2.1 The groups

Let $\operatorname {\mathrm {Sp}}_{2l}$ denote the symplectic group, realized as the subgroup of matrices $g\in \operatorname {\mathrm {SL}}_{2l}$ such that ${}^tg\left (\begin {smallmatrix}&J_l\\-J_l\end {smallmatrix}\right )g=\left (\begin {smallmatrix}&J_l\\-J_l\end {smallmatrix}\right )$ , where ${}^tg$ is the transpose of g and $J_l$ is the $l\times l$ permutation matrix with $1$ along the antidiagonal. We fix the diagonal torus $T_l$ and the Borel subgroup $B_l=T_l\ltimes N_l$ of upper triangular matrices in $\operatorname {\mathrm {Sp}}_{2l}$ . For a parabolic subgroup P of $\operatorname {\mathrm {Sp}}_{2l}$ , $\delta _P$ denotes the modulus character. If U is a subgroup of $N_l$ , $U^-$ denotes the opposite unipotent subgroup. Denote the Weyl group of $T_l$ in $\operatorname {\mathrm {Sp}}_{2l}$ by $W_{\operatorname {\mathrm {Sp}}_{2l}}$ .

In the group $\operatorname {\mathrm {GL}}_d$ , fix the Borel subgroup $B_{\operatorname {\mathrm {GL}}_d}=T_{\operatorname {\mathrm {GL}}_d}\ltimes N_{\operatorname {\mathrm {GL}}_d}$ of upper triangular invertible matrices, where $T_{\operatorname {\mathrm {GL}}_d}$ is the diagonal torus. A composition of a positive integer d is an ordered sequence of positive integers whose sum is d. For a composition $\beta =(\beta _1,\ldots ,\beta _l)$ of d, let $P_{\beta }=M_{\beta }\ltimes V_{\beta }$ denote the standard parabolic subgroup with $M_{\beta }=\operatorname {\mathrm {GL}}_{\beta _1}\times \ldots \times \operatorname {\mathrm {GL}}_{\beta _l}$ , and $V_{\beta }^-$ be the opposite unipotent subgroup. A block-diagonal matrix in $\operatorname {\mathrm {GL}}_d$ will be denoted $\operatorname {\mathrm {diag}}(a_1,\ldots ,a_l)$ ( $a_i\in \operatorname {\mathrm {GL}}_{\beta _i}$ ).

Denote the set of roots of $\operatorname {\mathrm {GL}}_d$ by $\Phi _d$ , each $\alpha \in \Phi _d$ corresponds to a pair $(i,j)$ with $1\leq i\ne j\leq d$ . Let $\Phi _d^+\subset \Phi _d$ denote the positive roots (defined with respect to $B_{\operatorname {\mathrm {GL}}_d}$ ). The simple roots are $(i,i+1)$ , $1\leq i<d$ . The Weyl group of $T_{\operatorname {\mathrm {GL}}_d}$ in $\operatorname {\mathrm {GL}}_d$ is denoted $W_{\operatorname {\mathrm {GL}}_d}$ . Let $\ell (w)$ be the length of $w\in W_{\operatorname {\mathrm {GL}}_d}$ . We write $w_{\alpha }$ for the reflection along $\alpha $ . For $b\in \operatorname {\mathrm {GL}}_d$ , denote $b^{*}=J_d{}^tb^{-1}J_d$ .

For two elements x and y in a group X, ${}^xy=xyx^{-1}$ , and if $Y<X$ , that is, Y is a subgroup of X, denote ${}^xY=\{{}^{x}y:y\in Y\}$ .

All fields in this work are of characteristic $0$ . Over a local field F, by convention if H is a linear algebraic group defined over F, we denote its F-points also by H, that is, $H=H(F)$ . When F is a number field, $\mathbb {A}$ denotes its ring of adeles and we write F-points or $\mathbb {A}$ -points explicitly, for example, $H(\mathbb {A})$ . When the discussion applies to both local and global situations, we simply write H for both cases. For an integer m, $F^{*m}=\{a^m:a\in F^{*}\}$ and similarly denote $\mathbb {A}^{*m}$ .

Over a local field, representations of H are assumed to be complex and smooth. When the field is Archimedean, an admissible representation is understood to be admissible Fréchet of moderate growth. If $\pi $ is a representation of a unipotent subgroup $U<H$ on a space V, and $\psi $ is a character of U, the Jacquet module $J_{U,\psi }(\pi )$ is the quotient $V(U,\psi )\backslash V$ , where over non-Archimedean fields $V(U,\psi )\subset V$ is the subspace spanned by all vectors of the form $\pi (u)\xi -\psi (u)\xi $ , $u\in U$ and $\xi \in V$ , and over Archimedean fields $V(U,\psi )$ is the closure of this subspace.

For a local non-Archimedean field F, we let $\mathcal {O}$ denote its ring of integers, $\mathcal {P}$ be the maximal ideal of $\mathcal {O}$ , $\varpi $ be a uniformizer and $|\varpi |=q^{-1}$ . Then we take a hyperspecial maximal compact subgroup $K_{\operatorname {\mathrm {Sp}}_{2l}}=\operatorname {\mathrm {Sp}}_{2l}(\mathcal {O})$ of $\operatorname {\mathrm {Sp}}_{2l}$ , and additionally $K_{\operatorname {\mathrm {SL}}_d}=\operatorname {\mathrm {SL}}_d(\mathcal {O})$ and $K_{\operatorname {\mathrm {GL}}_d}=\operatorname {\mathrm {GL}}_d(\mathcal {O})$ . We also denote by $K_{\operatorname {\mathrm {Sp}}_{2l}}$ a maximal compact subgroup of $\operatorname {\mathrm {Sp}}_{2l}$ over Archimedean fields (and similarly $K_{\operatorname {\mathrm {GL}}_d}$ ). In a global context, the maximal compact subgroup is $K_{\operatorname {\mathrm {Sp}}_{2l}}=\prod _{\nu }K_{\operatorname {\mathrm {Sp}}_{2l},\nu }$ , where the product varies over all places of F.

Let $\text {Mat}_{n\times l}$ be the abelian group of $n\times l$ matrices (over a field or over $\mathbb {A}$ ), and set $\text {Mat}_{n}=\text {Mat}_{n\times n}$ . The trace map is denoted $\operatorname {\mathrm {tr}}$ .

In both local and global settings, for a representation $\pi $ , say, of $\operatorname {\mathrm {Sp}}_{2l}$ , its contragredient is denoted $\pi ^{\vee }$ . Induction of representations from parabolic subgroups is always assumed to be normalized. The action of a group by right-translation is denoted $\cdot $ , for example, for a function f on $\operatorname {\mathrm {Sp}}_{2l}$ and $g,h\in \operatorname {\mathrm {Sp}}_{2l}$ , $h\cdot f(g)=f(gh)$ .

2.2 Covering groups

We introduce our notation for covering groups and include several general observations to be used throughout. Our basic reference is [Reference MooreMoo68]; see also [Reference Gan, Gao and WeissmanGGW18]. Let $m\geq 1$ be an integer. Let F be a local field or a number field. Fix the group $\mu _m\subset \mathbb {C}^{*}$ of m-th roots of unity. Assume that $F^{*}$ contains m m-th roots of unity, then we may identify the subgroup of these roots with $\mu _m$ and for simplicity write $\mu _m\subset F^{*}$ . Let $(\cdot ,\cdot )_m$ be a Hilbert symbol of order m in F (for $m=1$ it is trivially $1$ ), which is the product of local symbols if F is a number field ([Reference WeilWei95, Ch. IX.5]).

Let G be a linear algebraic group defined over F, and denote its F-points (over a local field) or $\mathbb {A}$ -points also by G. A topological (central) covering group of G by $\mu _m$ is a short exact sequence of groups

$$ \begin{align*} 1\rightarrow \mu_m\xrightarrow{i} \widetilde{G}\xrightarrow{p} G\rightarrow 1, \end{align*} $$

where i and p are continuous, $i(\mu _m)$ is closed and belongs to the center of $\widetilde {G}$ , and p induces an isomorphism $i(\mu _m)\backslash \widetilde {G}\cong G$ of topological groups. We call $\widetilde {G}$ an m-fold covering group of G. A Borel measurable $\sigma :G\times G\rightarrow \mu _m$ such that

(2.1) $$ \begin{align} \sigma(g,g')\sigma(gg',g")=\sigma(g,g'g")\sigma(g',g")\qquad\forall g,g',g"\in G, \end{align} $$

and on the identity element e of G, $\sigma (e,e)=1$ , is called a $2$ -cocycle of G. In particular, $\sigma (e,g')=\sigma (g,e)=1$ . Let $\mathrm {Z}^2(G,\mu _m)$ denote the group of $2$ -cocycles. A Borel measurable map $\eta :G\rightarrow \mu _m$ such that $\eta (e)=1$ is called a $1$ -cochain; the corresponding $2$ -coboundary is given by the function $(g,g')\mapsto \eta (g)\eta (g')/\eta (gg')$ . Let $\mathrm {C}^1(G,\mu _m)$ (resp., $\mathrm {B}^2(G,\mu _m)$ ) be the group of $1$ -cochains (resp., $2$ -coboundaries). The $2$ -nd cohomology $\mathrm {H}^2(G,\mu _m)$ is by definition $\mathrm {B}^2(G,\mu _m)\backslash \mathrm {Z}^2(G,\mu _m)$ , and its elements parameterize the topological covering groups of G by $\mu _m$ . Given $\sigma \in \mathrm {Z}^2(G,\mu _m)$ , we can realize the group $\widetilde {G}$ as the set of elements $\langle g,\epsilon \rangle $ with $g\in G$ and $\epsilon \in \mu _m$ , and the product given by

$$ \begin{align*} \langle g,\epsilon\rangle\langle g',\epsilon'\rangle=\langle gg',\epsilon\epsilon'\sigma(g,g')\rangle. \end{align*} $$

If $\sigma ,\rho \in \mathrm {Z}^2(G,\mu _m)$ are cohomologous, that is, equal in $\mathrm {H}^2(G,\mu _m)$ , there exists $\eta \in \mathrm {C}^1(G,\mu _m)$ such that

(2.2) $$ \begin{align} \rho(g,g')=\frac{\eta(g)\eta(g')}{\eta(gg')}\sigma(g,g'),\qquad\forall g,g'\in G. \end{align} $$

In this case, replacing $\sigma $ with $\rho $ yields an isomorphic group $\widetilde {G}$ : $\langle g,1\rangle \mapsto \langle g,\eta (g)\rangle $ is a topological isomorphism, where the domain is realized using $\rho $ and the image by $\sigma $ .

If X is a closed subgroup of G, the restriction of $\sigma $ to $X\times X$ will be briefly referred to as the restriction to X. This is a $2$ -cocycle of X. The resulting covering $\widetilde {X}$ of X, which depends on the embedding of X in G, is by definition realized using the restriction of $\sigma $ to X. A section of X is a continuous map $x\mapsto \langle x,\eta (x)\rangle $ where $\eta :X\rightarrow \mu _m$ satisfies $\eta (e)=1$ , and we call it a splitting of X if it is also a homomorphism, that is,

$$ \begin{align*} \langle x,\eta(x)\rangle\langle x',\eta(x')\rangle=\langle xx',\eta(xx')\rangle,\qquad\forall x,x'\in X. \end{align*} $$

In this case, we say that $\widetilde {G}$ splits over X. If $x\mapsto \langle x,\eta (x)\rangle $ and $x\mapsto \langle x,\eta '(x)\rangle $ are two splittings, $x\mapsto \eta (x)\eta '(x)^{-1}$ is a homomorphism $X\rightarrow \mu _m$ . There will be $2$ recurring cases where we can conclude $\eta =\eta '$ : first, if X is perfect (because then $\operatorname {\mathrm {Hom}}(X,\mu _m)$ is trivial) and, second, when X is the F- or $\mathbb {A}$ -points of an algebraic unipotent subgroup because such a homomorphism implies that there is a homomorphism of the local field or $\mathbb {A}$ into $\mu _m$ , hence must be trivial.

With the notation of Equation (2.2), if $\sigma $ and $\rho $ are trivial on X, that is, $\sigma (x,x')=\rho (x,x')=1$ for all $x,x'\in X$ , $\eta $ becomes a homomorphism which may then also be trivial, depending on X.

In a local-global context, $G(\mathbb {A})$ is the restricted direct product with respect to a family of compact open subgroups $K_{G,\nu }$ defined at almost all places $\nu $ . Assume that we have a family $(\sigma _{\nu },\rho _{\nu },\eta _{\nu })$ where $\sigma _{\nu },\rho _{\nu }\in \mathrm {Z}^2(G(F_{\nu }),\mu _m)$ and $\eta _{\nu }\in \mathrm {C}^1(G(F_{\nu }),\mu _m)$ , related by Equation (2.2). If we know the r.h.s. of Equation (2.2) is trivial on $K_{G,\nu }$ for almost all $\nu $ , we can define $\rho =\prod _{\nu }\rho _{\nu }\in \mathrm {Z}^2(G(\mathbb {A}),\mu _m)$ .

In the opposite direction, let $X<G$ be a closed algebraic subgroup such that $X(\mathbb {A})$ is the restricted direct product $\prod ^{\prime }_{\nu } X(F_{\nu })$ with respect to a family $\{X_{0,\nu }\}_{\nu }$ , where for almost all $\nu $ , $X_{0,\nu }=X(F_{\nu })\cap K_{G,\nu }$ and there is a unique homomorphism $X_{0,\nu }\rightarrow \mu _m$ . Assume $\sigma ,\rho \in \mathrm {Z}^2(X(\mathbb {A}),\mu _m)$ , write $\sigma =\prod _{\nu }\sigma _{\nu }$ , $\rho =\prod _{\nu }\rho _{\nu }$ , and assume that we have $\eta _{\nu }\in \mathrm {C}^1(X(F_{\nu }),\mu _m)$ as in Equation (2.2) for all $\nu $ , and both $\rho _{\nu }$ and $\sigma _{\nu }$ are trivial on $X_{0,\nu }$ for almost all $\nu $ . Then almost everywhere, $\eta _{\nu }$ becomes a homomorphism $X_{0,\nu }\rightarrow \mu _m$ , thus $\eta _{\nu }(X_{0,\nu })=1$ . Under these assumptions, we can then define $\eta =\prod _{\nu }\eta _{\nu }\in \mathrm {C}^1(X(\mathbb {A}),\mu _m)$ . Moreover, if $x\mapsto \langle x,\zeta (x)\rangle $ is a splitting of $X(\mathbb {A})$ in $\widetilde {X}(\mathbb {A})$ realized using $\sigma $ ,

(2.3) $$ \begin{align} \rho(x,x')=\frac{\eta(x)\eta(x')}{\eta(xx')}\frac{\zeta(xx')}{\zeta(x)\zeta(x')},\qquad\forall x,x'\in X(\mathbb{A}). \end{align} $$

Thus, $x\mapsto \langle x,\zeta (x)/\eta (x)\rangle $ is a splitting of $X(\mathbb {A})$ when $\widetilde {X}(\mathbb {A})$ is realized using $\rho $ .

Since $\mu _m$ belongs to the center of $\widetilde {G}$ , G acts on $\widetilde {G}$ by conjugation, which is also a homeomorphism. In fact, ${}^{\langle x,\epsilon '\rangle }\langle y,\epsilon \rangle =\langle {}^xy,\sigma (x,y)\sigma (xy,x^{-1})\sigma (x,x^{-1})^{-1}\epsilon \rangle $ (independent of $\epsilon '$ ).

For a topological automorphism $\chi :G\rightarrow G$ , a lift of $\chi $ to $G^{(m)}$ is a topological automorphism $\widetilde {\chi }:G^{(m)}\rightarrow G^{(m)}$ making the following diagram commute:

$$ \begin{align*} \begin{array}{ccccccccc} 1& \rightarrow & \mu_m & \rightarrow & G^{(m)} & \rightarrow & G & \rightarrow &1\\ & & \mathrm{id}\downarrow & & \widetilde{\chi}\downarrow & & \chi\downarrow & &\\ 1& \rightarrow & \mu_m & \rightarrow & G^{(m)} & \rightarrow & G & \rightarrow &1. \end{array} \end{align*} $$

One can disregard topological considerations and consider $\chi $ and $\widetilde {\chi }$ abstractly. In this weaker sense, $\operatorname {\mathrm {Hom}}(G,\mu _m)$ , where G is regarded as an abstract group, acts transitively on the set of lifts ([Reference WeissWei69, § V.1, Propositions 1, 4]; see also [Reference KableKab99, § 2]). Thus, if G is perfect and $\widetilde {\chi }$ (an abstract lift or in particular, a topological lift) exists, it is unique.

Let $\chi $ and $\widetilde {\chi }$ be as above (again, topological), and Y be a closed subgroup of G. Assume that $y\mapsto \langle y,\eta (y)\rangle $ is the unique splitting of Y and ${}^{\chi }y\mapsto \langle {}^{\chi }y,\eta ({}^{\chi }y)\rangle $ is a splitting of ${}^{\chi }Y$ . Then $y\mapsto {}^{(\widetilde {\chi })^{-1}}\langle {}^{\chi }y,\eta ({}^{\chi }y)\rangle $ is a splitting of Y, hence by uniqueness ${}^{(\widetilde {\chi })^{-1}}\langle {}^{\chi }y,\eta ({}^{\chi }y)\rangle =\langle y,\eta (y)\rangle $ . We deduce that in this case

(2.4) $$ \begin{align} {}^{\widetilde{\chi}}\langle y,\eta(y)\rangle=\langle {}^{\chi}y,\eta({}^{\chi}y)\rangle. \end{align} $$

When there is no risk of confusion, we will denote $\widetilde {\chi }$ also by $\chi $ .

For a parabolic subgroup $P<G$ , induction from $\widetilde {P}$ to $\widetilde {G}$ is implicitly normalized by $\delta _P^{1/2}$ .

Given a faithful character $\varepsilon :\mu _m\rightarrow \mathbb {C}^{*}$ , an $\varepsilon $ -genuine representation $\pi $ of $\widetilde {G}$ is a representation where $\mu _m$ acts by $\varepsilon $ . We will usually assume that this character is fixed and omit it from the definition and notation. An $\varepsilon ^{-1}$ -genuine representation will then be called antigenuine; for example, since $\pi $ is genuine, $\pi ^{\vee }$ is antigenuine. See [Reference Kaplan and SzpruchKS23] for the extension of several fundamental results from the theory of representations of groups G over non-Archimedean fields, to their m-fold covering groups $\widetilde {G}$ , when G is a connected reductive linear algebraic group. For example, any genuine irreducible representation of $\widetilde {G}$ is admissible ([Reference Kaplan and SzpruchKS23, Theorem 3.3]).

For $G=\operatorname {\mathrm {Sp}}_{2l}$ let $G^{(m)}$ denote the m-fold covering group $\widetilde {G}$ of G, over a local field or over $\mathbb {A}$ , defined by [Reference MatsumotoMat69] (following [Reference MooreMoo68, Reference SteinbergSte68]) with the Steinberg symbol constructed from $(\cdot ,\cdot )_m^{-1}$ . It is a locally compact group, and an l-group ([Reference Bernstein and ZelevinskyBZ76] 1.1) over local non-Archimedean fields. At almost all places $\nu $ of a global field F, the covering $G^{(m)}$ of $G(F_{\nu })$ splits over $K_{G,\nu }$ ([Reference MooreMoo68, Lemma 11.3] and the proof of [Reference MooreMoo68, Theorem 12.2]), uniquely because $K_{G,\nu }$ is perfect ([Reference MooreMoo68, Lemma 11.1]). Since $G^{(m)}$ is split canonically over a maximal unipotent subgroup (for more general statements, see [Reference SteinbergSte62, Reference MatsumotoMat69, Reference Banks, Levy and SepanskiBLS99, Reference McNamaraMcN12] and [Reference Mœglin and WaldspurgerMW95, Appendix I]), notions involving unipotent orbits transfer immediately to covering groups.

Similarly, denote by $\operatorname {\mathrm {SL}}_{l}^{(m)}$ the m-fold covering group $\widetilde {\operatorname {\mathrm {SL}}}_{l}$ of $\operatorname {\mathrm {SL}}_l$ , defined using $(\cdot ,\cdot )_m^{-1}$ .

We point out that one can redefine, once and for all, $(\cdot ,\cdot )_m$ to be $(\cdot ,\cdot )_m^l$ for any $l\in \mathbb {Z}$ coprime to m, our arguments are independent of this choice (e.g., $G^{(m)}$ can be constructed from $(\cdot ,\cdot )_m^l$ ).

Henceforth throughout most of this work, G will be the symplectic group.

2.3 Embedding $G\times G$ in H

Let n, k and m be positive integers, and set $c=2n$ . If m is even, put $r=m/2$ , otherwise $r=m$ . Let $G=\operatorname {\mathrm {Sp}}_{c}$ and $H=\operatorname {\mathrm {Sp}}_{2rkc}$ . Denote by $P=M_P\ltimes U_P$ the standard maximal parabolic subgroup of H with $M_P=\operatorname {\mathrm {GL}}_{rkc}$ , that is, the Siegel parabolic subgroup. Let $Q=M_Q\ltimes U_Q$ be the standard parabolic subgroup of H, whose Levi part $M_Q$ is isomorphic to $\operatorname {\mathrm {GL}}_{c}\times \ldots \operatorname {\mathrm {GL}}_{c}\times \operatorname {\mathrm {Sp}}_{2c}$ , where $\operatorname {\mathrm {GL}}_{c}$ appears $rk-1$ times. Put $U=U_Q$ .

For a fixed nontrivial additive character $\psi $ , which is a character of a local field F, or a character of $F\backslash \mathbb {A}$ when F is a number field, define a character of U as follows. In a local context, it is a character of $U(F)$ , in a global context it is a character of $U(\mathbb {A})$ which is trivial on $U(F)$ . For $v\in V_{(c^{rk-1})}$ , write $v=(v_{i,j})_{i,j}$ , where $v_{i,j}\in \text {Mat}_c$ , then define

(2.5) $$ \begin{align} \psi(v)=\psi(\sum_{i=1}^{rk-2}\operatorname{\mathrm{tr}}(v_{i,i+1})). \end{align} $$

For $x\in \text {Mat}_{((rk-1)c)\times 2c}$ , if $rk>1$ , write the bottom $c\times 2c$ block in the form

$$ \begin{align*} \left(\begin{smallmatrix} Y_1&Z_1&Y_2\\ Y_3&Z_2&Y_4 \end{smallmatrix}\right),\qquad Y_i\in \text{Mat}_{n},Z_j\in\text{Mat}_{n\times c}. \end{align*} $$

Then define

(2.6) $$ \begin{align} \psi_U(\left(\begin{smallmatrix}v&x&y\\&I_{2c}&x'\\&&v^{*}\end{smallmatrix}\right))=\psi(v)\psi(\operatorname{\mathrm{tr}}(Y_1)+\operatorname{\mathrm{tr}}(Y_4)), \qquad v\in V_{(c^{rk-1})}. \end{align} $$

The stabilizer of $\psi _U$ in $M_Q$ contains $G\times G$ . The embedding $G\times G\hookrightarrow H$ is defined by

$$ \begin{align*} (g_1,g_2)\mapsto\operatorname{\mathrm{diag}}(g_1,\ldots,g_1,\left(\begin{smallmatrix} g_{1,1}&&g_{1,2}\\ &g_2&\\ g_{1,3}&&g_{1,4}\end{smallmatrix}\right),g_1^{*},\ldots,g_1^{*}),\qquad g_1=\left(\begin{smallmatrix} g_{1,1} & g_{1,2} \\ g_{1,3} & g_{1,4} \end{smallmatrix}\right),\qquad g_{1,j}\in\text{Mat}_n. \end{align*} $$

Here, on the r.h.s. $g_1^{*}$ appears $rk-1$ times. Oftentimes, it will be convenient to refer to each of the copies separately. We write for $g\in G$ , $\mathfrak {e}_1(g)=(g,I_c)$ and $\mathfrak {e}_2(g)=(I_c,g)$ . We call $\mathfrak {e}_1(G)$ the left copy of G in H, and $\mathfrak {e}_2(G)$ is the right copy.

2.4 Local covering

We proceed with the notation of § 2.2 and § 2.3. Consider a local field F. Let $H^{(m)}$ denote the m-fold covering of $H=H(F)$ of [Reference MatsumotoMat69] defined with the Steinberg symbol constructed from $(\cdot ,\cdot )_m^{-1}$ (see also [Reference SavinSav04, p. 114]). Since H is a subgroup of $\operatorname {\mathrm {SL}}_{2rkc}$ , it is convenient to use the $2$ -cocycle $\sigma _{2rkc}$ of $\operatorname {\mathrm {GL}}_{2rkc}$ of Banks et al. [Reference Banks, Levy and SepanskiBLS99, § 3] for local computations.

First, we recall several properties of this $2$ -cocycle, used throughout. For any integer $d>1$ , let $\sigma _{\operatorname {\mathrm {SL}}_{d+1}}\in \mathrm {Z}^2(\operatorname {\mathrm {SL}}_{d+1},\mu _m)$ be the $2$ -cocycle of [Reference Banks, Levy and SepanskiBLS99, § 2] which represents $\operatorname {\mathrm {SL}}_{d+1}^{(m)}$ in $\mathrm {H}^2(\operatorname {\mathrm {SL}}_{d+1},\mu _m)$ (see § 2.2). Let $\sigma _d$ be the $2$ -cocycle of $\operatorname {\mathrm {GL}}_d$ defined in [Reference Banks, Levy and SepanskiBLS99, § 3] for $b,b'\in \operatorname {\mathrm {GL}}_d$ by

$$ \begin{align*} \sigma_d(b,b')=(\det b,\det b')_m\sigma_{\operatorname{\mathrm{SL}}_{d+1}}(\operatorname{\mathrm{diag}}(b,\det b^{-1}),\operatorname{\mathrm{diag}}(b',\det {b'}^{-1})). \end{align*} $$

The image of $\operatorname {\mathrm {SL}}_d$ in $\operatorname {\mathrm {SL}}_{d+1}$ under the embedding $b\mapsto \operatorname {\mathrm {diag}}(b,1)$ is standard in the sense of [Reference Banks, Levy and SepanskiBLS99, § 2], hence by [Reference Banks, Levy and SepanskiBLS99, § 2, Theorem 7] the restriction of $\sigma _d$ to $\operatorname {\mathrm {SL}}_d$ is $\sigma _{\operatorname {\mathrm {SL}}_{d}}$ . By [Reference Banks, Levy and SepanskiBLS99, § 3, Lemma 1], for $t=\operatorname {\mathrm {diag}}(t_1,\ldots ,t_d)\in T_{\operatorname {\mathrm {GL}}_d}$ and $t'\in T_{\operatorname {\mathrm {GL}}_d}$ (with similar notation),

(2.7) $$ \begin{align} \sigma_d(t,t')=\prod_{i<j}(t_i,t^{\prime}_j)_m. \end{align} $$

According to [Reference Banks, Levy and SepanskiBLS99, § 3, Lemma 4], $\sigma _{d}$ is trivial on $N_{\operatorname {\mathrm {GL}}_d}$ (i.e., on $N_{\operatorname {\mathrm {GL}}_d}\times N_{\operatorname {\mathrm {GL}}_d}$ , see § 2.2), and for all $b,b'\in \operatorname {\mathrm {GL}}_{d}$ and $v,v'\in N_{\operatorname {\mathrm {GL}}_d}$ ,

(2.8) $$ \begin{align} &\sigma_{d}(b,v')=\sigma_{d}(v,b')=1, \end{align} $$

(2.9) $$ \begin{align} &\sigma_{d}(vb,b'v')=\sigma_{d}(b,b'). \end{align} $$

It follows that if ${}^bv\in N_{\operatorname {\mathrm {GL}}_d}$ ,

(2.10) $$ \begin{align} {}^b\langle v,1\rangle=\langle {}^bv,1\rangle. \end{align} $$

Also, if $u^-\mapsto \langle u^-,\varsigma (u^-)\rangle $ is the splitting of $N_{\operatorname {\mathrm {GL}}_d}^-$ and for $u^-\in N_{\operatorname {\mathrm {GL}}_d}^-$ we have ${}^bu^-\in N_{\operatorname {\mathrm {GL}}_d}$ ,

(2.11) $$ \begin{align} {}^b\langle u^-,\varsigma(u^-)\rangle=\langle {}^bu^-,1\rangle. \end{align} $$

This follows from Equation (2.4) with $Y=N_{\operatorname {\mathrm {GL}}_d}$ and $\chi ={}^b(\cdot )$ .

One of the key properties of $\sigma _d$ is the block-compatibility formula [Reference Banks, Levy and SepanskiBLS99, Theorem 11]: For $0<l<d$ , $a,a'\in \operatorname {\mathrm {GL}}_l$ and $b,b'\in \operatorname {\mathrm {GL}}_{d-l}$ ,

(2.12) $$ \begin{align} \sigma_{d}(\operatorname{\mathrm{diag}}(a,b),\operatorname{\mathrm{diag}}(a',b'))=(\det a,\det b')_m\sigma_l(a,a')\sigma_{d-l}(b,b'). \end{align} $$

We also mention that $\sigma _1$ is trivial and $\sigma _2$ coincides with the $2$ -cocycle of Kubota [Reference KubotaKub67]:

(2.13) $$ \begin{align} \sigma_2(g,g')=\Big(\frac{\gamma(gg')}{\gamma(g)},\frac{\gamma(gg')}{\gamma(g')\det g}\Big)_m, \qquad\gamma\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)=\begin{cases}c&c\ne0,\\d&c=0.\end{cases} \end{align} $$

In addition, there is a well-defined action of $W_{\operatorname {\mathrm {GL}}_d}$ on the genuine smooth admissible representations of $\widetilde {T}_{\operatorname {\mathrm {GL}}_d}$ (e.g., [Reference McNamaraMcN12, § 13.6]).

To compute conjugations of torus elements by Weyl elements, we will repeatedly use the set $\mathfrak {W}_d\subset \operatorname {\mathrm {GL}}_d$ as in [Reference Banks, Levy and SepanskiBLS99]. For any $\alpha \in \Phi _d$ , $w_{\alpha }$ was defined to be the reflection along $\alpha $ . For the simple roots $\alpha =(i,i+1)$ , we fix concrete representatives $w_{\alpha }=\operatorname {\mathrm {diag}}(I_{i-1},\left (\begin {smallmatrix}&-1\\1\end {smallmatrix}\right ),I_{d-i-1})$ . Any $w\in W_{\operatorname {\mathrm {GL}}_d}$ can be written as a product $w_{\alpha _1}\cdot \ldots \cdot w_{\alpha _{\ell (w)}}$ , where each $w_{\alpha _i}$ is a simple reflection. Define the set $\mathfrak {W}_d=\{w_{\alpha _1}\cdot \ldots \cdot w_{\alpha _{\ell (w)}}:w\in W_{\operatorname {\mathrm {GL}}_d}\}\subset \operatorname {\mathrm {SL}}_d$ ; it is not a group. Any permutation matrix $w\in \operatorname {\mathrm {GL}}_d$ can be written uniquely in the form $w=t_0w'$ , where $t_0\in T_{\operatorname {\mathrm {GL}}_d}$ and its coordinates are $\pm 1$ , and $w'\in \mathfrak {W}_d$ . For any $t\in T_{\operatorname {\mathrm {GL}}_d}$ , by [Reference Banks, Levy and SepanskiBLS99, § 3, Theorem 7(b), (d)],

(2.14) $$ \begin{align} &\sigma_d(t,w')=1, \end{align} $$

(2.15) $$ \begin{align} &\sigma_d(w',t)=\prod_{(i,j)=\alpha\in\Phi_d^+:w\alpha<0}(-t_j,t_i)_m. \end{align} $$

Using Equation (2.1) (with $g={}^{w'}t$ , $g'=w'$ , $g"={w'}^{-1}$ ) and because $\sigma ({}^{w'}t,w')=1$ ,

(2.16) $$ \begin{align} \sigma_d({}^{w'}tw',{w'}^{-1})= \sigma_d({}^{w'}t,w'{w'}^{-1})\sigma_d(w',{w'}^{-1})=\sigma_d(w',{w'}^{-1}). \end{align} $$

Hence, combining Equation (2.16) with Equation (2.15),

(2.17) $$ \begin{align} {}^{w'}\langle t,1\rangle=\langle {}^{w'}t,\sigma_d(w',t)\sigma_d(w't,{w'}^{-1})\sigma_d(w',{w'}^{-1})^{-1}\rangle= \langle {}^{w'}t,\prod_{(i,j)=\alpha\in\Phi_d^+:w\alpha<0}(-t_j,t_i)_m\rangle. \end{align} $$

Then ${}^{w}\langle t,1\rangle ={}^{t_0}({}^{w'}\langle t,1\rangle )$ can be computed from this and Equation (2.7). It can be a lengthy computation to find $t_0$ ; things are simplified when we know (a priori) that $t_0$ and ${}^{w'}t$ commute in the cover (e.g., when $-1\in F^{*m}$ ).

Example 1. Consider the permutation matrix $w=\left (\begin {smallmatrix}&I_d\\I_d\end {smallmatrix}\right )\in \operatorname {\mathrm {GL}}_{2d}$ . Denote $\alpha _i=(i,i+1)$ for $1\leq i<2d$ . Then

$$ \begin{align*} w=\operatorname{\mathrm{diag}}((-1)^{d}I_{d},I_{d})(w_{\alpha_{d}}\cdot\ldots\cdot w_{\alpha_{2d-1}}) (w_{\alpha_{d-1}}\cdot\ldots\cdot w_{\alpha_{2d-2}})\cdot\ldots\cdot (w_{\alpha_{1}}\cdot\ldots\cdot w_{\alpha_{d}}), \end{align*} $$

and the product is reduced: To see this, note that if $w_{0,d}$ is the longest Weyl element of $W_{\operatorname {\mathrm {GL}}_d}$ , $\ell (w)=\ell (w_{0,2d})-2\ell (w_{0,d})=d^2$ . Hence, $t_0=\operatorname {\mathrm {diag}}((-1)^{d}I_{d},I_{d})$ and $t_0w=w'\in \mathfrak {W}_{2d}$ .

According to the Bruhat decomposition $\operatorname {\mathrm {GL}}_d=\bigsqcup _{w"\in \mathfrak {W}_{d}}N_{\operatorname {\mathrm {GL}}_d}T_{\operatorname {\mathrm {GL}}_d}w"N_{\operatorname {\mathrm {GL}}_d}$ . For $g\in \operatorname {\mathrm {GL}}_d$ , write $g=n't'w"n"$ with $n',n"\in N_{\operatorname {\mathrm {GL}}_d}$ , $t'\in T_{\operatorname {\mathrm {GL}}_d}$ and $w"\in \mathfrak {W}_{d}$ . Define $\mathbf {t}(g)=t'$ (see [Reference Banks, Levy and SepanskiBLS99, p. 151]). By Equations (2.8) and (2.14),

(2.18) $$ \begin{align} \langle g,1\rangle=\langle n',1\rangle \langle t',1\rangle \langle w",1\rangle \langle n",1\rangle. \end{align} $$

As mentioned above $\mathfrak {W}_d$ is not a group. It will be convenient here to work with the following group $\mathfrak {W}^+_d$ , defined to be the group generated by $\mathfrak {W}_d$ and the diagonal matrices $\operatorname {\mathrm {diag}}(t_1,\ldots ,t_d)$ with $t_i=\pm 1$ for each i. In particular, $\mathfrak {W}^+_d$ contains the permutation matrices.

When $(-1,-1)_m=1$ in F, for example, if $-1\in F^{*m}$ , $\sigma _{d}$ is trivial on $\mathfrak {W}^+_d$ . This follows from the proof of [Reference Banks, Levy and SepanskiBLS99, § 3, Theorem 7], noting that for $g\in \mathfrak {W}^+_d$ , the coordinates of $\mathbf {t}(g)$ are $\pm 1$ (see also [Reference Banks, Levy and SepanskiBLS99, § 5]). In particular, if we identify $W_{\operatorname {\mathrm {GL}}_d}$ with the full subgroup of permutation matrices in $\operatorname {\mathrm {GL}}_d$ , $w\mapsto \langle w,1\rangle $ is a splitting of $W_{\operatorname {\mathrm {GL}}_d}$ .

We say that F is unramified if it is non-Archimedean, $|m|=1$ and $q>3$ . (If $m>2$ , the assumption $\mu _m\subset F^{*}$ implies $q>3$ .) Assume this is the case. Since $|m|=1$ , we have $(\mathcal {O}^{*},\mathcal {O}^{*})_m=1$ (see, e.g., [Reference WeilWei95, Ch. XIII.5]), whence $\sigma _d$ is trivial on $T_{\operatorname {\mathrm {GL}}_d}\cap K_{\operatorname {\mathrm {GL}}_d}$ , that is, on torus elements with coordinates in $\mathcal {O}^{*}$ (by Equation (2.7)), and on $\mathfrak {W}^+_d$ . By [Reference MooreMoo68, Lemma 11.3], there is a splitting of $K_{\operatorname {\mathrm {SL}}_{d+1}}$ in $\operatorname {\mathrm {SL}}_{d+1}^{(m)}$ , and in fact there is a unique splitting because the assumption $q>3$ implies that $K_{\operatorname {\mathrm {SL}}_{d+1}}$ is perfect ([Reference MooreMoo68, Lemma 11.1]). Restricting to $K_{\operatorname {\mathrm {GL}}_{d}}$ , we obtain a splitting of $K_{\operatorname {\mathrm {GL}}_{d}}$ denoted by $\langle y,\eta _d(y)\rangle $ . The splitting of $K_{\operatorname {\mathrm {GL}}_{d}}$ is not unique, and note that $\sigma _d$ is not trivial on $K_{\operatorname {\mathrm {GL}}_d}$ . Since $\sigma _d$ is trivial on $N_{\operatorname {\mathrm {GL}}_{d}}(\mathcal {O})$ , $T_{\operatorname {\mathrm {GL}}_d}\cap K_{\operatorname {\mathrm {GL}}_d}$ and $\mathfrak {W}^+_d$ , $\eta _d$ is a homomorphism of these subgroups, then the explicit description of $\eta _2$ of [Reference KubotaKub69, p. 19], namely $\eta _2(\left (\begin {smallmatrix}a&b\\c&d\end {smallmatrix}\right ))=1$ if $|c|=0,1$ and $(c,d(ad-bc)^{-1})_m$ otherwise, implies (in light of Equation (2.13)) that $\eta _d$ is trivial on $N_{\operatorname {\mathrm {GL}}_{d}}(\mathcal {O})$ , $T_{\operatorname {\mathrm {GL}}_d} \cap K_{\operatorname {\mathrm {GL}}_d}$ and $\mathfrak {W}^+_d$ (see [Reference Kazhdan and PattersonKP84, Proposition 0.I.3] and [Reference TakedaTak14, (1.3) and p. 183]).

Again, consider an arbitrary F, that is, not necessarily unramified. The image of H in $\operatorname {\mathrm {SL}}_{2rkc}$ is standard (as defined in [Reference Banks, Levy and SepanskiBLS99, p. 143]), hence by [Reference Banks, Levy and SepanskiBLS99, § 2, Theorem 7], the restriction of $\sigma _{2rkc}$ to H represents $H^{(m)}$ in $\mathrm {H}^2(H,\mu _m)$ . In fact this restriction is $\sigma _{\operatorname {\mathrm {Sp}}_{2rkc}}$ of [Reference Banks, Levy and SepanskiBLS99, § 2], but the description of $\sigma _{2rkc}$ in [Reference Banks, Levy and SepanskiBLS99, § 3] is more convenient for matrices. When we apply Equation (2.7) to $T_{rkc}<T_{\operatorname {\mathrm {GL}}_{2rkc}}$ , if $t=\operatorname {\mathrm {diag}}(t_1,\ldots ,t_{rkc},t_{rkc}^{-1},\ldots ,t_{1}^{-1})\in T_{rkc}$ and $t'\in T_{rkc}$ (with similar notation),

(2.19) $$ \begin{align} \sigma_{2rkc}(t,t')=\prod_{i=1}^{rkc}(t_i,t^{\prime}_i)_m^{-1}. \end{align} $$

Since $N_{rkc}<N_{\operatorname {\mathrm {GL}}_{2rkc}}$ , we can use Equations (2.8)-(2.10) for $b,b'\in H$ and $v,v'\in N_{rkc}$ . When F is unramified, $K_{H}=H(\mathcal {O})$ is perfect ([Reference MooreMoo68, Lemma 11.1]). Regarding $\eta _{2rkc}$ as a function on $K_H$ by restriction, it is the unique $1$ -cochain such that

(2.20) $$ \begin{align} \sigma_{2rkc}(y,y')=\frac{\eta_{2rkc}(yy')}{\eta_{2rkc}(y)\eta_{2rkc}(y')},\qquad\forall y,y'\in K_{H}. \end{align} $$

Indeed, if $\eta ':K_H\rightarrow \mu _m$ is another such mapping, $y\mapsto \eta _{2rkc}(y)\eta '(y)^{-1}$ is a character of $K_{H}$ .

Consider the involution $g\mapsto g^{*}=J_c{}^tg^{-1}J_c$ of $\operatorname {\mathrm {GL}}_c$ . Define

$$ \begin{align*} \sigma^{*}_{c}(g,g')=\sigma_{c}(g^{*},{g'}^{*}). \end{align*} $$

This is again a $2$ -cocycle of $\operatorname {\mathrm {GL}}_c$ and of $\operatorname {\mathrm {SL}}_c$ or G by restriction because $g\mapsto g^{*}$ is an automorphism and a homeomorphism.

Proof. The proof is similar to [Reference KableKab99, Lemma 2]. Let $T_{\operatorname {\mathrm {SL}}_c}=T_{\operatorname {\mathrm {GL}}_c}\cap \operatorname {\mathrm {SL}}_c$ . By [Reference MooreMoo68, p. 54, Corollary 2], restriction $\mathrm {H}^2(\operatorname {\mathrm {SL}}_c,\mu _m)\rightarrow \mathrm {H}^2(T_{\operatorname {\mathrm {SL}}_c},\mu _m)$ is injective. Now, by Equation (2.7), for $t,t'\in T_{\operatorname {\mathrm {SL}}_c}$ ,

$$ \begin{align*} \sigma_c(t,t')=\prod_{i=1}^{c-1}\prod_{j=1}^i(t_i,t^{\prime}_j)_m^{-1},\qquad \sigma^{*}_c(t,t')=\prod_{i=1}^{c-1}\prod_{j=i}^{c-1}(t_i,t^{\prime}_j)_m^{-1}. \end{align*} $$

Therefore, if we define $\eta \in \mathrm {C}^1(T_{\operatorname {\mathrm {SL}}_c},\mu _m)$ by $\eta (t)=\prod _{1\leq i\leq j\leq c-1}(t_i,t_j)_m$ ,

$$ \begin{align*} \sigma_c(t,t')\sigma^{*}_c(t,t')^{-1}=\prod_{i=1}^{c-1}\prod_{j=1}^i(t_i,t^{\prime}_j)_m^{-1} \prod_{i=1}^{c-1}\prod_{j=i}^{c-1}(t_i,t^{\prime}_j)_m=\frac{\eta(tt')}{\eta(t)\eta(t')}. \end{align*} $$

Hence, $\sigma ^{*}_{c}=\sigma _{c}$ in $\mathrm {H}^2(T_{\operatorname {\mathrm {SL}}_c},\mu _m)$ , thereby also in $\mathrm {H}^2(\operatorname {\mathrm {SL}}_c,\mu _m)$ .

We then have $\varsigma _{*,c}\in \mathrm {C}^1(\operatorname {\mathrm {SL}}_c,\mu _m)$ such that

(2.21) $$ \begin{align} \sigma^{*}_{c}(b,b')=\frac{\varsigma_{*,c}(b)\varsigma_{*,c}(b')}{\varsigma_{*,c}(bb')}\sigma_{c}(b,b'),\qquad \forall b,b'\in \operatorname{\mathrm{SL}}_c. \end{align} $$

We mention that $\sigma ^{*}_{c}\ne \sigma _c$ in $\mathrm {Z}^2(\operatorname {\mathrm {SL}}_c,\mu _m)$ (e.g., consider $g=\left (\begin {smallmatrix}a&\\&a^{-1}\end {smallmatrix}\right )$ and $g'=\left (\begin {smallmatrix}&-b^{-1}\\b\end {smallmatrix}\right )$ and use Equation (2.13)); for another example see below. We introduce a minor correction to $\sigma ^{*}_{c}$ using $\varsigma _{*,c}$ : Define $\sigma ^{*,rk}_{c}\in \mathrm {Z}^2(\operatorname {\mathrm {SL}}_c,\mu _m)$ by

(2.22) $$ \begin{align} \sigma^{*,rk}_{c}(b,b')=\left(\frac{\varsigma_{*,c}(b)\varsigma_{*,c}(b')}{\varsigma_{*,c}(bb')}\right)^{rk}\sigma^{*}_{c}(b,b') =\left(\frac{\varsigma_{*,c}(b)\varsigma_{*,c}(b')}{\varsigma_{*,c}(bb')}\right)^{rk+1}\sigma_{c}(b,b'). \end{align} $$

By definition $\sigma ^{*}_{c}=\sigma ^{*,rk}_{c}=\sigma _{c}$ in $\mathrm {H}^2(\operatorname {\mathrm {SL}}_c,\mu _m)$ , in particular in $\mathrm {H}^2(G,\mu _m)$ . Also, if $m|rk$ (e.g., when $r=m$ ), $\sigma ^{*,rk}_{c}=\sigma ^{*}_{c}$ in $\mathrm {Z}^2(\operatorname {\mathrm {SL}}_c,\mu _m)$ because $\varsigma _{*,c}^{rk}$ is trivial (being an m-th root of unity).

Example 6. Consider $c=4$ , $t=\operatorname {\mathrm {diag}}(t_1,t_2,t_2^{-1},t_1^{-1})$ and $w=\operatorname {\mathrm {diag}}(1,\left (\begin {smallmatrix}&-1\\1\end {smallmatrix}\right ),1)$ . Then $\sigma _4(w,t)=(-t_2^{-1},t_2)_m=1$ by Equation (2.15). To compute $\sigma _4^{*}(w,t)=\sigma _4(w^{*},t)$ , write $w^{*}=t_0w'$ , where $t_0=\operatorname {\mathrm {diag}}(1,-1,-1,1)$ and $w'$ belongs to the set $\mathfrak {W}_4$ . By Equation (2.1),

$$ \begin{align*} \sigma_4(t_0,w')\sigma_4(w^{*},t)=\sigma_4(t_0,w' t)\sigma_4(w',t). \end{align*} $$

Now, $\sigma _4(t_0,w')=1$ and $\sigma _4(w',t)=(-t_1^{-1},t_1)_m=1$ by Equations (2.14) and (2.15), and

$$ \begin{align*} \sigma_4(t_0,w' t)=\sigma_4(t_0,{}^{w'}tw')=\sigma_4(t_0,{}^{w'}t)=(-1,t_2)_m^{-1}, \end{align*} $$

where we used [Reference Banks, Levy and SepanskiBLS99, § 3, (5)] and Equation (2.7). Therefore, $\sigma _4(w^{*},t)=(-1,t_2)_m^{-1}\ne \sigma _4(w,t)$ .

We describe the restriction of the $2$ -cocycle $\sigma _{2rkc}$ to the image of $G\times G$ in H. Recall that for $g_1,g_2\in G$ , $\mathfrak {e}_1(g_1)\mathfrak {e}_2(g_2)=(g_1,g_2)$ .

Proposition 7. For all $g_i,g_i^{\prime }\in G$ ,

(2.23) $$ \begin{align} \sigma_{2rkc}(\mathfrak{e}_1(g_1)\mathfrak{e}_2(g_2),\mathfrak{e}_1(g_1')\mathfrak{e}_2(g_2'))=\sigma_{2rkc}((g_1,g_2),(g_1',g_2'))&=\sigma^{*,rk}_{c}(g_1,g_1')^{-1}\sigma_{c}(g_2,g_2'). \end{align} $$

In particular, $\mathfrak {e}_1(G)$ and $\mathfrak {e}_2(G)$ commute in $H^{(m)}$ (!).

Proof. For $g=\left (\begin {smallmatrix} g_1 & g_2 \\ g_3 & g_4 \end {smallmatrix}\right )\in G$ with $g_i\in \text {Mat}_n$ , denote

$$ \begin{align*} &\mathfrak{e}_1^{\circ}(g)=\operatorname{\mathrm{diag}}(g,\ldots,g,I_{2c},g^{*},\ldots,g^{*}),\qquad \mathfrak{e}_1^{\bullet}(g)=\operatorname{\mathrm{diag}}(I_{(rk-1)c},\left(\begin{smallmatrix} g_{1}&&g_{2}\\ &I_{c}&\\ g_{3}&&g_{4}\end{smallmatrix}\right),I_{(rk-1)c}). \end{align*} $$

Then $\mathfrak {e}_1(g)=\mathfrak {e}_1^{\circ }(g)\mathfrak {e}_1^{\bullet }(g)$ . The subgroup $\mathfrak {e}_2(G)$ is standard in the sense of [Reference Banks, Levy and SepanskiBLS99, § 2, Theorem 7] and so is $\mathfrak {e}_1^{\bullet }(G)$ (see the proof of [Reference TakedaTak16, Lemma 3.5]), but not $\mathfrak {e}_1^{\circ }(G)$ . Thus, by [Reference Banks, Levy and SepanskiBLS99, § 2, Theorem 7],

$$ \begin{align*} \sigma_{2rkc}(\mathfrak{e}_1^{\bullet}(g_1)\mathfrak{e}_2(g_2),\mathfrak{e}_1^{\bullet}(g_1')\mathfrak{e}_2(g_2'))&=\sigma_{2rkc}(\mathfrak{e}_1^{\bullet}(g_1),\mathfrak{e}_1^{\bullet}(g_1'))\sigma_{2rkc}(\mathfrak{e}_2(g_2),\mathfrak{e}_2(g_2')) =\sigma_{c}(g_1,g_1')\sigma_{c}(g_2,g_2'). \end{align*} $$

Also, by Equation (2.12),

$$ \begin{align*} \sigma_{2rkc}((g_1,g_2),(g_1',g_2'))&=\sigma_{2rkc}( \mathfrak{e}_1^{\circ}(g_1)\mathfrak{e}_1^{\bullet}(g_1)\mathfrak{e}_2(g_2) ,\mathfrak{e}_1^{\circ}(g_1')\mathfrak{e}_1^{\bullet}(g_1')\mathfrak{e}_2(g_2'))\\ &=\prod_{i=1}^{rk-1}\sigma_{c}(g_1,g_1')\sigma_{c}(g_1^{*},{g_1'}^{*})\sigma_{2rkc}(\mathfrak{e}_1^{\bullet}(g_1)\mathfrak{e}_2(g_2),\mathfrak{e}_1^{\bullet}(g_1')\mathfrak{e}_2(g_2')). \end{align*} $$

Hence,

$$ \begin{align*} \sigma_{2rkc}((g_1,g_2),(g_1',g_2'))=\prod_{i=1}^{rk-1}\sigma_{c}(g_1,g_1')\sigma_{c}(g_1^{*},{g_1'}^{*})\sigma_{c}(g_1,g_1')\sigma_{c}(g_2,g_2'). \end{align*} $$

It remains to observe that

$$ \begin{align*} \prod_{i=1}^{rk-1}\sigma_{c}(g_1,g_1')\sigma_{c}(g_1^{*},{g_1'}^{*}) & =\sigma_{c}^{2rk-2}(g_1,g_1')\left(\frac{\varsigma_{*,c}(g_1)\varsigma_{*,c}(g_1')}{\varsigma_{*,c}(g_1g_1')} \right)^{rk-1}\\ & =\sigma_{c}^{-2}(g_1,g_1')\left(\frac{\varsigma_{*,c}(g_1)\varsigma_{*,c}(g_1')}{\varsigma_{*,c}(g_1g_1')} \right)^{rk-1}. \end{align*} $$

Here, we used the fact that $\sigma _{c}^{2r}$ is trivial because the image of $\sigma _{c}$ is in $\mu _m$ .

We realize the left copy of G using the $2$ -cocycle $\sigma _c^{*,rk}$ and the right copy using $\sigma _c$ . Then by Equation (2.23), we can lift the embedding $G\times G \hookrightarrow H$ to an embedding (of topological groups)

$$ \begin{align*} \{(\epsilon_1,\epsilon_2)\in\mu_m^2:\epsilon_1=\epsilon_2\}\backslash G^{(m)}\times G^{(m)} \hookrightarrow H^{(m)} \end{align*} $$

( $\mu _m^2=\mu _m\times \mu _m$ ), via

(2.24) $$ \begin{align} \langle g,\epsilon\rangle\mapsto\langle \mathfrak{e}_1(g),\epsilon^{-1}\rangle,\qquad \langle g,\epsilon\rangle\mapsto\langle \mathfrak{e}_2(g),\epsilon\rangle. \end{align} $$

In addition, we have

(2.25) $$ \begin{align} \langle\mathfrak{e}_1(g_1),\epsilon_1^{-1}\rangle\langle \mathfrak{e}_2(g_2),\epsilon_2\rangle=\langle (g_1,g_2),\epsilon_1^{-1}\epsilon_2\rangle. \end{align} $$

While it is more natural to work with $\sigma _c$ for both copies of $G^{(m)}$ , if we realize the left copy using $\sigma _c$ , then by Equation (2.22), the embedding (2.24) of the left copy changes into

(2.26) $$ \begin{align} \langle g,\epsilon\rangle\mapsto\langle \mathfrak{e}_1(g),\varsigma_{*,c}^{rk+1}(g)\epsilon^{-1}\rangle, \end{align} $$

which complicates matters on the $H^{(m)}$ side. We combine both alternatives. Momentarily denote the realization of $G^{(m)}$ using $\sigma _c$ by $G^{(m)}[\sigma _c]$ , and similarly denote $G^{(m)}[\sigma _c^{*,rk}]$ . The map

(2.27) $$ \begin{align} G^{(m)}[\sigma_c^{*,rk}]\rightarrow G^{(m)}[\sigma_c],\qquad \langle g,\epsilon\rangle\mapsto \langle g,\varsigma_{*,c}^{rk+1}(g)\epsilon\rangle \end{align} $$

is an isomorphism, which is canonical in the sense that it is the only isomorphism (G is perfect) which projects to the identity map of G. Dualizing, for a function $\varphi $ on $G^{(m)}[\sigma _c]$ the function $\varphi ^{\varsigma _{*,c}^{rk+1}}$ on $G^{(m)}[\sigma _c^{*,rk}]$ is defined by

(2.28) $$ \begin{align} \varphi^{\varsigma_{*,c}^{rk+1}}(\langle g,\epsilon\rangle)=\varphi(\langle g,\varsigma_{*,c}^{rk+1}(g)\epsilon\rangle). \end{align} $$

Proposition 8. Let $\varphi _1,\varphi _2$ be continuous genuine functions on $G^{(m)}$ , realized using $\sigma _c$ , and f be a continuous genuine function on $H^{(m)}$ . The integral

$$ \begin{align*} \int\limits_{G\times G}\varphi_1^{\varsigma_{*,c}^{rk+1}}(\langle g_1,1\rangle)\overline{\varphi_2(\langle g_2,1\rangle)}f(\langle(g_1,g_2),1\rangle)\,dg_1\,dg_2 \end{align*} $$

is well defined in the sense that the integrand factors through $G\times G$ and the integral is a right- $(G\times G)$ -invariant functional, provided it is absolutely convergent.

Proof. First, note that since $\varphi _i(\langle g_i,\epsilon _i\rangle )=\epsilon _i\varphi _i(\langle g_i,1\rangle )$ , and Formulas (2.24)-(2.25) imply

$$ \begin{align*} f(\langle \mathfrak{e}_1(g_1),\epsilon_1^{-1}\rangle\langle \mathfrak{e}_2(g_2),\epsilon_2\rangle)=\epsilon_1^{-1}\epsilon_2 f(\langle (g_1,g_2),1\rangle), \end{align*} $$

the integrand factors through $G\times G$ . Also,

$$ \begin{align*} &\varphi_1^{\varsigma_{*,c}^{rk+1}}( \langle g_1,1\rangle\langle h_1,1\rangle)=\varphi_1^{\varsigma_{*,c}^{rk+1}}(\langle g_1h_1,\sigma_{c}^{*,rk}(g_1,h_1)\rangle),\\ &\varphi_2(\langle g_2,1\rangle\langle h_2,1\rangle)=\varphi_2(\langle g_2h_2,\sigma_{c}(g_2,h_2)\rangle) \end{align*} $$

and

$$ \begin{align*} &\langle (g_1,g_2),1\rangle\langle (h_1,h_2),1\rangle =\langle (g_1h_1,g_2h_2),\sigma_{c}^{*,rk}(g_1,h_1)^{-1}\sigma_{c}(g_2,h_2)\rangle. \end{align*} $$

Therefore, the integral is a right- $(G\times G)$ -invariant functional. This completes the proof.

Consider the subgroup $\{(g,g):g\in G\}$ of H. By Equation (2.23) and since $\sigma _c=\sigma ^{*,rk}_{c}$ in $\mathrm {H}^2(G,\mu _m)$ , the restriction of $H^{(m)}$ to this subgroup is the identity in $\mathrm {H}^2(G,\mu _m)$ , hence $H^{(m)}$ splits over $\{(g,g):g\in G\}$ . This splitting is unique (because G is perfect) and is defined as follows.

Proof. Let $g,g'\in G$ . By Proposition 7,

$$ \begin{align*} &\langle(g,g),1\rangle\langle(g',g'),1\rangle=\langle (gg',gg'),\sigma^{*,rk}_{c}(g,g')^{-1}\sigma_{c}(g,g')\rangle. \end{align*} $$

Now, Equation (2.22) implies that

$$ \begin{align*} \varsigma_{*,c}^{rk+1}(g)\varsigma_{*,c}(g')^{rk+1}\sigma^{*,rk}_{c}(g,g')^{-1} =\varsigma_{*,c}(gg')^{rk+1}\sigma_{c}(g,g')^{-1}, \end{align*} $$

therefore

$$ \begin{align*} &\langle(g,g),\varsigma_{*,c}^{rk+1}(g)\rangle\langle(g',g'),\varsigma_{*,c}(g')^{rk+1}\rangle=\langle (gg',gg'),\varsigma_{*,c}(gg')^{rk+1}\rangle. \end{align*} $$

The prescribed map is continuous because $\sigma ^{*,rk}_{c}\in \mathrm {C}^1(G,\mu _m)$ .

2.5 Global covering

Let F be a number field containing $\mu _m$ , and let $\nu $ be a place of F. Let $\sigma _{2rkc,\nu }$ be the local $2$ -cocycle of § 2.4, regarded as a $2$ -cocycle of $H(F_{\nu })$ by restriction from $\operatorname {\mathrm {GL}}_{2rkc}(F_{\nu })$ . Denote $\sigma _{\nu }=\sigma _{2rkc,\nu }$ . The product $\sigma =\prod _{\nu }\sigma _{\nu }$ is trivial on $N_{rkc}(\mathbb {A})$ . It is also well defined on $B_{rkc}(\mathbb {A})$ : To see this, use Equation (2.9), Equation (2.7) and note that for $t\in T_{rkc}(\mathbb {A})$ , at almost all places, all coordinates of t belong to $\mathcal {O}_{\nu }^{*}$ and $(\mathcal {O}_{\nu }^{*},\mathcal {O}_{\nu }^{*})_m=1$ . While it is well known that $\sigma $ is undefined on $H(\mathbb {A})$ , we can define $\rho \in \mathrm {Z}^2(H(\mathbb {A}),\mu _m)$ such that at any $\nu $ , $\rho _{\nu }=\sigma _{\nu }$ in $\mathrm {H}^2(H(F_{\nu }),\mu _m)$ . Thus, $\rho $ represents $H^{(m)}(\mathbb {A})$ in $\mathrm {H}^2(H(\mathbb {A}),\mu _m)$ (see § 2.2).

To achieve this, first note that for almost all $\nu $ there is a unique splitting $y\mapsto \langle y,\eta _{\nu }(y)\rangle $ of $K_{H,\nu }$ , where $\eta _{\nu }=\eta _{2rkc,\nu }$ (see § 2.4). We can extend $\eta _{\nu }$ to an element of $\mathrm {C}^1(H(F_{\nu }),\mu _m)$ (though Equation (2.20) will no longer hold for arbitrary $y,y'\in H(F_{\nu })$ ). Now, define, for almost all $\nu $ ,

(2.29) $$ \begin{align} \rho_{\nu}(h,h')=\frac{\eta_{\nu}(h)\eta_{\nu}(h')}{\eta_{\nu}(hh')}\sigma_{\nu}(h,h'),\qquad \forall h,h'\in H(F_{\nu}). \end{align} $$

Then $\rho _{\nu }$ is trivial on $K_{H,\nu }$ . For the remaining places, we can simply take $\rho _{\nu }=\sigma _{\nu }$ and $\eta _{\nu }=1$ . Then $\rho =\prod _{\nu }\rho _{\nu }$ is well defined on $H(\mathbb {A})$ .

We realize $H^{(m)}(\mathbb {A})$ using $\rho $ then use the embedding $G(\mathbb {A})\times G(\mathbb {A})\hookrightarrow H(\mathbb {A})$ to realize the $2$ -cocycles on the copies of $G(\mathbb {A})$ . Let

(2.30) $$ \begin{align} \rho_L(g,g')=\rho^{-1}(\mathfrak{e}_1(g),\mathfrak{e}_1(g')),\qquad\rho_R(g,g')=\rho(\mathfrak{e}_2(g),\mathfrak{e}_2(g')). \end{align} $$

Now, the product on the left copy of $G^{(m)}(\mathbb {A})$ is defined by $\rho _L(g,g')$ , that is,

$$ \begin{align*} \langle g,\epsilon\rangle\langle g',\epsilon'\rangle=\langle gg',\epsilon\epsilon'\rho_L(g,g')\rangle, \end{align*} $$

and on the right copy by $\rho _R(g,g')$ . In addition,

(2.31) $$ \begin{align} \langle \mathfrak{e}_1(g_1),1\rangle \langle \mathfrak{e}_2(g_2),1\rangle = \langle \mathfrak{e}_2(g_2),1\rangle \langle \mathfrak{e}_1(g_1),1\rangle,\qquad\forall g_1,g_2\in G(\mathbb{A}) \end{align} $$

because this holds locally by Proposition 7 (this does not mean the global version of Equation (2.23) holds with $\rho $ ). Therefore, we may lift the embedding $G(\mathbb {A})\times G(\mathbb {A}) \hookrightarrow H(\mathbb {A})$ to an embedding

$$ \begin{align*} \{(\epsilon_1,\epsilon_2)\in\mu_m^2:\epsilon_1=\epsilon_2\}\backslash G^{(m)}(\mathbb{A})\times G^{(m)}(\mathbb{A}) \hookrightarrow H^{(m)}(\mathbb{A}), \end{align*} $$

and we obtain the global analog of the embeddings (2.24), namely, $\langle g,\epsilon \rangle \mapsto \langle \mathfrak {e}_1(g),\epsilon ^{-1}\rangle $ for the left copy and $\langle g,\epsilon \rangle \mapsto \langle \mathfrak {e}_2(g),\epsilon \rangle $ for the right. Also, by Proposition 7, $\rho _{L,\nu }=\sigma ^{*,rk}_{c,\nu }=\sigma _{c,\nu }=\rho _{R,\nu }$ in $\mathrm {H}^2(G(F_{\nu }),\mu _m)$ , whence both copies of $G^{(m)}(\mathbb {A})$ are cohomologous.

As in the local setting, we would like to explicate the relation between the copies of $G^{(m)}(\mathbb {A})$ so that we can work with the same $2$ -cocycle for both yet still use the global embedding (2.31). At any place $\nu $ , Equation (2.23) implies that

$$ \begin{align*} \sigma_{\nu}(\mathfrak{e}_1(g),\mathfrak{e}_1(g'))=\sigma^{*,rk}_{c,\nu}(g,g')^{-1},\qquad \sigma_{\nu}(\mathfrak{e}_2(g),\mathfrak{e}_2(g'))=\sigma_{c,\nu}(g,g'). \end{align*} $$

Whence by Equation (2.22),

$$ \begin{align*} \sigma_{\nu}(\mathfrak{e}_1(g),\mathfrak{e}_1(g'))=\left(\frac{\varsigma_{*,c,\nu}(gg')}{\varsigma_{*,c,\nu}(g) \varsigma_{*,c,\nu}(g')}\right)^{rk+1}\sigma_{\nu}(\mathfrak{e}_2(g),\mathfrak{e}_2(g'))^{-1}. \end{align*} $$

Then for all $\nu $ ,

$$ \begin{align*} &\rho_{\nu}(\mathfrak{e}_1(g),\mathfrak{e}_1(g'))\\&=\frac{\eta_{\nu}(\mathfrak{e}_1(g))\eta_{\nu}(\mathfrak{e}_1(g'))}{\eta_{\nu}(\mathfrak{e}_1(gg'))} \sigma_{\nu}(\mathfrak{e}_1(g),\mathfrak{e}_1(g'))\\ &=\frac{\eta_{\nu}(\mathfrak{e}_1(g))\eta_{\nu}(\mathfrak{e}_1(g'))}{\eta_{\nu}( \mathfrak{e}_1(gg'))}\frac{\eta_{\nu}(\mathfrak{e}_2(g))\eta_{\nu}(\mathfrak{e}_2(g'))}{\eta_{\nu}(\mathfrak{e}_2(gg'))} \left(\frac{\varsigma_{*,c,\nu}(gg')}{\varsigma_{*,c,\nu}(g)\varsigma_{*,c,\nu}(g')}\right)^{rk+1}\rho_{\nu}( \mathfrak{e}_2(g),\mathfrak{e}_2(g'))^{-1}. \end{align*} $$

Hence, if we define $\eta ^{\times }_{\nu }\in \mathrm {C}^1(G(F_{\nu }),\mu _m)$ by

$$ \begin{align*} \eta^{\times}_{\nu}(g)&=\eta_{\nu}(\mathfrak{e}_1(g))\eta_{\nu}(\mathfrak{e}_2(g))/\varsigma_{*,c,\nu}^{rk+1}(g),\\\rho_{\nu}(\mathfrak{e}_1(g),\mathfrak{e}_1(g'))&=\frac{\eta^{\times}_{\nu}(g)\eta^{\times}_{\nu}(g')}{\eta^{\times}_{\nu}(gg')}\rho_{\nu}(\mathfrak{e}_2(g),\mathfrak{e}_2(g'))^{-1}.\end{align*} $$

Now, $\eta ^{\times }=\prod _{\nu }\eta ^{\times }_{\nu }\in \mathrm {C}^1(G(\mathbb {A}),\mu _m)$ is well defined since for almost all $\nu $ ,

$$ \begin{align*} \rho_{\nu}(\mathfrak{e}_1(y),\mathfrak{e}_1(y'))=\rho_{\nu}(\mathfrak{e}_2(y),\mathfrak{e}_2(y'))=1,\qquad\forall y,y'\in K_{G,\nu}, \end{align*} $$

and then for these places $\eta ^{\times }_{\nu }$ becomes a homomorphism of $K_{G,\nu }$ . Thus, we deduce the global relation, for all $g,g'\in G(\mathbb {A})$ , that is,

(2.32) $$ \begin{align} \rho(\mathfrak{e}_1(g),\mathfrak{e}_1(g'))= \frac{\eta^{\times}(g)\eta^{\times}(g')}{\eta^{\times}(gg')}\rho(\mathfrak{e}_2(g),\mathfrak{e}_2(g'))^{-1}. \end{align} $$

Then by definition

(2.33) $$ \begin{align} \rho_R(g,g')=\frac{\eta^{\times}(g)\eta^{\times}(g')}{\eta^{\times}(gg')}\rho_L(g,g'). \end{align} $$

Now, we can state the global analogs of Equations (2.27) and (2.28). First, if $G^{(m)}(\mathbb {A})[\rho _R]$ denotes the realization of $G^{(m)}(\mathbb {A})$ using $\rho _R$ and similarly for $G^{(m)}(\mathbb {A})[\rho _L]$ , we have the (canonical) isomorphism

(2.34) $$ \begin{align} G^{(m)}(\mathbb{A})[\rho_L]\rightarrow G^{(m)}(\mathbb{A})[\rho_R],\qquad \langle g,\epsilon\rangle\mapsto \langle g,(\eta^{\times})^{-1}(g)\epsilon\rangle. \end{align} $$

Then for a function $\varphi $ on $G^{(m)}(\mathbb {A})[\rho _R]$ , the function $\varphi ^{(\eta ^{\times })^{-1}}$ on $G^{(m)}(\mathbb {A})[\rho _L]$ is defined by

(2.35) $$ \begin{align} \varphi^{(\eta^{\times})^{-1}}(\langle g,\epsilon\rangle)= \varphi(\langle g,(\eta^{\times})^{-1}(g)\epsilon\rangle). \end{align} $$

Note that locally, ignoring the correction using $\eta _{\nu }$ , $(\eta _{\nu }^{\times })^{-1}(g_{\nu })$ becomes $\varsigma _{*,c,\nu }^{rk+1}(g_{\nu })$ from Equation (2.28).

Next, we state the analog of Proposition 8.

Proposition 10. Let $\varphi _1,\varphi _2$ be continuous genuine functions on $G^{(m)}(\mathbb {A})$ , realized using $\rho _R$ , and f be a continuous genuine function on $H^{(m)}(\mathbb {A})$ . The integral

$$ \begin{align*} \int\limits_{G(\mathbb{A})\times G(\mathbb{A})}\varphi_1^{(\eta^{\times})^{-1}}(\langle g_1,1\rangle)\overline{\varphi_2(\langle g_2,1\rangle)}f( \langle \mathfrak{e}_1(g_1),1\rangle\langle \mathfrak{e}_2(g_2),1\rangle)\,dg_1\,dg_2 \end{align*} $$

is well defined: The integrand factors through $G(\mathbb {A})\times G(\mathbb {A})$ and the integral is a right- $(G(\mathbb {A})\times G(\mathbb {A}))$ -invariant functional, if it is absolutely convergent.

Proof. The integrand factors through $G(\mathbb {A})\times G(\mathbb {A})$ by the global analog of the embeddings (2.24). To see that it is a right- $(G(\mathbb {A})\times G(\mathbb {A}))$ -invariant functional observe that by Equations (2.30) and (2.35),

$$ \begin{align*} &\varphi_1^{(\eta^{\times})^{-1}}(\langle g_1,1\rangle\langle h_1,1\rangle)=\rho_L(g_1,h_1)\varphi_1^{(\eta^{\times})^{-1}}(\langle g_1h_1,1\rangle)=\rho^{-1}(\mathfrak{e}_1(g_1),\mathfrak{e}_1(h_1))\varphi_1^{(\eta^{\times})^{-1}}(\langle g_1h_1,1\rangle),\\ &\varphi_2(\langle g_2,1\rangle\langle h_2,1\rangle)=\rho_R(g_2,h_2)\varphi_2(\langle g_2h_2,1\rangle) =\rho(\mathfrak{e}_2(g_2),\mathfrak{e}_2(h_2))\varphi_2(\langle 1,g_2h_2\rangle). \end{align*} $$

Now, although Equation (2.25) no longer holds, still by Equation (2.31),

$$ \begin{align*} &\langle \mathfrak{e}_1(g_1),1\rangle\langle \mathfrak{e}_2(g_2),1\rangle \langle \mathfrak{e}_1(h_1),1\rangle\langle \mathfrak{e}_2(h_2),1\rangle \\&= \langle \mathfrak{e}_1(g_1),1\rangle\langle \mathfrak{e}_1(h_1),1\rangle\langle \mathfrak{e}_2(g_2),1\rangle \langle \mathfrak{e}_2(h_2),1\rangle \\&= \rho(\mathfrak{e}_1(g_1),\mathfrak{e}_1(h_1))\rho(\mathfrak{e}_2(g_2),\mathfrak{e}_2(h_2))\langle \mathfrak{e}_1(g_1h_1),1\rangle\langle\mathfrak{e}_2(g_2h_2),1\rangle. \end{align*} $$

We see that both $\rho (\mathfrak {e}_i(g_i),\mathfrak {e}_i(h_i))$ are cancelled (we integrate against $\overline {\varphi _2}$ ). The result follows.

Now, consider the subgroup $\{(g,g):g\in G\}$ of H. Locally, the covering splits over this group (see Corollary 9), and because $K_{G,\nu }$ is perfect for almost all $\nu $ , we deduce that $H^{(m)}(\mathbb {A})$ is split over $\{(g,g):g\in G(\mathbb {A})\}$ (see § 2.2). Again, we determine the splitting.

Proof. Let $g,g'\in G(\mathbb {A})$ . Since

$$ \begin{align*} \langle(g,g),1\rangle=\langle \mathfrak{e}_1(g),1\rangle \langle\mathfrak{e}_2(g),\rho(\mathfrak{e}_1(g),\mathfrak{e}_2(g))^{-1}\rangle, \end{align*} $$

by Equation (2.31) we have

$$ \begin{align*} &\langle(g,g),\rho(\mathfrak{e}_1(g),\mathfrak{e}_2(g))\rangle\langle(g',g'),\rho(\mathfrak{e}_1(g'),\mathfrak{e}_2(g'))\rangle\\&= \langle \mathfrak{e}_1(gg'),\rho(\mathfrak{e}_1(g),\mathfrak{e}_1(g'))\rangle \langle \mathfrak{e}_2(gg'),\rho(\mathfrak{e}_2(g),\mathfrak{e}_2(g'))\rangle \\&=\langle (gg',gg'),\rho(\mathfrak{e}_1(gg'),\mathfrak{e}_2(gg'))\rho(\mathfrak{e}_1(g),\mathfrak{e}_1(g'))\rho(\mathfrak{e}_2(g),\mathfrak{e}_2(g'))\rangle. \end{align*} $$

Hence, by Equation (2.32),

$$ \begin{align*} &\langle(g,g),(\eta^{\times})^{-1}(g)\rho(\mathfrak{e}_1(g),\mathfrak{e}_2(g))\rangle\langle(g',g'),(\eta^{\times})^{-1}(g')\rho(\mathfrak{e}_1(g'),\mathfrak{e}_2(g'))\rangle \\&=\langle (gg',gg'),(\eta^{\times})^{-1}(gg')\rho(\mathfrak{e}_1(gg'),\mathfrak{e}_2(gg'))\rangle. \end{align*} $$

The continuity follows since $\eta ^{\times }$ and $g\mapsto \rho (\mathfrak {e}_1(g),\mathfrak {e}_2(g))$ belong to $\mathrm {C}^1(G(\mathbb {A}),\mu _m)$ .

As mentioned above, $\sigma =\prod _{\nu }\sigma _{\nu }$ is defined on certain subgroups of $H(\mathbb {A})$ , for example, on $N_{rkc}(\mathbb {A})$ . It is also defined on $H(F)$ : Indeed for $h,h'\in H(F)$ , $\sigma _{\nu }(h_{\nu },h^{\prime }_{\nu })=1$ for almost all $\nu $ , because the local $2$ -cocycle is written as a finite product of Hilbert symbols $(x,x')_{m,\nu }$ , with elements $x,x'\in F^{*}$ that are independent of $\nu $ , and $(x,x')_{m,\nu }=1$ for almost all $\nu $ (see the proof of [Reference Banks, Levy and SepanskiBLS99, § 3, Theorem 7], and also [Reference Kazhdan and PattersonKP84, § 0.2] and [Reference TakedaTak14, Proposition 1.7]). Moreover, $\eta _{\nu }$ given by Equation (2.29) is trivial on $h_{\nu }$ for almost all $\nu $ . This was shown in [Reference TakedaTak14, Proposition 1.8], and we follow the argument. Write $h=n't'w'n"$ according to the Bruhat decomposition in $H(F)$ , that is, $n',n"\in N_{rkc}(F)$ , $t'\in T_{rkc}(F)$ and $w'\in H(F)\cap \mathfrak {W}_{2rkc}^+$ (with $\mathfrak {W}_{2rkc}^+$ defined over F). For almost all $\nu $ , we have $n^{\prime }_{\nu },t^{\prime }_{\nu },w^{\prime }_{\nu },n^{\prime \prime }_{\nu }\in K_{G,\nu }$ , then by Equation (2.18), $\eta _{\nu }(h_{\nu })=1$ because $\eta _{\nu }(n^{\prime }_{\nu })=\eta _{\nu }(t^{\prime }_{\nu })=\eta _{\nu }(w^{\prime }_{\nu })=\eta _{\nu }(n^{\prime \prime }_{\nu })=1$ (see § 2.4). Therefore, we can define $\eta =\prod _{\nu }\eta _{\nu }$ on $H(F)$ and deduce that $\sigma =\rho $ in $\mathrm {H}^2(H(F),\mu _m)$ . Furthermore, because of the product formula $\prod _{\nu }(x,x')_{m,\nu }=1$ for any $x,x'\in F^{*}$ , $\sigma $ is in fact trivial on $H(F)$ , thus $\langle h,\eta ^{-1}(h)\rangle $ is the splitting of $H(F)$ in $H^{(m)}(\mathbb {A})$ (see [Reference TakedaTak14, Proposition 1.7]).

Since $G(F)\times G(F)<H(F)$ , we can use $\eta $ to define splittings of $G(F)$ . A direct verification shows that $g\mapsto \langle g,\eta (\mathfrak {e}_1(g))\rangle $ is the splitting of $G(F)$ in the covering $G^{(m)}(\mathbb {A})$ realized using $\rho _L$ , and $g\mapsto \langle g,\eta ^{-1}(\mathfrak {e}_2(g))\rangle $ is the splitting when the covering is realized via $\rho _R$ (i.e., the right copy). Having fixed these splittings, we can now consider the spaces of automorphic forms on $G^{(m)}(\mathbb {A})$ and $H^{(m)}(\mathbb {A})$ , which are in particular functions on $G(F)\backslash G^{(m)}(\mathbb {A})$ and $H(F)\backslash H^{(m)}(\mathbb {A})$ (resp.). We must specify whether we are considering the left or right copy of $G^{(m)}(\mathbb {A})$ because the $2$ -cocycles differ (up to a $2$ -coboundary) and so do the splittings of $G(F)$ (see Equation (2.3)). Our next goal is to show that the map (2.35) preserves the notion of automorphic forms.

Observe that Equation (2.23) also implies that $\sigma _{\nu }(\mathfrak {e}_1(g),\mathfrak {e}_2(g))=1$ for $g\in G(F_{\nu })$ . Therefore,

(2.36) $$ \begin{align} \rho_{\nu}(\mathfrak{e}_1(g),\mathfrak{e}_2(g))= \frac{\eta_{\nu}(\mathfrak{e}_1(g))\eta_{\nu}(\mathfrak{e}_2(g))}{\eta_{\nu}(\mathfrak{e}_1(g)\mathfrak{e}_2(g))} =\frac{\eta^{\times}_{\nu}(g)\varsigma_{*,c,\nu}^{rk+1}(g)}{\eta_{\nu}(\mathfrak{e}_1(g)\mathfrak{e}_2(g))},\qquad\forall g\in G(F_{\nu}). \end{align} $$

Now, $\eta $ is well defined on $\{(g,g):g\in G(F)\}$ because it is defined on $H(F)$ ; $\rho $ is well defined on $G(F)$ since it is defined on $H(\mathbb {A})$ , and $\eta ^{\times }$ is well defined on $G(F)$ because it is defined on $G(\mathbb {A})$ . Thus, if $g\in G(F)$ , then for almost all $\nu $ , by Equation (2.36), $\varsigma _{*,c,\nu }^{rk+1}(g_{\nu })=1$ , whence $\varsigma _{*,c}^{rk+1}=\prod _{\nu }\varsigma _{*,c,\nu }^{rk+1}$ is well defined on $G(F)$ (if $m\nmid rk$ , $\varsigma _{*,c}$ might not be). Therefore, we can write globally

(2.37) $$ \begin{align} \rho(\mathfrak{e}_1(g),\mathfrak{e}_2(g))=\frac{\eta^{\times}(g)\varsigma_{*,c}^{rk+1}(g)}{\eta(\mathfrak{e}_1(g)\mathfrak{e}_2(g))},\qquad\forall g\in G(F). \end{align} $$

Proof. For each $\nu $ , raising Equation (2.21) to the power $rk+1$ we have

$$ \begin{align*} (\sigma^{*}_{c,\nu})^{rk+1}(g,g')=\left(\frac{\varsigma_{*,c,\nu}(g)\varsigma_{*,c,\nu}(g')}{\varsigma_{*,c,\nu}(gg')}\right)^{rk+1}\sigma_{c,\nu}^{rk+1}(g,g'). \end{align*} $$

Since $\sigma _c$ is trivial on $G(F)$ and $g\mapsto g^{*}$ is an involution of $G(F)$ , $\sigma _c^{*}$ is also trivial on $G(F)$ . Also, $\varsigma _{*,c}^{rk+1}$ is well defined on $G(F)$ . Therefore, we can globalize this equality and deduce that $\varsigma _{*,c}^{rk+1}:G(F)\rightarrow \mu _m$ is a homomorphism, which must be trivial because $G(F)$ is perfect.

Proof. By Proposition 12, $\eta ^{\times }(y)=\eta (\mathfrak {e}_1(y))\eta (\mathfrak {e}_2(y))$ for all $y\in G(F)$ , hence for any $h\in G^{(m)}(\mathbb {A})$ ,

$$ \begin{align*} \varphi_1^{(\eta^{\times})^{-1}}(\langle y,\eta(\mathfrak{e}_1(y))\rangle h) =\varphi_1(\langle y,(\eta^{\times})^{-1}(y)\eta(\mathfrak{e}_1(y))\rangle h) =\varphi_1(\langle y,\eta^{-1}(\mathfrak{e}_2(y))\rangle h)=\varphi_1(h), \end{align*} $$

where we used the left-invariance of $\varphi _1$ .

Proposition 14. Let $\varphi _1,\varphi _2$ be continuous genuine functions on $G(F)\backslash G^{(m)}(\mathbb {A})$ , where $G^{(m)}(\mathbb {A})$ is realized using $\rho _R$ . Let f be a continuous genuine function on the image of

$$ \begin{align*} G(F)\times G(F)\backslash G^{(m)}(\mathbb{A})\times G^{(m)}(\mathbb{A}) \end{align*} $$

in $H(F)\backslash H^{(m)}(\mathbb {A})$ , with the above identifications (e.g., f on $H(F)\backslash H^{(m)}(\mathbb {A})$ ). Then the integral

$$ \begin{align*} \int\limits_{G(F)\times G(F)\backslash G(\mathbb{A})\times G(\mathbb{A})}\varphi_1^{(\eta^{\times})^{-1}}(\langle g_1,1\rangle)\overline{\varphi_2(\langle g_2,1\rangle)}f(\langle\mathfrak{e}_1(g_1),1\rangle\langle\mathfrak{e}_2(g_2),1\rangle)\,dg_1\,dg_2 \end{align*} $$

is well defined: The integrand factors through the quotient and the integral is a right- $(G(\mathbb {A})\times G(\mathbb {A}))$ -invariant functional, provided it is absolutely convergent.

Proof. By Proposition 10, it remains to show that the integrand factors through the quotient. Let $y_1,y_2\in G(F)$ and $g_1,g_2\in G(\mathbb {A})$ . Put $\epsilon _i=\eta _i(\mathfrak {e}_i(y_i))$ . Then

$$ \begin{align*} &\varphi_1^{(\eta^{\times})^{-1}}(\langle y_1g_1,1\rangle)\overline{\varphi_2(\langle y_2g_2,1\rangle)}f( \langle\mathfrak{e}_1(y_1g_1),1\rangle\langle\mathfrak{e}_2(y_2g_2),1\rangle) \\&=\varphi_1^{(\eta^{\times})^{-1}}(\langle y_1,1\rangle \langle g_1,1\rangle )\overline{\varphi_2(\langle y_2,1\rangle \langle g_2,1\rangle )} f( \langle\mathfrak{e}_1(y_1),1\rangle\langle\mathfrak{e}_1(g_1),1\rangle\langle\mathfrak{e}_2(y_2),1\rangle\langle\mathfrak{e}_2(g_2),1\rangle)\nonumber \\&=\varphi_1^{(\eta^{\times})^{-1}}(\langle y_1,\epsilon_1\rangle \langle g_1,1\rangle) \overline{\varphi_2(\langle y_2,\epsilon_2^{-1}\rangle\langle g_2,1\rangle)}f( \langle\mathfrak{e}_1(y_1),\epsilon_1^{-1}\rangle\langle\mathfrak{e}_1(g_1),1\rangle\langle\mathfrak{e}_2(y_2), \epsilon_2^{-1}\rangle\langle\mathfrak{e}_2(g_2),1\rangle)\nonumber \\&=\varphi_1^{(\eta^{\times})^{-1}}(\langle y_1,\epsilon_1\rangle \langle g_1,1\rangle) \overline{\varphi_2(\langle y_2,\epsilon_2^{-1}\rangle\langle g_2,1\rangle)}f( \langle\mathfrak{e}_1(y_1),\epsilon_1^{-1}\rangle\langle\mathfrak{e}_2(y_2), \epsilon_2^{-1}\rangle\langle\mathfrak{e}_1(g_1),1\rangle\langle\mathfrak{e}_2(g_2),1\rangle)\\ &=\varphi_1^{(\eta^{\times})^{-1}}(\langle g_1,1\rangle) \overline{\varphi_2(\langle g_2,1\rangle)}f(\langle\mathfrak{e}_1(g_1),1\rangle\langle\mathfrak{e}_2(g_2),1\rangle)\nonumber. \end{align*} $$

For the last equality, we used the left invariance under $G(F)$ (resp., $H(F)$ ) of $\varphi _i$ (resp., f) and note that this left invariance is with respect to the particular section for each copy of $G(F)$ .

As explained above, on $H(F)$ we have global definitions of $\sigma $ and $\eta $ so that the global analog of Equation (2.29) is valid, and $h\mapsto \langle h,\eta ^{-1}(h)\rangle $ is the splitting of $H(F)$ under $\rho $ . A simpler argument applies to $N_{rkc}(\mathbb {A})$ : Since $\eta _{\nu }$ is trivial on $N_{rkc}(\mathcal {O}_{\nu })$ for all $\nu $ where $F_{\nu }$ is unramified (see § 2.4), $\eta \in \mathrm {C}^1(N_{rkc}(\mathbb {A}),\mu _m)$ , and because $\sigma _{\nu }$ is trivial on $N_{rkc}(F_{\nu })$ for all $\nu $ , the global analog of Equation (2.29) holds and $u\mapsto \langle u,\eta ^{-1}(u)\rangle $ is the splitting of $N_{rkc}(\mathbb {A})$ in $H^{(m)}(\mathbb {A})$ .

The following lemma is the extension of Equation (2.10) to the global cover.

Corollary 16. For any $h\in H(F)$ and $u\in N_{rkc}(\mathbb {A})$ ,

$$ \begin{align*} \langle h,\eta^{-1}(h)\rangle\langle u,\eta^{-1}(u)\rangle= \langle hu,\eta^{-1}(hu)\rangle,\qquad \langle u,\eta^{-1}(u)\rangle\langle h,\eta^{-1}(h)\rangle= \langle uh,\eta^{-1}(uh)\rangle. \end{align*} $$

Proof. By Equation (2.8), $\sigma (h^{\prime }_{\nu },u^{\prime }_{\nu })=1$ for any $h'\in H(F)$ and $u'\in N_{rkc}(\mathbb {A})$ . Hence, Equation (2.29) gives

$$ \begin{align*} \rho_{\nu}(h^{\prime}_{\nu},u^{\prime}_{\nu})=\eta_{\nu}(h^{\prime}_{\nu})\eta_{\nu}(u^{\prime}_{\nu})/\eta_{\nu}(h^{\prime}_{\nu}u^{\prime}_{\nu}). \end{align*} $$

Since the left-hand side (l.h.s.) is $1$ for almost all $\nu $ and so are $\eta _{\nu }(h^{\prime }_{\nu })$ and $\eta _{\nu }(u^{\prime }_{\nu })$ , we deduce that $\eta _{\nu }(h^{\prime }_{\nu }u^{\prime }_{\nu })=1$ almost everywhere. Therefore, $\eta (hu)$ is well defined and we have

$$ \begin{align*} \rho(h,u)=\frac{\eta(h)\eta(u)}{\eta(hu)}, \end{align*} $$

proving the first equality. The symmetric argument (for $\rho (u,h)$ ) implies the second formula.

2.6 The lift of the involution ${}^{\iota }$ of G

For the construction of the integral, we will repeatedly use the outer involution ${}^{\iota }$ of G, given locally and globally by $g\mapsto {}^{\iota }g=\iota g\iota ^{-1}$ , where $\iota =\left (\begin {smallmatrix}&I_{c/2}\\I_{c/2}\end {smallmatrix}\right )$ . In this section, we discuss its lift to $G^{(m)}$ .

First, consider the local setting and realize $G^{(m)}$ with $\sigma _c$ . Since ${}^{\iota }$ is also a (continuous) involution of $\operatorname {\mathrm {GL}}_c$ , we can define $\sigma _c^{\iota }\in \mathrm {Z}^2(\operatorname {\mathrm {GL}}_c,\mu _m)$ by $\sigma _c^{\iota }(g,g')=\sigma _c({}^{\iota }g,{}^{\iota }g')$ for $g,g'\in \operatorname {\mathrm {GL}}_c$ .