1)^{Q(\alpha^\vee)}=s_\alpha (\alpha^\vee(\tau^{-1}))\cdot \{\tau, \tau^{-2}\}^{Q(\alpha^\vee)}. \end{align*} $$

We now have

$$ \begin{align*} s_{-\alpha}(-\alpha^\vee(\tau))\cdot s_\alpha(\alpha^\vee(\tau))= \{\tau,\tau^{-1}\}^{Q(\alpha^\vee)} \cdot (-1)^{Q(\alpha^\vee)} \cdot \{\tau, \tau^{-2}\}^{Q(\alpha^\vee)}. \end{align*} $$

From this we deduce that

$$ \begin{align*} s_Q(\alpha^\vee) \cdot s_Q(-\alpha^\vee) = 1. \end{align*} $$

We write $\alpha ^\vee $ as a sum of simple coroots:

$$ \begin{align*} \alpha^\vee = \alpha_{i_1}^\vee + \cdots + \alpha_{i_n}^\vee. \end{align*} $$

We say that this expression has property $(\ast )$ if the following holds:

(4) $$ \begin{align} (\ast): \alpha_{i_1}^\vee, \ \alpha_{i_1}^\vee+\alpha_2^\vee, \ \cdots, \ \alpha_{i_1}^\vee + \cdots + \alpha_{i_n}^\vee \in \Phi^\vee. \end{align} $$

Lemma 2.6. Assume that $\epsilon _{\alpha ,\beta }^{Q(\beta ^\vee )}=1$ and $\langle \alpha ,\beta ^\vee \rangle = -1$ for all $\alpha ,\beta $ . If we write $\alpha ^\vee $ as a sum of simple coroots such that property ( $\ast $ ) holds, then

$$ \begin{align*} s_{Q}(\alpha^\vee)=s_{Q}(\alpha_{i_n}^\vee)\cdots s_{Q}(\alpha_{i_2}^\vee) \cdot s_{Q}(\alpha_{i_1}^\vee). \end{align*} $$

2.7. General reductive groups

Let $\mathbf {G}$ be a connected reductive group over F, with a fixed Chevalley system of épinglage for $(\mathbf {G},\mathbf {T})$ . Let $X^{sc}\subset X\otimes _{\mathbb {Z}} \mathbb Q$ be the dual lattice of $Y^{sc}$ . Then the quadruple $(X^{sc}, \Delta , Y^{sc}, Y)$ is the root system of the simply connected cover $\mathbf {G}^{sc}$ of the derived group $\mathbf {G}^{der}$ of $\mathbf {G}$ and one has a natural map

$$ \begin{align*} q:\mathbf{G}^{sc}\to \mathbf{G}^{der} \to \mathbf{G}. \end{align*} $$

Let $\mathbf {T}^{sc}$ be the preimage of $\mathbf {T}$ in $\mathbf {G}^{sc}$ . It is a maximal F-torus of $\mathbf {G}^{sc}$ with cocharacter group $Y^{sc}\subset Y$ , so that one has a commutative diagram

The restriction $Q^{sc}:=Q|_{Y^{sc}}$ gives an element $\overline {\mathbf {G}}^{sc}\in \mathbf {CExt}(\mathbf {G}^{sc},\mathbf {K}_2)$ . It also gives the extension $\mathcal {E}_{Q|_{Y^{sc}}}^{sc}$ . For simplicity, we just write $\mathcal {E}_{Q^{sc}}$ with no confusion caused.

3.1. BD covering groups

Start with a multiplicative $\mathbf {K}_2$ -torsor $\overline {\mathbf {G}}$ on $\mathbf {G}$ , with associated BD data $(Q,\mathcal {E},f)$ . Since $H^1(F,\mathbf {K}_2)=0$ , by taking F-points, we obtain a short exact sequence of abstract groups

$$ \begin{align*} 1\to \mathbf{K}_2(F) \to \overline{\mathbf{G}}(F) \to G=\mathbf{G}(F) \to 1. \end{align*} $$

Let $\mu (F)$ denote the set of roots of unity contained in the local field $F\neq \mathbb C$ ; when $F=\mathbb C$ , we let $\mu (F)$ be the trivial group. Then the Hilbert symbol gives a map

$$ \begin{align*} (-,-)_F: \mathbf{K}_2(F) \to \mu(F). \end{align*} $$

For any n dividing $\# \mu (F)$ , one has the nth Hilbert symbol

$$ \begin{align*} (-,-)_n : (-,-)_F^{\#\mu(F)/n}: \mathbf{K}_2(F) \to \mu_n(F). \end{align*} $$

By pushing out the exact sequence via the Hilbert symbol $\mathbf {K}_2(F)\to \mu _n(F)$ , we obtain an exact sequence of locally compact topological groups

$$ \begin{align*} 1\to \mu_n(F) \to \overline{G} \to G \to 1. \end{align*} $$

We call this the BD covering group associated to the BD data $(Q,\mathcal {E},f,n)$ .

3.2. Unipotent subgroups

Let $\mathcal {N}_G$ be the set of all unipotent elements of G. Because a BD extension is uniquely split over any unipotent subgroup, one has unique splittings

$$ \begin{align*} \tilde x_\alpha: F\to \overline{U}_\alpha \text{ for each }\alpha \in \Phi. \end{align*} $$

Indeed, as shown in [Reference Mœglin and Waldspurger22] Appendix I and [Reference Li18] Proposition 2.2.1, there is a unique section

$$ \begin{align*} i:\mathcal{N}_G \to \overline{G} \end{align*} $$

satisfying the following:

• • for each unipotent subgroup $\mathbf {U}\subset \mathbf {G}$ , the restriction of i to $U=\mathbf {U}(F)$ is a group homomorphism;

• • the map i is G-equivariant.

3.3. Tori

The following result is a consequence of [Reference Brylinski and Deligne1] Proposition 3.13.

Proposition 3.1. Let L be any field containing F over which $\mathbf {T}$ splits. Let $\overline {\mathbf {T}}(L)$ be the resulting central extension

$$ \begin{align*} 1\to \mathbf{K}_2(L)\to \overline{\mathbf{T}}(L)\to \mathbf{T}(L)\to 1. \end{align*} $$

Then the commutator of this extension satisfies

$$ \begin{align*} \mathrm{Comm} (y_1(u_1), y_2(u_2))=\{u_1,u_2\}^{B_Q(y_1,y_2)}, \end{align*} $$

for all $y_1,y_2\in \operatorname {\mathrm {Hom}}(\mathbf {G}_m,\mathbf {T}_L)$ and all $u_1,u_2\in L^{\times }$ .

We would like to note the following useful observation. If $\mathbf {T}=\mathbf {T}_1\times \mathbf {T}_2$ , then $Y=Y_1\oplus Y_2$ . We have the following consequence.

Let $\mathbf {G}=\mathbf {G}_1\times \mathbf {G}_2$ with maximal torus $\mathbf {T}=\mathbf {T}_1\times \mathbf {T}_2$ . Then there is a corresponding decomposition of cocharacter lattice $Y=Y_1\oplus Y_2$ . Let $\overline {\mathbf {G}}\in \mathbf {CExt}(\mathbf {G},\mathbf {K}_2)$ . This gives $\overline {\mathbf {G}}_i \in \mathbf {CExt}(\mathbf {G}_i,\mathbf {K}_2)$ for $i=1,2$ . We have inclusions $\overline {G}_i\to \overline {G}, i= 1, 2$ .

3.4. The tame case

We now discuss the splitting of maximal compact subgroups at unramified places.

Let F be a non-Archimedean field with ring of integers $\mathcal {O}$ . Suppose that the group $\mathbf {G}$ has an integral model $\underline {G}$ over $\mathcal {O}$ . The $\mathbf {K}_2$ -extension $\overline {\mathbf {G}}$ might not be defined over $\mathcal {O}$ . If it is, then there is a natural splitting of $\overline {G}$ over $K=\mathbf {G}(\mathcal {O})$ .

The $\mathbf {K}_2$ -extension $\overline {\mathbf {G}}$ yields a short exact sequence

$$ \begin{align*} 1\to \mathbf{K}_2(F)\to \overline{\mathbf{G}}(F)\to \mathbf{G}(F)\to 1. \end{align*} $$

If $p\nmid n$ , then this ‘tameness’ gives an exact sequence

$$ \begin{align*} 1\to \mathbf{K}_2(\mathcal{O})\to \mathbf{K}_2(F)\xrightarrow{\mathrm{Hilb}_n}\mu_n\to 1. \end{align*} $$

This gives a commutative diagram

Thus, the central extension $1\to \mu _n\to \overline {G}\to G\to 1$ is endowed with a splitting over the hyperspecial maximal compact subgroup $\mathbf {G}(\mathcal {O})$ .

3.5. Adelic BD covering

In this section, F is a global field. For a place v of F, we write $F_v$ for the completion of F at v.

Starting with a BD extension $\overline {\mathbf {G}}$ over $\mathrm {Spec}(F)$ and a positive integer n such that $|\mu _n(F)|=n$ , Brylinski and Deligne showed using results of [Reference Moore20] that one inherits the following data:

• • for each place v of k, a local BD covering group $\overline {G}_v$ of degree n;

• • for almost all v, a splitting $s_v:\mathbf {G}(\mathcal {O}_v)\to \overline {G}_v$ ;

• • a restricted direct product $\prod ^{\prime }_v \overline {G}_v$ with respect to the family of subgroups $s_v(\mathbf {G}(\mathcal {O}_v))$ , from which one can define

  $$ \begin{align*} \overline{G}(\mathbb A):=\prod_v{}^{'} \overline{G}_v/\{(\zeta_v)\in\oplus_v \mu_n(k_v):\prod_v \zeta_v=1\}, \end{align*} $$
  which gives a topological central extension
  $$ \begin{align*} 1\to \mu_n(F)\to \overline{G}(\mathbb A) \to G(\mathbb A) \to 1, \end{align*} $$
  called the adelic or global BD covering group;

• • a natural inclusion

  
  for each place v of k;

• • a natural splitting

  $$ \begin{align*} i:\mathbf{G}(F)\to \overline{G}(\mathbb A), \end{align*} $$
  which allows one to consider the space of automorphic forms on $\overline {G}(\mathbb A)$ .

In this article, we fix an embedding $\epsilon : \mu _n\to \mathbb C^\times $ . We say a representation $\pi $ of $\overline {G}(\mathbb A)$ is $\epsilon $ -genuine if $\mu _n$ acts via $\epsilon $ .

We briefly recall how the splitting i is obtained. Let $X=\mathrm {Spec}(\mathcal {O}_F)$ . Let $S_1$ be a finite set of finite places of F. We assume that $S_1$ is large enough so that the conclusion of [Reference Brylinski and Deligne1] Lemma 10.6 holds. Write $S=S_1\cup \{\text {infinite places}\}$ . This gives a central extension

$$ \begin{align*} 1 \to \mathrm{H}^0(X-S_1,\mathbf{K}_2) \to E_1 \to G(X-S_1) \to 1. \end{align*} $$

For v a place of F, it maps to the local central extension

$$ \begin{align*} 1 \to \mu_n \to \overline{G}_v \to G_v \to 1. \end{align*} $$

For an unramified place v, the map factors through a central extension

$$ \begin{align*} 1\to \mathbf{K}_2(\mathcal{O}_v) \to \overline{\mathbf{G}(\mathcal{O}_v)} \to \mathbf{G}(\mathcal{O}_v) \to 1. \end{align*} $$

If $p\nmid n$ , the exact sequence

$$ \begin{align*} 1\to \mathbf{K}_2(\mathcal{O}_v)\to \mathbf{K}_2(F_v)\to F_v^\times \to 1 \end{align*} $$

shows that $\mathbf {K}_2(\mathcal {O}_v)$ maps trivially to $\mu _n$ . We obtain a trivialisation of $\overline {G}_v$ over $\mathbf {G}(\mathcal {O}_v)$ .

We now have a commutative diagram

provided that for all v in $X-S_1$ , $p\nmid n$ . This holds for $S_1$ is large enough. The first vertical map, composed with the reciprocity map $\prod \zeta _v$ with values in $\mu _n$ , vanishes. We hence obtain

Taking direct limit over S gives the desired natural splitting:

4.1. Pushout

We now recall the definition of pushout.

Definition 4.1. For a central extension

$$ \begin{align*} 1\to A\xrightarrow{i} E\xrightarrow{p} G\to 1 \end{align*} $$

and a homomorphism $f:A\to B$ of abelian groups, we define

$$ \begin{align*} f_\ast(E):=(B\times E)/\langle (f(a), i(a)^{-1}): a\in A \rangle. \end{align*} $$

The maps $B\to f_\ast (E), b\mapsto (b,1)$ and $f_{\ast }(E) \to G, (b,e)\mapsto p(e)$ define an exact sequence

$$ \begin{align*} 1\to B \to f_\ast(E) \to G \to 1. \end{align*} $$

This exact sequence is called the pushout by f.

4.2. Baer sum

Another method to construct new exact sequences is the Baer sum. In this article, we only consider the Baer sum of n copies of an exact sequence.

Consider an exact sequence

(6) $$ \begin{align} 1\to A \to E \to G \to 1 \end{align} $$

with A abelian. By taking the direct sum of n copies of the exact sequence (6), we obtain

$$ \begin{align*} 1\to \oplus_{i=1}^n A \to \oplus_{i=1}^n E \to \oplus_{i=1}^n G \to 1. \end{align*} $$

By pushing out the exact sequence via the product map

$$ \begin{align*} \mathrm{pr}:\prod_{i=1}^n A\to A, \qquad (x_i)\mapsto \prod_{i=1}^n x_i, \end{align*} $$

we obtain an exact sequence:

$$ \begin{align*} 1\to A \to \mathrm{pr}_\ast(\oplus_{i=1}^n E) \to \oplus_{i=1}^n G \to 1. \end{align*} $$

Now we pull back this exact sequence via the diagonal map

$$ \begin{align*} d:G \to \oplus_{i=1}^n G,\qquad x\mapsto (x,\cdots,x) \end{align*} $$

to obtain

$$ \begin{align*} 1\to A \to d^\ast(p_{\ast} \oplus_{i=1}^n E) \to G \to 1. \end{align*} $$

This exact sequence is the Baer sum of n copies of (6).

We now claim that this is also the same as pushing out (6) by the map $[n]: A\to A,x\mapsto x^n$ .

Lemma 4.2. The following commutative diagram gives an isomorphism of exact sequences:

Proof. Recall that $[n]_\ast (E)=(A\times E)/\langle x^n,i^{-1}(x)\mid x\in A \rangle $ and

$$ \begin{align*} p_{\ast} (\oplus_{i=1}^n E )= A \times (\oplus_{i=1}^n E)/\langle \prod_{i=1}^n x_i,(i^{-1}(x_1),\cdots,i^{-1}(x_n)) \mid (x_1,\cdots,x_n)\in \oplus_{i=1}^n A \rangle. \end{align*} $$

We now define $[n]_\ast (E) \to d^\ast (p_{\ast } \oplus _{i=1}^n E)$ by

$$ \begin{align*} (a,e)\mapsto (a,(e,\cdots,e)). \end{align*} $$

It is straightforward to check that this is well-defined and is an isomorphism of exact sequences.

4.3. Functoriality of pullback

Let $f:\mathbf {G}\to \mathbf {H}$ be a morphism of connected reductive groups. Let

$$ \begin{align*} 1\to \mathbf{K}_2\to \overline{\mathbf{H}}\to \mathbf{H}\to 1 \end{align*} $$

be a multiplicative $\mathbf {K}_2$ -torsor on $\mathbf {H}$ . By pulling back via f, we obtain a multiplicative $\mathbf {K}_2$ -torsor on $\mathbf {G}$ :

$$ \begin{align*} 1\to \mathbf{K}_2\to f^\ast(\overline{\mathbf{H}})\to \mathbf{G}\to 1. \end{align*} $$

For ease of notation, let us write $\overline {\mathbf {G}}=f^{\ast }(\overline {\mathbf {H}})$ . Thus, this fits into a commutative diagram

and gives a functor

(7) $$ \begin{align} \mathbf{CExt}(\mathbf{H},\mathbf{K}_2)\to \mathbf{CExt}(\mathbf{G},\mathbf{K}_2). \end{align} $$

At every local place v, the pullback determines the following data:

• • At every local place v, we have a commutative diagram

  

• • The commutative diagram is compatible with the lift of unipotent elements. In other words, the following diagram commutes:

  

• • In the tame case, the commutative diagram is compatible with the natural lift of maximal compact subgroups:

  

We now move to the global setup. So from now on, F is a global field. The local homomorphisms glue to

$$ \begin{align*} \prod_v \overline{G}_v \to \prod_v \overline{H}_v. \end{align*} $$

As $f_v(s_v(\mathbf {G}(\mathcal {O}_v)))\subset s_v(\mathbf {H}(\mathcal {O}_v))$ , we obtain a homomorphism

$$ \begin{align*} \prod_v{}^{\prime}{\overline{G}}_v\to \prod_v{}^{\prime}{\overline{H}}_v\to {\overline{H}}(\mathbb A). \end{align*} $$

This map factors through

$$ \begin{align*} f_{\mathbb A}:\overline{G}(\mathbb A) \to \overline{H}(\mathbb A). \end{align*} $$

From the construction of the natural splitting $H(F) \to \overline {H}(\mathbb A)$ , it is not hard to check that this is compatible with the splitting over rational points. In other words, the diagram

commutes.

We now describe the functor (7) in terms of the BD data. Let $\mathbf {T}_G$ and $\mathbf {T}_H$ be maximal F-tori of $\mathbf {G}$ and $\mathbf {H}$ , respectively. We assume that $f(\mathbf {T}_G) \subset \mathbf {T}_H$ . The map $\mathbf {T}_G \to \mathbf {T}_H$ induces a map $Y_G \to Y_H$ , which gives $\mathcal {E}_G$ as the pullback of $\mathcal {E}_H$ via $Y_G\to Y_H$ :

(8)

The homomorphism $\mathbf {G} \to \mathbf {H}$ also determines a map $\mathbf {G}^{sc} \to \mathbf {H}^{sc}$ . This gives a map $\mathcal {E}_{\mathbf {G}^{sc}} \to \mathcal {E}_{\mathbf {H}^{sc}}$ . It is easy to check that the image of $\mathcal {E}_{\mathbf {G}^{sc}} \to \mathcal {E}_{\mathbf {H}^{sc}} \to \mathcal {E}_H$ agrees with the image of $\mathcal {E}_G \to \mathcal {E}_H$ , which gives a commutative diagram

(9)

Proposition 4.3. With notation as above, the functor (7) can be described in terms of BD data:

$$ \begin{align*} \mathbf{BD}(\mathbf{H},\mathbf{T}_H) \to \mathbf{BD}(\mathbf{G},\mathbf{T}_G),\qquad (Q_H,\mathcal{E}_H,f_H)\mapsto (Q_G,\mathcal{E}_G,f_G), \end{align*} $$

where

• • $Q_G=Q_H|_{Y_G}$ ;

• • $\mathcal {E}_G$ is given by the top row of (8);

• • $f_G$ is given by the commutative diagram in (9).

4.4. Functoriality of pushout

The pushout action is functorial, so it can be glued to a construction of sheaves. Let $f\in \operatorname {\mathrm {End}}(\mathbf {K}_2)$ . Then for a multiplicative $\mathbf {K}_2$ -torsor $\overline {\mathbf {G}}$ , one can push it out via f to obtain a new multiplicative $\mathbf {K}_2$ -torsor. In this article, we consider the following map:

$$ \begin{align*} [m]:\mathbf{K}_2\to \mathbf{K}_2,\qquad x\mapsto x^m \end{align*} $$

for an integer m. We have a natural map $\overline {\mathbf {G}}\to \overline {\mathbf {G}}^\natural $ which fits into the commutative diagram

This defines a functor

$$ \begin{align*} \mathbf{CExt}(\mathbf{G},\mathbf{K}_2)\to \mathbf{CExt}(\mathbf{G},\mathbf{K}_2). \end{align*} $$

At every local place v, we obtain the following:

• • We have the following commutative diagram:

  
  where the first vertical map is $x\mapsto x^m$ .

• • the map $\mathcal {N}_{G_v} \to \overline {G}_v \to \overline {G}_v^{\natural }$ is the canonical unipotent section for $\overline {G}_v^{\natural }$ ;

• • in the tame case, the map $\mathbf {G}(\mathcal {O}_v)\to \overline {G}_v \to \overline {G}_v^{\natural }$ is the natural splitting of maximal compact subgroups.

Globally, we can glue the local maps to obtain a global map. The map $G(F) \to \overline {G}(\mathbb A) \to \overline {G}^{\natural }(\mathbb A)$ is the natural splitting for the multiplicative $\mathbf {K}_2$ -torsor $\overline {\mathbf {G}}^\natural $ .

We now describe the functor $\mathbf {CExt}(\mathbf {G},\mathbf {K}_2)\to \mathbf {CExt}(\mathbf {G},\mathbf {K}_2)$ in terms of BD data.

Proposition 4.4. The functor

$$ \begin{align*} \mathbf{CExt}(\mathbf{G},\mathbf{K}_2)\to \mathbf{CExt}(\mathbf{G},\mathbf{K}_2), \qquad \overline{\mathbf{G}} \mapsto \overline{\mathbf{G}}^{\natural} \end{align*} $$

in terms of the BD data is given by

$$ \begin{align*} \mathbf{BD}(\mathbf{G},\mathbf{T})\mapsto \mathbf{BD}(\mathbf{G},\mathbf{T}),\qquad (Q,\mathcal{E},f)\mapsto (Q^{\natural},\mathcal{E}^{\natural},f^{\natural}), \end{align*} $$

where

• • $Q^{\natural }=mQ$ ;

• • $\mathcal {E}^{\natural }$ is obtained from $\mathcal {E}$ by pushing out via the map $[m]: F_s^\times \to F_s^\times , x\mapsto x^m$ ;

• • $f^{\natural }$ is obtained by pushing out the commutative diagram in (5) via the map $[m]$ .

Observe that if $m\equiv -1 \mod n$ , then

$$ \begin{align*} [m]:\mathbf{K}_2 \to \mathbf{K}_2, \qquad \zeta \mapsto \zeta^{m} \end{align*} $$

becomes $\mu _n\to \mu _n, \zeta \mapsto \zeta ^{-1}$ after taking the Hilbert symbol. Thus,

$$ \begin{align*} \overline{G}_v^\natural =(\mu_n\times \overline{G}_v)/\langle (\zeta,\zeta):\zeta\in \mu_n\rangle. \end{align*} $$

We have a commutative diagram

where the first vertical map is the inverse map and the second vertical map $\overline {G}_v\to \overline {G}_v^\natural $ is a group isomorphism. This works globally as well.

5.1. Classical groups

For the definition of classical groups, we follow the setup in [Reference Yamana29].

By an involution of an algebra D whose centre E contains F, we mean an arbitrary anti-automorphism $\rho $ of D of order 2 under which F is the fixed subfield of E. We denote the restriction of $\rho $ to E also by $\rho $ . We take a couple of $(D,\rho )$ belonging to the following five types:

1. (a) $D=E=F$ and $\rho $ is the identity map;

2. (b) D is a division quaternion algebra over $E=F$ and $\rho $ is the main involution of D;

3. (c) D is a division algebra central over a quadratic extension E of F and $\rho $ generates $\mathrm {Gal}(E/F)$ ;

4. (d) $D=\operatorname {\mathrm {M}}_2(E)$ , $E=F$ and $\begin {pmatrix} a & b \\ c & d \end {pmatrix}^\rho = \begin {pmatrix} d & -b \\ -c & a \end {pmatrix}$ ;

5. (e) $D=\mathbf {D}\oplus \mathbf {D}^{\mathrm {op}}$ , $E=F\oplus F$ and $(x,y)^\rho =(y,x)$ , where $\mathbf {D}$ is a division algebra central over F and $\mathbf {D}^{\mathrm {op}}$ is its opposite algebra.

If F is local, we fix a nontrivial additive character $\psi _F$ of F; if F is global, we fix a nontrivial additive character $\psi _F$ of $F\backslash \mathbb A$ . If $E=F$ , we set $\psi =\psi _F$ ; if $E/F$ is an étale quadratic algebra, we set $\psi =\psi _F\circ \mathrm {tr}_{E/F}$ . The global version is defined similarly. If x is a square matrix with coordinates in D, then $\nu (x)\in E$ and $\tau (x)\in E$ stand for its reduced norm and reduced trace to the centre E of D.

The rank of D as a module over E is a square of a natural number which will be denoted by d. We assume D to be division if F is a number field, so that D is of type (d) (respectively (e)) will appear in our later discussion as a localisation of a global D of type (b) (respectively (c)).

Let $\epsilon $ be either $1$ or $-1$ . We fix once and for all of the triple $(D,\rho ,\epsilon )$ .

Let W be a free left D-module of rank m. By an $\epsilon $ -skew Hermitian space we mean a structure $\mathcal {W}=(W,\langle \ ,\ \rangle )$ , where $\langle \ , \ \rangle $ is an $\epsilon $ -skew Hermitian form on W; that is, an F-bilinear map $\langle \ , \ \rangle : W\times W\to D$ such that

$$ \begin{align*} \langle x, y\rangle^\rho = -\epsilon\langle y , x \rangle,\qquad \langle ax,by\rangle=a\langle x,y\rangle b^\rho, \ (a,b\in D; \ x,y \in W). \end{align*} $$

Such a form is called nondegenerate if $\langle x ,W\rangle =0$ implies that $x=0$ . We assume that $\langle \ , \ \rangle $ is nondegenerate.

We denote the ring of all D-linear endomorphisms of W by $\operatorname {\mathrm {End}}_D(W)$ and set $\mathbf {GL}_D(W)=\operatorname {\mathrm {End}}_D(W)^\times $ . Note that $\mathbf {GL}_D(W)$ acts on W on the right. We sometimes write $\mathbf {GL}_{W;D}$ for $\mathbf {GL}_D(W)$ for ease of notation. Let

$$ \begin{align*} \mathbf{G}= \{ g\in \mathbf{GL}_D(W): \langle xg,yg\rangle = \langle x,y\rangle \text{ for all }x,y\in W \} \end{align*} $$

be the unitary group of $(W,\langle \ , \ \rangle )$ , which is a reductive algebraic group defined over F. It is important to realise that $\mathbf {G}$ always comes together with a space W and a form $\langle \ , \ \rangle $ . We usually just speak of $\mathbf {G}$ and the data $\mathcal {W}=(W,\langle \ ,\ \rangle )$ will be implicitly understood. We write $\mathbf {G}=\mathbf {G}(\mathcal {W})$ when the dependence of $\mathbf {G}$ on $\mathcal {W}$ needs to be stressed.

5.2. Doubling homomorphism

Let $\mathcal {W}=(W,\langle \ , \ \rangle )$ be one of the $\epsilon $ -skew Hermitian forms described above. Let k be a fixed positive integer. We would like to define the following in this section:

(10) $$ \begin{align} (\mathbf{G},\mathbf{G}^{\square,k},\iota,\mathbf{P}, \mathbf{N}_{\mathcal{W},k}^{\bullet}, \psi_{\mathcal{W},k}^{\bullet}). \end{align} $$

Put $W^{\square ,k}=W^{\oplus 2k}$ . We usually write

$$ \begin{align*} W^{\square,k} = W_{1,+}\oplus W_{2,+}\oplus \cdots \oplus W_{k,+}\oplus W_{k,-}\oplus \cdots \oplus W_{2,-} \oplus W_{1,-} \end{align*} $$

to distinguish the copies of W in $W^{\square ,k}$ . We write an element in $W^{\square ,k}$ as

$$ \begin{align*} (\boldsymbol{x};\boldsymbol{y})=(x_1,\cdots, x_k; y_{k},\cdots, y_{1}),\qquad x_i\in W_{i,+}, \ y_i\in W_{i,-}. \end{align*} $$

Define an $\epsilon $ -skew Hermitian form $\langle \ , \ \rangle ^{\square ,k} $ on $W^{\square ,k}$ by

$$ \begin{align*} \langle (\boldsymbol{x};\boldsymbol{y}), (\boldsymbol{x}';\boldsymbol{y}') \rangle^{\square,k} = \sum_{i=1}^k(\langle x_i,x^{\prime}_i\rangle-\langle y_{i},y^{\prime}_{i}\rangle) \qquad (x_i,x^{\prime}_i\in W_{i,+}; y_{i},y^{\prime}_{i}\in W_{i,-}). \end{align*} $$

Let $\mathbf {G}^{\square ,k}$ denote the unitary group of $(W^{\square ,k}, \langle \ , \ \rangle ^{\square ,k})$ .

For $W^\square =W_+\oplus W_-$ , let

$$ \begin{align*} W^\nabla=\{(x,-x)\in W_{+}\oplus W_{-}: x\in W\} \end{align*} $$

be the graph of minus the identity map from W to W and

$$ \begin{align*} W^{\Delta}=\{(x,x)\in W_{+}\oplus W_{-}: x\in W\} \end{align*} $$

be the graph of the identity map. Given $x\in W$ , we write

$$ \begin{align*} x^\Delta=(x ,x)\in W^\Delta \text{ and } x^\nabla=(x,-x)\in W^\nabla. \end{align*} $$

We have the following observations:

1. (1) For each i, $W_i^\square =W_{i,+}\oplus W_{i,-}=W^\Delta _{i}+ W_i^\nabla.$ Both $W_i^\Delta $ and $W_i^\nabla $ are totally isotropic in $W^{\square ,k}$ .

2. (2) The space $W^\Delta $ is isomorphic to W as vector spaces via

   $$ \begin{align*} W^\Delta \simeq W, \qquad (x,x)\mapsto x. \end{align*} $$
   The space $W^\nabla $ is identified with W via $(x,-x)\mapsto 2x$ . Thus, we can view $\mathbf {G}(\mathcal {W})$ as a subgroup of $\mathbf {GL}_D(W^\Delta )$ or $\mathbf {GL}_D(W^\nabla )$ and identify $\operatorname {\mathrm {Hom}}_D(W_i^\nabla ,W_j^\nabla )$ with $\operatorname {\mathrm {End}}_D(W)$ .

Define

$$ \begin{align*} W^{\Delta,k}=W_1^\Delta \oplus \cdots \oplus W_k^\Delta,\qquad W^{\nabla,k}=W_1^\nabla \oplus \cdots \oplus W_k^\nabla. \end{align*} $$

Both spaces are totally isotropic in $W^{\square ,k}$ and $W^{\square ,k} =W^{\Delta ,k}+W^{\nabla ,k}$ . This is a complete polarisation of $W^{\square ,k}$ . Unless otherwise specified, we write $\mathbf {P}=\mathbf {P}(W^{\Delta ,k})$ .

We first construct a Fourier coefficient for the group $\mathbf {G}^{\square ,k}$ . We choose the following flag of totally isotropic subspaces in $W^{\square ,k}$ :

(11) $$ \begin{align} 0\subset W_k^\nabla \subset W_{k-1}^\nabla \oplus W_{k}^\nabla \subset \cdots \subset W_2^\nabla \oplus \cdots \oplus W_{k}^\nabla. \end{align} $$

Let $\mathbf {P}_{\mathcal {W},k}^{\bullet }=\mathbf {M}_{\mathcal {W},k}^{\bullet }\cdot \mathbf {N}_{\mathcal {W},k}^{\bullet }$ be the corresponding parabolic subgroup. Then

$$ \begin{align*} \mathbf{M}_{\mathcal{W},k}^{\bullet}\simeq \mathbf{GL}_D(W_{k}^\nabla)\times \cdots \times \mathbf{GL}_D(W_{2}^\nabla) \times \mathbf{G}(\mathcal{W}_1^\square). \end{align*} $$

The character is defined on the group $\mathbf {N}_{\mathcal {W},k}^{\bullet }$ .

We reindex the flag in (11) as

$$ \begin{align*} 0\subset Y_1\subset \cdots \subset Y_{k-1} \end{align*} $$

and extend it to

$$ \begin{align*} 0\subset Y_1\subset \cdots \subset Y_{k-1} \subset Y_{k-1}^{\perp}\subset \cdots \subset Y_1^\perp\subset W^{\square,k}. \end{align*} $$

Note that except for $Y_{k-1}^\perp /Y_{k-1}=W_1^\square $ , the quotient between two successive terms is isomorphic to either $W^\Delta $ or $W^\nabla $ . For convenience, we write $Y_0=0$ and $Y_k:=Y_{k-1}^\perp $ .

To describe the character of $\mathbf {N}_{\mathcal {W},k}^{\bullet }(F)\backslash \mathbf {N}_{\mathcal {W},k}^{\bullet }(\mathbb A)$ , we have to specify elements

$$ \begin{align*} A_i\in \operatorname{\mathrm{Hom}}_D(Y_i/Y_{i-1},Y_{i+1}/Y_i)\simeq \operatorname{\mathrm{End}}_D(W^\nabla),\qquad i=1,\cdots, k-2, \end{align*} $$

and

$$ \begin{align*} A_{k-1}\in \operatorname{\mathrm{Hom}}_D(Y_{k-1}/Y_{k-2},Y_{k-1}^{\perp}/Y_{k-1})\simeq \operatorname{\mathrm{Hom}}_D(W^\nabla,W^\square). \end{align*} $$

We choose $A_1,\cdots ,A_{k-2}$ to be the identity map in $\operatorname {\mathrm {End}}_D(W^{\nabla })$ . The map

$$ \begin{align*} Y_{k-1}/Y_{k-2}\xrightarrow{A_{k-1}} Y_{k-1}^{\perp}/Y_{k-1}\xrightarrow{A_k} Y_{k-2}^{\perp}/Y_{k-1}^{\perp} \end{align*} $$

is translated from

$$ \begin{align*} W^\nabla\to W_+\oplus W_- \to W^\Delta,\qquad x^{\nabla}\mapsto (2x,0)\mapsto 2x^{\Delta}. \end{align*} $$

Note that $A_{k}\circ A_{k-1}$ is an isomorphism.

An element $u\in \mathbf {N}_{\mathcal {W},k}^{\bullet }(F)\backslash \mathbf {N}_{\mathcal {W},k}^{\bullet }(\mathbb A)$ induces

$$ \begin{align*} u_i:(Y_{i+1}/Y_i)\otimes (F\backslash \mathbb A) \to (Y_i/Y_{i-1})\otimes (F\backslash \mathbb A). \end{align*} $$

Then we define

$$ \begin{align*} \psi_{\mathcal{W},k}^{\bullet}:\mathbf{N}_{\mathcal{W},k}^{\bullet}(F)\backslash \mathbf{N}_{\mathcal{W},k}^{\bullet}(\mathbb A) \to \mathbb C,\qquad u\mapsto \psi\left(\sum_{i=1}^{k-1}\tau(u_i\circ A_i)\right). \end{align*} $$

Given $(g_1,g_2)\in \mathbf {G}\times \mathbf {G}$ , we define its action on $W_{1,+}\oplus W_{2,+}\oplus \cdots \oplus W_{k,+}\oplus W_{k,-}\oplus \cdots \oplus W_{2,-} \oplus W_{1,-}$ via

$$ \begin{align*} (x_1,\cdots, x_k; y_{k},\cdots, y_2,y_{1})(g_1,g_2)=(x_1 g_1,\cdots, x_kg_1; y_{k}g_1,\cdots, y_2g_1,y_{1}g_2). \end{align*} $$

This extends to an action of $\mathbf {G}\times \mathbf {G}$ on $W^{\square ,k}$ and gives a map

$$ \begin{align*} \iota=\iota_k: \mathbf{G}\times \mathbf{G} \to \mathbf{G}^{\square,k}. \end{align*} $$

It is in fact a homomorphism and, in particular, the images of these two copies of $\mathbf {G}$ commute in $\mathbf {G}^{\square ,k}$ . It is straightforward to check that $\iota (G\times G)$ lies in the stabiliser of $\psi _{\mathcal {W},k}^{\bullet }$ in $\mathbf {G}^{\square ,k}$ .

Recall that for a subgroup J of G, we define $J^{\diamondsuit }=\{(g,g)\in G\times G \mid g\in J\}$ . We have the following results from [Reference Cai5].

5.3. The case of special orthogonal groups

We now discuss the case of special orthogonal groups. Since the group $\mathbf {O}(W)$ is disconnected, to consider multiplicative $\mathbf {K}_2$ -torsors, it would be better to consider its connected component $\mathbf {SO}(W)$ . We now explain the modifications in order to develop the twisted doubling integrals. For the group $\mathbf {O}(W)$ and a fixed positive integer k, we have defined a list of input $(\mathbf {G},\mathbf {G}^{\square },\iota , \mathbf {P},\mathbf {N}_{\mathcal {W},k}^{\bullet },\psi _{\mathcal {W},k}^{\bullet })$ . We now explain how to define it for the group $\mathbf {SO}(W)$ .

The doubling homomorphism $\iota :\mathbf {O}(W) \times \mathbf {O}(W) \to \mathbf {O}(W^{\square ,k})$ restricts to

$$ \begin{align*} \iota: \mathbf{SO}(W) \times \mathbf{SO}(W) \to \mathbf{SO}(W^{\square,k}). \end{align*} $$

Note that $\mathbf {P}, \mathbf {N}_{\mathcal {W},k}^{\bullet } \subset \mathbf {SO}(W^{\square ,k})$ . Thus, we can still use $\psi _{\mathcal {W},k}^{\bullet }$ and $\mathbf {N}_{\mathcal {W},k}^{\bullet }$ to define a Fourier coefficient of an automorphic form on $\mathbf {SO}(W^{\square ,k})(\mathbb A)$ . Thus, we take

$$ \begin{align*} (\mathbf{SO}(W), \mathbf{SO}(W^{\square,k}),\iota, \mathbf{P},\mathbf{N}_{\mathcal{W},k}^{\bullet}, \psi_{\mathcal{W},k}^{\bullet}) \end{align*} $$

to be the input in the case of special orthogonal groups. To unify our discussion, if $\mathbf {G}=\mathbf {SO}(W)$ , then we take $\mathbf {G}^{\square ,k}=\mathbf {SO}(W^{\square ,k})$ .

The case of inner forms of orthogonal groups can be treated similarly.

6.1. Degenerate Whittaker models

We first recall the definition of degenerate Whittaker models. One can attach a degenerate Whittaker model for a Whittaker pair (see [Reference Gomez, Gourevitch and Sahi12, Reference Mœglin and Waldspurger21]). Given an admissible representation $\pi $ , an important question is to find the largest nilpotent orbits that support degenerate Whittaker models for $\pi $ . Locally, this determination is related to other nilpotent invariants such as the wavefront set. We refer the reader to the introduction of [Reference Gomez, Gourevitch and Sahi12] for a comprehensive account of discussion.

In this article, we consider only a subclass of degenerate Whittaker models for $\operatorname {\mathrm {GL}}_{n,D}$ . As explained in [Reference Cai5] Subsection 2.4, this is sufficient for determining nilpotent invariants and fits into our examples later. As unipotent subgroups split canonically over covering groups, these notions transfer from the linear case to the covering group case automatically. In the following, we only define these in the linear case.

Let $R=F$ if F is a local field and $R=\mathbb A/F$ if F is a number field. Fix a nontrivial additive character $\psi _F:R \to \mathbb C^\times $ . Let D be a central division algebra as in Subsection 5.1. Let W be a free left D-module of rank m and consider the group $\mathbf {GL}_D(W)$ . Recall that we sometimes write $\mathbf {GL}_{W;D}$ for $\mathbf {GL}_D(W)$ for ease of notation. Let

$$ \begin{align*} \mathcal{Y}:0\subset Y_1\subset Y_2\subset \cdots \subset Y_k\subset W \end{align*} $$

be a flag of distinct subspaces of W. We sometimes write $Y_0=\{0\}$ and $Y_{k+1}=W$ for convenience. The stabiliser of $\mathcal {Y}$ is a parabolic subgroup $\mathbf {P}(\mathcal {Y})=\mathbf {M}(\mathcal {Y})\cdot \mathbf {N}(\mathcal {Y})$ with Levi component $\mathbf {M}(\mathcal {Y})$ . Then as algebraic groups,

$$ \begin{align*} \mathbf{N}(\mathcal{Y})^{ab}\cong \prod_{i=1}^k \operatorname{\mathrm{Hom}}_D(Y_{i+1}/Y_i,Y_i/Y_{i-1}),\qquad u\mapsto (u_i)_{i=1}^{k}. \end{align*} $$

To give a character of $\mathbf {N}(\mathcal {Y})(R)$ , we specify an element in

$$ \begin{align*} \mathcal{A}=(A_1,\cdots, A_k)\in \prod_{i=1}^k \operatorname{\mathrm{Hom}}_D(Y_{i}/Y_{i-1},Y_{i+1}/Y_i). \end{align*} $$

More concretely, given such an $\mathcal {A}$ , we define a character $\psi _{\mathcal {A}}$ of $N(\mathcal {Y})(R)$ by

$$ \begin{align*} \psi_{\mathcal{A}}(u)=\psi\left(\sum_{i=1}^k \tau(u_i\circ A_i)\right). \end{align*} $$

Here, $\psi =\psi _F\circ \mathrm {tr}_{E/F}$ .

Assume now we have a pair $(\mathbf {N}(\mathcal {Y}),\psi _{\mathcal {A}})$ . Globally, for an irreducible automorphic representation $\pi $ of $\overline {\mathrm {GL}}_{W;D}(\mathbb A)$ , we define the $(N(\mathcal {Y}),\psi _{\mathcal {A}})$ -Fourier coefficient of $\phi \in \pi $ as

$$ \begin{align*} \phi^{N(\mathcal{Y}),\psi_{\mathcal{A}}}(g)=\int\limits_{[N(\mathcal{Y})]} \phi(ug)\psi_{\mathcal{A}}(u) \ du. \end{align*} $$

Locally, we consider the space $\operatorname {\mathrm {Hom}}_{N(\mathcal {Y})(F)}(\pi ,\psi _{\mathcal {A}})$ of $(N(\mathcal {Y}),\psi _{\mathcal {A}})$ -functional for an admissible representation $\pi $ of $\overline {\mathrm {GL}}_{W;D}(F)$ .

6.2. Representations of type $(k,m)_D$

The purpose of this section is to introduce the notion of representations of type $(k,m)_D$ , both locally and globally. These representations are supported on a suitable nilpotent orbit and admit unique models of degenerate type. In the linear case and when D is a field, the generalised Speh representations are examples of such representations.

In this section, we assume that $\dim _D W=km$ .

Definition 6.1. We say a pair $(N(\mathcal {Y}),\psi _{\mathcal {A}})$ is in the orbit $(k^m)_D$ if $\mathcal {Y}$ is of the form

$$ \begin{align*} 0\subset Y_1 \subset \cdots \subset Y_{k-1} \subset W \end{align*} $$

and for $i=1,\cdots , k-1$ , $\dim _D Y_i=mi$ and $A_i$ is an isomorphism.

The stabiliser of a coefficient in the orbit $(k^m)_D$ is isomorphic to $\operatorname {\mathrm {GL}}_{m,D}$ .

Definition 6.2. We say a pair $(N(\mathcal {Y}),\psi _{\mathcal {A}})$ lies in an orbit higher than $(k^m)_D$ if

$$ \begin{align*} A_{i+k-1}\circ \cdots \circ A_{i}\neq 0 \end{align*} $$

for some i.

Note that this implies that there are at least k proper subspaces in the flag $\mathcal {Y}$ .

6.3. Invariance under stabiliser

This section is a straightforward adaption of [Reference Cai5] Subsection 2.4.2 to the case of covering groups. We collect necessary results but omit the proofs.

We now assume that $\dim _D W=km$ and the representation $\theta $ of $\overline {\mathrm {GL}}_{W;D}$ is of type $(k,m)_D$ . We consider the following situation:

$$ \begin{align*} 0\subset Y_1 \subset \cdots \subset Y_{k-1}\subset W \end{align*} $$

such that $\dim _D Y_i=mi$ , $A_2,\cdots , A_{k-1}$ are isomorphisms and the rank of $A_{1}$ is $a>0$ (which might not be of full rank).

The Fourier coefficients defined by such a pair enjoy an extra invariance property. We start with the case of $a=m$ . Recall that the stabiliser $\mathrm {St}_{\mathcal {A}}$ of a pair $(N(\mathcal {Y}),\psi _{\mathcal {A}})$ that lies in the orbit $(k^m)_D$ is isomorphic to $\operatorname {\mathrm {GL}}_{m,D}$ . We start with the local version.

Here is the global version.

Lemma 6.7 [Reference Cai5] Lemma 2.15

Let $\theta $ be an irreducible unitary $\epsilon $ -genuine automorphic representation of $\operatorname {\mathrm {GL}}_{W;D}(\mathbb A)$ . Then there is a character $\overline {\chi }_\theta : \overline {\mathrm {St}}_{\mathcal {A}}(F\backslash \mathbb A) \to \mathbb C^\times $ such that, for any $\phi \in \theta $ ,

$$ \begin{align*} \phi^{N(\mathcal{Y}),\psi_{\mathcal{A}}}(gh) =\overline{\chi}_{\theta}(g)\phi^{N(\mathcal{Y}),\psi_{\mathcal{A}}}(h) \end{align*} $$

for any $g\in \overline {\mathrm {St}}_{\mathcal {A}}(F\backslash \mathbb A)$ and $h\in \overline {\mathrm {GL}}_{W;D}(\mathbb A)$ .

We now consider the case $a<m$ . Define $S_{\mathcal {A}}$ to be the subgroup of $M(\mathcal {Y})$ :

$$ \begin{align*} N(\mathrm{Ker}(A_1))\times \{1\} \times\cdots \times \{1\} \subset \operatorname{\mathrm{GL}}(Y_1)\times \operatorname{\mathrm{GL}}(Y_2/Y_1) \times \cdots \times \operatorname{\mathrm{GL}}(W/Y_{k-1}). \end{align*} $$

Here, $N(\mathrm {Ker}(A_1))$ is the unipotent radical of the parabolic subgroup of $\operatorname {\mathrm {GL}}(Y_1)$ stabilising $\mathrm {Ker}(A_1)$ . Then the unipotent group $S_{\mathcal {A}}$ is in the stabiliser of the pair $(N(\mathcal {Y}),\psi _{\mathcal {A}})$ . (Note that $S_{\mathcal {A}}$ is not the full stabiliser.) The Fourier coefficient $\phi ^{N(\mathcal {Y}),\psi _{\mathcal {A}}}(g)$ is left-invariant under $[S_{\mathcal {A}}]$ .

Proposition 6.8 [Reference Cai5] Proposition 2.17

For $\phi \in \theta $ ,

$$ \begin{align*} \phi^{N(\mathcal{Y}),\psi_{\mathcal{A}}}(gh)=\phi^{N(\mathcal{Y}),\psi_{\mathcal{A}}}(h) \end{align*} $$

for any $g\in S_{\mathcal {A}}(F\backslash \mathbb A)$ and $h\in \overline {\mathrm {GL}}_{W;D}(\mathbb A)$ .

8.1. The local doubling homomorphism

We now discuss the local and global consequences using results from Section 4. The homomorphism

$$ \begin{align*} \iota:\overline{G}_v^\natural\times \overline{G}_v\to \overline{G^{\square,kn_Q}_v} \end{align*} $$

is a lift of the local doubling homomorphism $\iota :G_v\times G_v\to G^{\square ,kn_Q}_v$ . Here are the consequences:

• • The images of $\overline {G}_v$ and $\overline {G}_v^\natural $ commute in $\overline {G^{\square ,kn_Q}_v}$ .

• • We have a commutative diagram

  
  where the first vertical map is given by
  $$ \begin{align*} \mu_n\times \mu_n \to \mu_n,\qquad (\zeta_1,\zeta_2)\mapsto \zeta_1^{-1}\zeta_2. \end{align*} $$

8.2. Splitting over the diagonal copy

We now discuss an important consequence of Lemma 8.4. We write $\overline {(G\times G)}_v$ to be the pullback of $\overline {G^{\square ,kn_Q}_v}$ via $\iota :G_v\times G_v \to G^{\square ,kn_Q}_v$ .

Proof. We know that the homomorphism

$$ \begin{align*} \overline{G}_v \times \overline{G}_v \to \overline{(G\times G)}_v \to \overline{G^{\square,kn_Q}_v} \end{align*} $$

restricts to $\mu _n\times \mu _n\to \mu _n,(\zeta _1,\zeta _2)\mapsto \zeta _1^{-1}\zeta _2$ . Thus, we obtain

$$ \begin{align*} \overline{(G\times G)}_v=(\overline{G}_v\times \overline{G}_v)/\mu_n^\diamondsuit. \end{align*} $$

This implies that the image of $\overline {G}_v^\diamondsuit \subset \overline {G}_v\times \overline {G}_v$ in $\overline {(G\times G)}_v$ is $\overline {G}_v^\diamondsuit /\mu _n^\diamondsuit \simeq G_v^{\diamondsuit }$ . Thus, we have a natural splitting $G_v^{\diamondsuit }\to \overline {(G\times G)}_v$ .

8.3. The global doubling homomorphism

The local doubling homomorphism glues to

$$ \begin{align*} \iota_{\mathbb A}:\prod_v \overline{G}_v \times \prod_v \overline{G}_v \to \prod_v \overline{G^{\square,kn_Q}_v}. \end{align*} $$

As $\iota _v(s_v(\mathbf {G}(\mathcal {O}_v))\times s_v(\mathbf {G}(\mathcal {O}_v))) \subset s_v(\mathbf {G}^{\square ,kn_Q}(\mathcal {O}_v))$ for almost all v, we obtain a homomorphism

$$ \begin{align*} \iota_{\mathbb A}:\prod_v{}^{\prime} \ \overline{G}_v \times \prod_v {}^{\prime} \ \overline{G}_v \to \prod_v{}^{\prime}\ \overline{G^{\square,kn_Q}_v}\to \overline{G^{\square,kn_Q}}(\mathbb A). \end{align*} $$

This map factors through

$$ \begin{align*} \iota_{\mathbb A}:\overline{G}(\mathbb A)\times \overline{G}(\mathbb A) \to \overline{G^{\square,kn_Q}}(\mathbb A). \end{align*} $$

To summarise, we have obtained a global doubling homomorphism so that

• • the restriction to $\mu _n\times \mu _n$ is given by

  $$ \begin{align*} \iota_{\mathbb A}:\mu_n\times \mu_n\to \mu_n, \qquad (\zeta_1,\zeta_2)\mapsto \zeta_1^{-1}\zeta_2. \end{align*} $$

• • this map is a lift of the linear doubling homomorphism $G(\mathbb A)\times G(\mathbb A) \to G^{\square ,kn_Q}(\mathbb A)$ .

• • this map is also compatible with the local doubling homomorphism.

The section over the rational points is also compatible with the doubling homomorphism.

Proposition 8.7. We have a commutative diagram

We can also check that the cover splits over $\iota _{\mathbb A}(G(\mathbb A)^{\diamondsuit })$ .

9.1. Petersson inner product

Let $\pi $ be an irreducible $\epsilon $ -genuine cuspidal automorphic representation of $\overline {G}(\mathbb A)$ realised on a space $V_\pi \subset L_{\epsilon }^2(G(F)\backslash \overline {G}(\mathbb A))$ , where we fix an embedding $\pi \hookrightarrow V_\pi \subset \mathcal {A}(\overline {G}(\mathbb A))$ . The contragredient representation $\pi ^\vee $ is $\epsilon ^{-1}$ -genuine and is realised on the complex conjugate $\overline {V_\pi }$ of $V_\pi $ . The Petersson pairing $\mathcal {P}=\mathcal {P}_{\pi }: V_\pi \boxtimes \overline {V_\pi }\to \mathbb C$ is defined by

$$ \begin{align*} \mathcal{P}_{\pi}(\xi_1\boxtimes \xi_2)=\dfrac{1}{n}\int\limits_{G(F)\backslash \overline{G}(\mathbb A)}\xi_1(g)\xi_2(g) \ dg. \end{align*} $$

The integrand is trivial on both $\mu _n$ and $G(F)$ and the pairing is $\overline {G}(\mathbb A)$ -invariant.

The Petersson inner product admits a slightly different formula. The function $\xi _1\boxtimes \xi _2\in \pi \boxtimes \pi ^\vee $ is a function on $\overline {G}(\mathbb A)\times \overline {G}(\mathbb A)$ which is trivial on $\mu _n^\diamondsuit $ . Thus, this descends to a function on $\overline {G\times G}(\mathbb A)$ , which will be denoted as $\overline {\xi _1\boxtimes \xi _2}$ . By Lemma 8.8, the image of $\overline {G}(\mathbb A)$ under $\overline {G}(\mathbb A)\times \overline {G}(\mathbb A)\to \overline {G\times G}(\mathbb A)$ is $G(\mathbb A)$ . It is easy to check that

$$ \begin{align*} \mathcal{P}_{\pi}(\xi_1\boxtimes \xi_2)=\int\limits_{G(F)\backslash G(\mathbb A)} \overline{\xi_1\boxtimes \xi_2}((g,g)) \ dg. \end{align*} $$

9.2. Metaplectic restricted tensor product

We now recall the notion of meteplectic restricted tensor product. Notation: $\pi =\tilde \otimes _v'\pi _v$ . We can view $\pi $ as a representation of $\prod _v' \overline {G}_v$ via the projection map $\prod _v' \overline {G}_v\to \overline {G}(\mathbb A)$ . The space of the representation stays the same. Thus, we can write $\pi $ as a restricted tensor product $\pi =\otimes ^{\prime }_v \pi _v$ , where $\pi _v$ is an admissible representation of $\overline {G}_v$ .

We fix isomorphisms $V_\pi \simeq \tilde \otimes ^{\prime }_v \pi _v$ and $\overline {V_\pi }\simeq \tilde \otimes ^{\prime }_v \pi _v^\vee $ .

Lemma 9.1. We can choose a standard local pairing $\mathcal {P}_{\pi _v}:\pi _v\boxtimes \pi _v^\vee \to \mathbb C$ at every local place v in order that

$$ \begin{align*} \mathcal{P}_{\pi}(\xi)=\prod_v \mathcal{P}_{\pi_v}(\xi_v) \end{align*} $$

for all $\xi =\otimes _v \xi _v\in V_\pi \boxtimes \overline {V_\pi }$ , where $\mathcal {P}_{\pi _v}(\xi _{0,v})=1$ for almost all $s_v(K_v)\times s_v(K_v)$ -invariant vectors $\xi _{0,v}\in \pi _v\boxtimes \pi _v^\vee $ used to define the restricted tensor products.

Proof. Let $\pi _v$ be an $\epsilon $ -genuine representation of $\overline {G}_v$ . The action on $\pi ^\vee $ is given via

$$ \begin{align*} \langle \xi_v,\pi_v^\vee(g)\xi_v^\vee \rangle=\langle \pi_v(g^{-1})\xi_v,\xi_v^\vee \rangle. \end{align*} $$

Thus, $\pi _v^\vee $ is an $\epsilon ^{-1}$ -genuine representation of $\overline {G}_v$ . Recall that $\operatorname {\mathrm {Hom}}_{\overline {G}_v}(\pi _v,\pi _v)=\mathbb C$ . This implies that $\operatorname {\mathrm {Hom}}_{\overline {G}_v}(\pi _v\boxtimes \pi _v^\vee ,\mathbb C)=\mathbb C$ .

The Petersson inner product defines a $\overline {G}(\mathbb A)$ -equivariant pairing on global representations

$$ \begin{align*} \pi\boxtimes \pi^\vee \to \mathbb C. \end{align*} $$

This (abstract) pairing is also $\prod _v'\overline {G}_v$ -equivariant. One can argue as in the linear case to prove the result (see [Reference Bump4] Subsection 3.5 for an analogous argument).

9.3. Eisenstein series

We now describe the Eisenstein series that appear in the global construction. Let $\theta $ be an irreducible unitary automorphic representation of $\overline {\mathrm {GL}}_{kmn_Q,D}(\mathbb A)$ of type $(kn_Q,m)_D$ . We define the normalised global induced representation $I(s,\theta )=\operatorname {\mathrm {Ind}}_{\overline {P}(\mathbb A)}^{\overline {G^{\square ,kmn_Q}}(\mathbb A)}(\theta \cdot \nu ^s)$ . Here $\nu $ is defined as $\overline {P}(\mathbb A)\to P(\mathbb A)\to M(\mathbb A) \to \mathbb C^\times $ .

For any holomorphic section $\tilde \phi ^{(s)}$ of $I(s,\theta )$ , we write $\phi ^{(s)}(g)=\tilde \phi ^{(s)}(g;1)$ to be the value at the identity. We form the associated Eisenstein series $E(\phi ^{(s)})$ on $G^{\square ,kmn_Q}(F)\backslash \overline {G^{\square ,kmn_Q}}(\mathbb A)$ by

$$ \begin{align*} E(\phi^{(s)})(g)=\sum_{\gamma\in P(F)\backslash G^{\square,kmn_Q}(F)} \phi^{(s)}(\gamma g). \end{align*} $$

The Eisenstein series converges for $\Re s\gg 0$ . By the theory of Eisenstein series, it can be continued to a meromorphic function in s on all of $\mathbb C$ satisfying a functional equation.

We now discuss the extra invariance property in this setup. Notation: we write $N_{\mathcal {W}}^{\bullet } = N_{\mathcal {W},kn_Q}^{\bullet }\subset G^{\square ,kmn_Q}$ and $\psi _{\mathcal {W}}^{\bullet }=\psi _{\mathcal {W},kn_Q}^{\bullet }$ for ease of notation.

Let

$$ \begin{align*} f^{(s)}(g)=f_{\phi}^{(s)}(g):=\int\limits_{[N_{\mathcal{W}}^{\bullet} \cap P]}\phi^{(s)}(ug) \psi_{\mathcal{W}}^{\bullet}(u) \ du=\int\limits_{[N_{\mathcal{W}}^{\bullet}\cap P]}\tilde\phi^{(s)}(g;u) \psi_{\mathcal{W}}^{\bullet}(u) \ du. \end{align*} $$

The pair $(\mathbf {N}_{\mathcal {W}}^{\bullet }\cap \mathbf {P},\psi _{\mathcal {W}}^{\bullet }|_{[N_{\mathcal {W}}^{\bullet }\cap P]})$ gives a Fourier coefficient in the orbit $((kn_Q)^m)_D$ and $\iota (G^{\diamondsuit })$ lies in the stabiliser. For fixed g, the function $u\mapsto \phi ^{(s)}(ug)$ on $(N_{\mathcal {W}}^{\bullet }\cap P)(\mathbb A)$ is an element of $\theta \cdot \nu ^s$ . Thus, one can view $f^{(s)}(g)$ as a $(kn_Q,m)_D$ -coefficient of an element in $\theta \cdot \nu ^s$ .

Lemma 9.2. There exists a character $\chi _\theta : [G] \to \mathbb C^\times $ such that for $\iota (g,g)\in \iota (G(\mathbb A)^\diamondsuit )$ and $h\in \overline {G^{\square ,kmn_Q}}(\mathbb A)$ ,

$$ \begin{align*} f^{(s)}(\iota(g,g)h)=\chi_\theta(\nu(g)) f^{(s)}(h). \end{align*} $$

Proof. Recall that the homomorphism

$$ \begin{align*} \overline{G}^\diamondsuit(\mathbb A) \to \overline{G}(\mathbb A)\times \overline{G}(\mathbb A) \to \overline{G^{\square,kmn_Q}}(\mathbb A) \end{align*} $$

is trivial on $\mu _n$ and thus descends to

$$ \begin{align*} G^{\diamondsuit}(\mathbb A)\to \overline{G^{\square,kmn_Q}}(\mathbb A). \end{align*} $$

This indeed gives a homomorphism

$$ \begin{align*} G^{\diamondsuit}(\mathbb A) \to \overline{P}(\mathbb A) \to \overline{M}(\mathbb A). \end{align*} $$

The image of $G^{\diamondsuit }(\mathbb A)$ lies in the stabiliser of the Fourier coefficient. Thus, the Fourier coefficient is left-equivariant under a character $\chi _\theta $ ; in other words,

$$ \begin{align*} f^{(s)}(\iota(g,g)h)=\chi_\theta(\nu(g))f^{(s)}(h), \end{align*} $$

for $\iota (g,g)\in \iota (G(\mathbb A)^\diamondsuit )$ and $h\in \overline {G^{\square ,kmn_Q}}(\mathbb A)$ .

9.4. The global integral

We view $\chi _\theta $ as a character of $\overline {G}(\mathbb A)$ via $\overline {G}(\mathbb A)\to G(\mathbb A) \xrightarrow {\chi _{\theta }} \mathbb C^\times $ , which we still denote as $\chi _\theta $ . We define the global integral to be

$$ \begin{align*} Z(\xi_1\boxtimes \xi_2,\phi^{(s)}) &=\dfrac{1}{n^2}\int\limits_{[\overline{G} \times \overline{G}]} \chi_\theta(\nu(g_2))^{-1}\xi_1(g_1) \xi_2(g_2)\\ &\quad {}\times \int\limits_{[N_{\mathcal{W}}^{\bullet}]}E(\phi^{(s)})(u\cdot \iota(g_1,g_2)) \psi_{\mathcal{W}}^{\bullet}(u) \ du \ dg_1 \ dg_2. \end{align*} $$

Since $\iota _{\mathbb A}(G(F)\times G(F))\subset G^{\square ,kmn_Q}(F)$ , this integral is well-defined.

Since the two cusp forms are rapidly decreasing on $G(F)\backslash \overline {G}(\mathbb A)$ and the Eisenstein series is only of moderate growth, we see that the integral converges absolutely for all s away from the poles of the Eisenstein series and is hence meromorphic in s.

9.5. Main global identity

We now state the main global identity.

Theorem 9.5. When $\Re s\gg 0$ , $Z(\xi _1\boxtimes \xi _2,\phi ^{(s)})$ equals

$$ \begin{align*} &\dfrac{1}{n^2}\int\limits_{G^\diamondsuit(F)\backslash (\overline{G}\times \overline{G})(\mathbb A)}\chi_\theta(\nu(g_2))^{-1} \xi_1(g_1)\xi_2(g_2)\\ &\quad{}\times\int\limits_{(N_{\mathcal{W}}^{\bullet}\cap P)(F)\backslash N_{\mathcal{W}}^{\bullet}(\mathbb A)} \phi^{(s)}(u \cdot \iota(g_1,g_2))\psi_{\mathcal{W}}^{\bullet}(u) \ du \ dg_1 \ dg_2. \end{align*} $$

Proof. The proof is similar to that of [Reference Cai5]. We will give a sketch here.

When $\Re s \gg 0$ , the global integral becomes

$$ \begin{align*} \begin{aligned} Z(\xi_1\boxtimes \xi_2,\phi^{(s)}) &=\dfrac{1}{n^2}\int\limits_{[\overline{G}\times \overline{G}]} \chi_\theta(\nu(g_2))^{-1} \xi_1(g_1)\xi_2(g_2)\\ &\quad\times \int\limits_{[N_{\mathcal{W}}^{\bullet}]} \sum_{\gamma\in P(F)\backslash G^{\square,kmn_Q}(F)} \phi^{(s)}(\gamma u \cdot \iota(g_1,g_2))\psi_{\mathcal{W}}^{\bullet}(u) \ du \ dg_1 \ dg_2.\\ \end{aligned} \end{align*} $$

We can rewrite the integral as a sum over $P(F)\backslash G^{\square ,kmn_Q}(F)/\iota (G\times G)N_{\mathcal {W}}^{\bullet }(F)$ . Our goal is to show that only the double coset $P(F)\iota (G\times G)N_{\mathcal {W}}^{\bullet }(F)$ supports nonzero contribution.

We first calculate the Fourier coefficient and deduce that

(13) $$ \begin{align} \begin{aligned} & \int\limits_{[N_{\mathcal{W}}^{\bullet}]} \ \sum_{\gamma\in P(F)\backslash G^{\square,kmn_Q}(F)} \phi^{(s)}(\gamma u \cdot \iota(g_1,g_2))\psi_{\mathcal{W}}^{\bullet}(u) \ du\\ =& \int\limits_{[N_{\mathcal{W}}^{\bullet}]} \ \sum_{\gamma\in P(F)\backslash G^{\square,kmn_Q}(F)/N_{\mathcal{W}}^{\bullet}(F)} \ \sum_{\gamma'\in \gamma^{-1}P(F)\gamma\cap N(F)\backslash N_{\mathcal{W}}^{\bullet}(F)} \phi^{(s)}(\gamma \gamma' u \cdot \iota(g_1,g_2))\psi_{\mathcal{W}}^{\bullet}(u) \ du\\ =& \sum_{\gamma\in P(F)\backslash G^{\square,kmn_Q}(F)/N_{\mathcal{W}}^{\bullet}(F)} \ \int\limits_{(N_{\mathcal{W}}^{\bullet}\cap \gamma^{-1}P\gamma)(F)\backslash N_{\mathcal{W}}^{\bullet}(\mathbb A)} \phi^{(s)}(\gamma u \cdot \iota(g_1,g_2))\psi_{\mathcal{W}}^{\bullet}(u) \ du. \end{aligned} \end{align} $$

For each $\gamma \in P(F)\backslash G^{\square ,kmn_Q}(F)/N_{\mathcal {W}}^{\bullet }(F)$ and $h \in \overline {G^{\square ,kmn_Q}}(\mathbb {A})$ , we write

$$ \begin{align*} \begin{aligned} I_\gamma(h)=& \int\limits_{(N_{\mathcal{W}}^{\bullet}\cap \gamma^{-1}P\gamma)(F)\backslash N_{\mathcal{W}}^{\bullet}(\mathbb A)} \phi^{(s)}(\gamma u h)\psi_{\mathcal{W}}^{\bullet}(u) \ du,\\ J_\gamma(h)=& \int\limits_{[N_{\mathcal{W}}^{\bullet}\cap \gamma^{-1}P\gamma]} \phi^{(s)}(\gamma u h)\psi_{\mathcal{W}}^{\bullet}(u) \ du. \end{aligned} \end{align*} $$

Thus,

$$ \begin{align*} I_\gamma(h)= \int\limits_{(N_{\mathcal{W}}^{\bullet}\cap \gamma^{-1}P\gamma\backslash N_{\mathcal{W}}^{\bullet})(\mathbb A)} J_\gamma(uh)\psi_{\mathcal{W}}^{\bullet}(u) \ du. \end{align*} $$

Recall that in Section 7 we defined two subsets $\Omega _1$ and $\Omega _2$ of $G^{\square ,kmn_Q}$ . If $\gamma \in \Omega _1$ , then $J_{\gamma }(h)$ contains the following inner integral:

(14) $$ \begin{align} \int\limits_{[N_{\mathcal{W}}^{\bullet}\cap \gamma^{-1}N\gamma]} \phi^{(s)}(\gamma u h)\psi_{\mathcal{W}}^{\bullet}(u) \ du. \end{align} $$

Note that $\phi ^{(s)}$ is left invariant under $[N]$ . Thus, as a function on $[N_{\mathcal {W}}^{\bullet }\cap \gamma ^{-1}N\gamma ]$ , $\phi ^{(s)}(\gamma u h)$ is a constant function. According to the definition of $\Omega _1$ , the restriction of $\psi _{\mathcal {W}}^{\bullet }(u)$ to $[N_{\mathcal {W}}^{\bullet }\cap \gamma ^{-1}N\gamma ]$ is a nontrivial character. Therefore, (14) vanishes and so do $J_{\gamma }(h)$ and $I_{\gamma }(h)$ .

If $\gamma \in \Omega _2$ , then (14) is constant as a function on $[N_{\mathcal {W}}^{\bullet }\cap \gamma ^{-1}N\gamma ]$ and $J_{\gamma }(h)$ becomes a Fourier coefficient of $\theta $ which is given by a pair that lies in an orbit higher than $((kn_Q)^{m})_D$ . We now have $J_{\gamma }(h)=0$ since $\theta $ is a representation of type $(kn_Q,m)_D$ . Thus, $I_{\gamma }(h)=0$ as well.

Therefore, we have shown that (13) equals

$$ \begin{align*} \sum_{\gamma\in P(F)\backslash \tilde G^{\square,kmn_Q}(F)/N_{\mathcal{W}}^{\bullet}(F)} \ I_{\gamma}(\iota(g_1,g_2)) \end{align*} $$

and, therefore,

$$ \begin{align*} &Z(\xi_1\boxtimes \xi_2,\phi^{(s)})\\ &\quad=\dfrac{1}{n^2}\int\limits_{[\overline{G}\times \overline{G}]} \chi_\theta(\nu(g_2))^{-1} \xi_1(g_1)\xi_2(g_2) \ \sum_{\gamma\in P(F)\backslash \tilde G^{\square,kmn_Q}(F)/N_{\mathcal{W}}^{\bullet}(F)} \ I_{\gamma}(\iota(g_1,g_2)) \ dg_1 \ dg_2. \end{align*} $$

We now break the sum and exchange it with integration again. This shows that the above equation equals

$$ \begin{align*} \dfrac{1}{n^2} \sum_{\gamma \in P(F)\backslash \tilde G^{\square,kmn_Q}(F)/\iota(G\times G)N_{\mathcal{W}}^{\bullet}(F)} K_{\gamma}, \end{align*} $$

where

$$ \begin{align*} K_{\gamma}=\int\limits_{\iota^{-1}(\gamma P\gamma^{-1}\cap \iota(G\times G))(F)\backslash (\overline{G}\times \overline{G})(\mathbb A)} \chi_\theta(\nu(g_1))^{-1}\xi_1(g_1)\xi_2(g_2) I_{\gamma}(\iota(g_1,g_2)) \ dg_1 \ dg_2. \end{align*} $$

We now apply results in Proposition 7.2. The double coset $P(F)\backslash \tilde G^{\square ,kmn_Q}(F)/\iota (G\times G)N_{\mathcal {W}}^{\bullet }(F)$ is finite and the only open coset is $P(F)\iota (G\times G)N_{\mathcal {W}}^{\bullet }(F)$ . For negligible double cosets, by Propositions 7.2 and 6.8, we know

$$ \begin{align*} J_{\gamma}(\iota(1,g_2)h)=J_{\gamma}(h) \end{align*} $$

for $g_2 \in N_-(\mathbb A)$ and $h\in \overline {G^{\square ,kmn_Q}}(\mathbb A)$ . The proof as in [Reference Cai5] Proposition 6.7 shows that

$$ \begin{align*} I_{\gamma}(\iota(1,g_2)h)=I_{\gamma}(h) \end{align*} $$

for $g_2 \in N_-(\mathbb A)$ and $h\in \overline {G^{\square ,kmn_Q}}(\mathbb A)$ . Here we need to use Lemma 5.2.

We conclude that for a negligible double coset, the contribution $K_{\gamma }$ contains the inner integral

$$ \begin{align*} \int\limits_{[N_-]}\xi_2(ug_2) \ du. \end{align*} $$

This is zero since $\pi ^\vee $ is cuspidal. Thus, only the main orbit has nonzero contribution. We have arrived at

$$ \begin{align*} Z(\xi_1\boxtimes \xi_2,\phi^{(s)})= \dfrac{1}{n^2}\int\limits_{G^\diamondsuit(F)\backslash (\overline{G}\times \overline{G})(\mathbb A)}\chi_\theta(\nu(g_2))^{-1} \xi_1(g_1)\xi_2(g_2) I_1(\iota(g_1,g_2))\ d g_1 \ d g_2. \end{align*} $$

This proves the result.

9.6. Euler product

We can indeed rewrite $Z(\xi _1\boxtimes \xi _2,\phi ^{(s)})$ as an Euler product using uniqueness of $(kn_Q,m)_D$ -models for $\theta $ .

Let $\mathbf {N}_{\mathcal {W}}^{\circ }=\mathbf {N}_{\mathcal {W}}^{\bullet }\cap \mathbf {N}(W^{\nabla ,kn_Q})$ . Then for any $h\in \overline {G^{\square ,kmn_Q}}(\mathbb A)$ ,

$$ \begin{align*} \begin{aligned} &\int\limits_{(N_{\mathcal{W}}^{\bullet}\cap P)(F)\backslash N_{\mathcal{W}}^{\bullet}(\mathbb A)} \phi^{(s)}( u \cdot \iota(g_1,g_2)h)\psi_{\mathcal{W}}^{\bullet}(u) \ du\\ &=\int\limits_{N_{\mathcal{W}}^\circ(\mathbb A)} \ \int\limits_{[N_{\mathcal{W}}^{\bullet}\cap P]} \phi^{(s)}( u u' \cdot \iota(g_1,g_2)h)\psi_{\mathcal{W}}^{\bullet}( uu') \ du \ du'\\ &=\int\limits_{N_{\mathcal{W}}^\circ(\mathbb A)} f^{(s)}( u' \cdot \iota(g_1,g_2)h)\psi_{\mathcal{W}}^{\bullet}( u') \ du'\\ &=\int\limits_{N_{\mathcal{W}}^\circ(\mathbb A)} f^{(s)}(\iota(g_2,g_2)\cdot u' \cdot \iota(g_2^{-1}g_1,1)h)\psi_{\mathcal{W}}^{\bullet}( u') \ du'\\ &=\chi_{\theta}(\nu(g_2))\int\limits_{N_{\mathcal{W}}^\circ(\mathbb A)} f^{(s)}(u \cdot \iota(g_2^{-1}g_1,1)h)\psi_{\mathcal{W}}^{\bullet}(u) \ du.\\ \end{aligned} \end{align*} $$

Observe that we use change of variable and Lemma 5.2 in the third equality and Lemma 9.2 in the last equality. We also use the fact that the canonical lift of $N_{\mathcal {W}}^{\bullet }(\mathbb A)$ is $G^{\square ,kmn_Q}(\mathbb A)$ -equivariant. As a consequence, we can write $Z(\xi _1\boxtimes \xi _2,\phi ^{(s)})$ as

$$ \begin{align*} \begin{aligned} &\dfrac{1}{n^2}\int\limits_{G^{\diamondsuit}(F)\backslash (\overline{G}\times \overline{G})(\mathbb A)}\xi_1(g_1)\xi_2(g_2)\int\limits_{N_{\mathcal{W}}^\circ(\mathbb A)} f^{(s)}(u \cdot \iota(g_2^{-1}g_1,1))\psi_{\mathcal{W}}^{\bullet}( u) \ du \ dg_2 \ dg_1\\ &=\dfrac{1}{n^2}\int\limits_{G^{\diamondsuit}(F)\backslash (\overline{G}\times \overline{G})(\mathbb A)}\xi_1(g_2g_1)\xi_2(g_2)\int\limits_{N_{\mathcal{W}}^\circ(\mathbb A)} f^{(s)}(u \cdot \iota(g_1,1))\psi_{\mathcal{W}}^{\bullet}( u) \ du\ dg_2 \ dg_1\\ &=\dfrac{1}{n^2}\int\limits_{\overline{G}(\mathbb A)} \ \int\limits_{[\overline{G}]}\xi_1(g_2g_1)\xi_2(g_2)\int\limits_{N_{\mathcal{W}}^\circ(\mathbb A)} f^{(s)}(u \cdot \iota(g_1,1))\psi_{\mathcal{W}}^{\bullet}( u) \ du\ dg_2 \ dg_1\\ &=\dfrac{1}{n}\int\limits_{\overline{G}(\mathbb A)}\mathcal{P}(\pi(g)\xi_1\boxtimes \xi_2) \int\limits_{N_{\mathcal{W}}^\circ(\mathbb A)} f^{(s)}(u \cdot \iota(g,1))\psi_{\mathcal{W}}^{\bullet}( u) \ du \ d g.\\ \end{aligned} \end{align*} $$

For decomposable data, it follows from uniqueness of $(kn_Q,m)_D$ -models for $\theta _v$ that

$$ \begin{align*} f^{(s)}(g)=\prod_v f_v^{(s)}(g_v). \end{align*} $$

If, furthermore, $\xi _i=\otimes _v \xi _{i,v}$ , then

$$ \begin{align*} Z(\xi_1\boxtimes\xi_2,\phi^{(s)})=\prod_v Z_v(\xi_{1,v}\boxtimes \xi_{2,v},f_v^{(s)}), \end{align*} $$

where

$$ \begin{align*} Z_v(\xi_{1,v}\boxtimes \xi_{2,v},f_v^{(s)}) = \dfrac{1}{n} \int\limits_{\overline{G}_v}\mathcal{P}_v(\pi(g)\xi_{1,v}\boxtimes \xi_{2,v})\int\limits_{N_{\mathcal{W}}^{\circ}}f_v^{(s)}(u\cdot \iota(g,1))\psi_{\mathcal{W}}^{\bullet}(u) \ du \ dg. \end{align*} $$

10.2. Strategy of the proofs

In this section, we explain the strategy of the proof of Proposition 10.4 and set up some notation that is commonly used.

Let $\mathbf {T}$ be a maximal F-torus of $\mathbf {G}$ . Then $\iota (\mathbf {T}\times \mathbf {T})\subset \iota (\mathbf {G}\times \mathbf {G})$ is an F-torus of $\mathbf {G}^{\square }$ . Let $\mathbf {T}^{\square }\supset \iota (\mathbf {T}\times \mathbf {T})$ be a maximal F-torus of $\mathbf {G}^{\square }$ . Note that $\mathbf {T}^{\square }=\iota (\mathbf {T}\times \mathbf {T})$ except in the case of odd $\mathbf {SO}(W)$ . Observe that this is not a maximal F-torus in $\mathbf {P}(W^\Delta )$ but only up to a conjugation over $F_s$ . We still use this nonsplit torus even if $\mathbf {G}^{\square }$ might be a split group over F.

The torus $\mathbf {T}$ splits over $F_s$ . Let Y be the cocharacter lattice of $\mathbf {T}$ over $F_s$ . Let $Y^{\square }$ be the cocharacter lattice of $\mathbf {T}^{\square }$ over $F_s$ . Then $Y^{\square }\supset Y\oplus Y$ . We first fix a Chevalley system of pinning for $(\mathbf {G}_{F_s}, \mathbf {T}_{F_s})$ and then choose one for $(\mathbf {G}^{\square }_{F_s}, \mathbf {T}^{\square }_{F_s})$ which is compatible with $\iota :\mathbf {G} \times \mathbf {G} \to \mathbf {G}^{\square }$ .

Notation: the BD data for $\overline {\mathbf {G}}$ are denoted $(Q,\mathcal {E},f)$ ; to distinguish the BD data for the two copies of $\overline {\mathbf {G}}$ , we use $(Q_+,\mathcal {E}_+,f_+)$ and $(Q_-, \mathcal {E}_-, f_-)$ when needed. The BD data for $\overline {\mathbf {G}}^{\square }$ are denoted $(Q^{\square },\mathcal {E}^{\square },f^{\square })$ .

Definition 10.8. We define the subcategory

$$ \begin{align*} \mathbf{BD}(\mathbf{G}, \mathbf{T})^{\Delta}\subset \mathbf{BD}(\mathbf{G},\mathbf{T})\times \mathbf{BD}(\mathbf{G},\mathbf{T}) \end{align*} $$

as follows: the pair of triples $(Q,\mathcal {E},f)$ and $(Q',\mathcal {E}',f')$ is in the subcategory if and only if $(Q,\mathcal {E},f)$ is isomorphic to $(Q',\mathcal {E}',f')$ .

It is easy to see that Proposition 10.4 is equivalent to the following result.

Proposition 10.9. The functor induced by pulling back via $\mathbf {G}\times \mathbf {G} \to \mathbf {G}^{\square }$

$$ \begin{align*} \mathbf{BD}(\mathbf{G}^{\square},\mathbf{T}^{\square}) \to \mathbf{BD}(\mathbf{G},\mathbf{T}) \times \mathbf{BD}(\mathbf{G},\mathbf{T}) \end{align*} $$

is essentially surjective on $\mathbf {BD}(\mathbf {G},\mathbf {T})^{\Delta }$ .

We will prove this result case by case.

10.3. A useful lemma

Given an exact sequence

$$ \begin{align*} 1\to F_s^\times \to \mathcal{E} \to Y \to 1, \end{align*} $$

we can push out the direct sum of two copies via the product map $\mathrm {pr}:F_s^\times \times F_s^\times \to F_s^\times $ to obtain

$$ \begin{align*} 1\to F_s^\times \to \mathrm{pr}_\ast(\mathcal{E} \oplus \mathcal{E}) \to Y\oplus Y \to 1. \end{align*} $$

We now give a useful criterion to compare elements in $\mathrm {pr}_{\ast }(\mathcal {E}\oplus \mathcal {E})$ . Recall that

$$ \begin{align*} \mathrm{pr}_\ast(\mathcal{E} \oplus \mathcal{E}) = F_s^\times \times (\mathcal{E} \oplus \mathcal{E}) /\langle (x_1x_2, x_1^{-1},x_2^{-1}): (x_1, x_2) \in F_s^\times \times F_s^\times \rangle. \end{align*} $$

Define

$$ \begin{align*} \mathrm{mul}:F_s^\times \times (\mathcal{E}\oplus \mathcal{E}) \to \mathcal{E}, \qquad (x,e_1,e_2)\mapsto xe_1e_2. \end{align*} $$

This gives a well-defined map $\mathrm {mul}:\mathrm {pr}_{\ast }(\mathcal {E}\oplus \mathcal {E}) \to \mathcal {E}$ .

10.4. Symplectic groups

We now discuss the case of symplectic groups or their inner forms. This is probably the easiest case since $\mathbf {G}$ is simply connected. A multiplicative $\mathbf {K}_2$ -torsor $\overline {\mathbf {G}}$ on $\mathbf {G}$ is determined by a Galois invariant Weyl group invariant quadratic form Q on Y. In [Reference Weissman27] Proposition 3.15, it is shown that for every integer a, there is a unique such quadratic form on Y such that its value on a short coroot is a. In other words, we have an equivalence of categories

$$ \begin{align*} \mathbf{CExt}(\mathbf{G},\mathbf{K}_2)\to \mathbf{BD}(\mathbf{G},\mathbf{T})\to \mathbb{Z}, \end{align*} $$

where the last functor sends a quadratic form to its value on a short coroot.

The torus $\mathbf {T}\times \mathbf {T}$ is a maximal F-torus of $\mathbf {G}^{\square }$ , so $\mathbf {T}^{\square }=\mathbf {T}\times \mathbf {T}$ . The cocharacter lattice of $\mathbf {T}^{\square }$ over $F_s$ is $Y^{\square }=Y\oplus Y$ . The functor

$$ \begin{align*} \mathbf{CExt}(\mathbf{G}^{\square},\mathbf{K}_2)\to \mathbf{CExt}(\mathbf{G},\mathbf{K}_2)\times \mathbf{CExt}(\mathbf{G},\mathbf{K}_2) \end{align*} $$

can be described in terms of

$$ \begin{align*} \mathbf{BD}(\mathbf{G}^{\square},\mathbf{T}^{\square}) \to \mathbf{BD}(\mathbf{G},\mathbf{T}) \times \mathbf{BD}(\mathbf{G},\mathbf{T}). \end{align*} $$

In terms of $\mathbb {Z}\to \mathbb {Z}\times \mathbb {Z}$ , it is simply $a\mapsto (a,a)$ . Proposition 10.9 follows trivially.

10.5. Special even orthogonal groups and inner forms

We now consider the case of special even orthogonal groups. We start with some basic results.

10.5.1. Preparation

Let $n=(d\cdot \dim _D W)/2$ . (The results in this section only involve multiplicative $\mathbf {K}_2$ -torsors. This n has no relation with the degree of the cover. It will not cause any confusion.) We choose a standard basis of $Y=\mathbb {Z}^n=\mathrm {Span}\{e_1^{\vee },\cdots , e_n^{\vee }\}$ so that the root lattice is given by

$$ \begin{align*} Y^{sc}=\mathrm{Span}\{\alpha_1^\vee,\cdots,\alpha_n^\vee\} =\mathrm{Span}\{e_1^{\vee}-e_2^{\vee},\cdots, e_{n-1}^{\vee}-e_n^{\vee},e_{n-1}^{\vee}+e_n^{\vee}\}. \end{align*} $$

Note that $Y^{sc}$ is a sublattice of Y with index $2$ . A W-invariant quadratic form Q is determined by its value on a coroot. Let $Q(\alpha _1^\vee )=a$ .

Lemma 10.11. We have $2\mid a$ .

Proof. We know that $Q(\alpha _{n-1}^\vee )=Q(\alpha _n^\vee )=a$ . Let $Q(e_n^{\vee })=b\in \mathbb {Z}$ . As $Q(2e_{n}^{\vee })=Q(\alpha _{n-1}^\vee )+Q(\alpha _n^\vee )$ , we have $4b=2a$ . This implies $2\mid a$ .

The torus $\mathbf {T}^{\square }=\mathbf {T} \times \mathbf {T}$ is a maximal F-torus of $\mathbf {G}^{\square }$ . Its cocharacter lattice over $F_s$ is $Y^{\square }=Y_+\oplus Y_-$ . We write

$$ \begin{align*} \begin{aligned} Y_+^{sc}&=\mathrm{Span}\{e_1^{\vee}-e_2^{\vee}, \cdots, e_{n-1}^{\vee}-e_n^{\vee}, e_{n-1}^{\vee}+e_n^{\vee}\}\\ Y_-^{sc}&=\mathrm{Span}\{e_{n+1}^{\vee}-e_{n+2}^{\vee}, \cdots, e_{2n-1}^{\vee}-e_{2n}^{\vee}, e_{2n-1}^{\vee}+e_{2n}^{\vee}\}\\ Y^{\square,sc}&=\mathrm{Span}\{e_1^{\vee}-e_2^{\vee},\cdots, e_{2n-1}^{\vee}-e_{2n}^{\vee},e_{2n-1}^{\vee}+e_{2n}^{\vee}\}. \end{aligned} \end{align*} $$

Thus, $Y_+^{sc}\oplus Y_-^{sc}$ is a subgroup of $Y^{\square ,sc}$ of index $2$ and $e_1^{\vee }-e_{n+1}^{\vee }\notin Y_+^{sc}\oplus Y_-^{sc}$ .

Let $Q^{\square }$ be a W-invariant quadratic form on $Y^{\square }$ .

Lemma 10.12. The restriction of $Q^{\square }$ to $Y_+\oplus Y_-$ is a direct sum of two quadratic forms $Q_+\oplus Q_-$ with $Q_+=Q_-$ . In particular, we have

$$ \begin{align*} B_{Q^{\square}}((y_+,0),(0,y_-))=0 \end{align*} $$

for $y_+\in Y_+$ and $y_-\in Y_-$ .

Conversely, for a W-invariant quadratic form Q on Y, there is a unique W-invariant quadratic form $Q^{\square }$ which restricts to $Q_+\oplus Q_-$ on $Y_+\oplus Y_-$ .

Proof. The proof is straightforward.