2.1.1.

Let F be a local field of characteristic zero or a number field. Denote by ${\mathcal {O}}_F$ the ring of integers of F. Let E be a quadratic extension of F. If F is a local (respectively global) field, denote by $\eta$ the quadratic character of $F^\times/NE^\times$ (respectively ${\mathbb {A}}^\times/F^\times$ ) attached to $E/F$ , where $NE^\times=N_{E/F} E^\times$ denotes the norm of $E^\times$ in $F^\times$ (respectively ${\mathbb {A}}={\mathbb {A}}_F$ denotes the ring of adèles of F).

Let G be a (connected) reductive group over F. Denote by $\mathrm{rk}_F(G)$ the F-rank of G. Let $\widehat{G}$ be the Langlands dual of G, which is a complex reductive group. For a topological group H, denote by $H^\circ$ the neutral component of H and let $\pi_0(H):=H/H^\circ$ . All algebraic groups (except $\widehat{G}$ ) and varieties are assumed to be defined over F until further notice. Denote by $G_{\mathrm{ad}}$ the adjoint group of G, by $G_{\mathrm{der}}$ the derived subgroup of G and by $G_{\mathrm{sc}}$ the simple connected cover of $G_{\mathrm{der}}$ . Denote by $Z_G$ the centre of G and by $C_G:=G/G_{\mathrm{der}}=Z_G/Z_{G_{\mathrm{der}}}$ the cocentre of G. Fix an algebraic closure $\overline{F}$ of F. Let $\Gamma:={\mathrm{Gal}}(\overline{F}/F)$ . If $\Gamma$ acts on a set S, denote by $S^\Gamma$ the subset of S consisting of all $\Gamma$ -fixed points. For an F-variety V, we sometimes abuse notation and also write V for $V(\overline{F})$ when there is no confusion.

We use a lowercase Fraktur letter to denote the Lie algebra of its corresponding algebraic group. For example, we write ${\mathfrak{g}}:={\mathrm{Lie}}(G)$ . Denote by ${\mathrm{Ad}}$ the adjoint action of G on itself or ${\mathfrak{g}}$ . If G acts on an F-variety V and $X\in V(F)$ , denote by $G_X$ the centraliser of X in G. If ${\mathfrak{v}}$ is an F-subvariety of ${\mathfrak{g}}$ , denote by ${\mathfrak{v}}_X$ the centraliser of $X\in{\mathfrak{g}}(F)$ in ${\mathfrak{v}}$ . If $\theta$ is an automorphism on G, denote by $G^\theta$ the subgroup of $\theta$ -fixed points of G.

Fix a Levi F-factor $M_0$ of a minimal parabolic F-subgroup of G. By a Levi subgroup of G, we mean a Levi F-factor of some parabolic F-subgroup of G. For a semi-standard (namely containing $M_0$ ) Levi subgroup M of G, denote by $\mathscr{F}^G(M)$ , $\mathscr{P}^G(M)$ and $\mathscr{L}^G(M)$ the sets of parabolic F-subgroups of G containing M, parabolic F-subgroups of G with Levi factor M and Levi subgroups of G containing M, respectively. For $P\in\mathscr{F}^G(M_0)$ , denote by $M_P$ the unique Levi factor containing $M_0$ and by $N_P$ the unipotent radical. Let $\overline{P}$ be the parabolic subgroup opposite to P.

For $M\in\mathscr{L}^G(M_0)$ , define the Weyl group of (G,M) by

\begin{align*} W^G(M):={\mathrm{Norm}}_{G(F)}(M)/M(F). \end{align*}

In particular, we also write $W_0^G:=W^G(M_0)$ . For $M, L\in\mathscr{L}^G(M_0)$ , denote

\begin{align*} {\mathrm{Tran}}_G(M,L):=\{w\in W_0^L\backslash W_0^G: {\mathrm{Ad}}(w)(M)\subseteq L\}. \end{align*}

Denote by $A_G$ the maximal F-split central torus of G. Let $X(G)_F$ be the group of F-rational characters of G. Define the ${\mathbb {R}}$ -linear space

\begin{align*} {\mathfrak{a}}_G:={\mathrm{Hom}}_{\mathbb {Z}}(X(G)_F,{\mathbb {R}}), \end{align*}

whose dual space is denoted by ${\mathfrak{a}}_G^\ast$ . Fix a scalar product on ${\mathfrak{a}}_{M_0}$ which is invariant under the action of $W_0^G$ , from which we deduce Haar measures on all subspaces of ${\mathfrak{a}}_{M_0}$ . Denote by ${\mathfrak{a}}_M^G$ the orthogonal complement of ${\mathfrak{a}}_G$ in ${\mathfrak{a}}_M$ .

Let D be a central division algebra over F. Denote by $\deg(D)$ the degree of D, i.e. $\dim_F(D)=\deg(D)^2$ . Denote by $GL_{n, D}$ the reductive group over F whose F-points are $GL_n(D)$ . For $x\in {\mathfrak{g}}{\mathfrak{l}}_n(D)$ , we write ${\mathrm{Nrd}} (x), {\mathrm{Trd}} (x)$ and ${\mathrm{Prd}}_x$ for its reduced norm, reduced trace and reduced characteristic polynomial, respectively. If $D=F$ , we also write them as $\det(x)$ , ${\mathrm{tr}}(x)$ and $\chi_x$ , respectively.

2.1.2.

Now suppose that F is a local field of characteristic zero. Denote by $|\cdot|_F$ the normalised absolute value on F. Define a homomorphism $H_G: G(F)\rightarrow{\mathfrak{a}}_G$ by

\begin{align*} \langle H_G(x),\chi\rangle=\log(|\chi(x)|_F) \end{align*}

for all $x\in G(F)$ and $\chi\in X(G)_F$ . Fix a maximal compact subgroup $K=K_G$ of G(F), which is admissible relative to $M_0$ in the sense of [Reference ArthurArt81, p. 9]. Unless otherwise stated (see § 7.3), we shall choose the standard maximal compact subgroup when $G(F)=GL_n(D)$ , where D is a central division algebra over a finite field extension of F. That is to say, if F is non-archimedean, then $K=GL_n({\mathcal {O}}_D)$ with ${\mathcal{O}}_D$ being the ring of integers of D (see [Reference WeilWei95, p. 191]), while if F is archimedean, K is the unitary group with respect to some hermitian form (see [Reference WeilWei95, p. 199]). For $P\in\mathscr{F}^G(M_0)$ , we may extend the function $H_{M_P}$ to a map $H_P: G(F)\rightarrow{\mathfrak{a}}_{M_P}$ using the decomposition $G(F)=M_P(F)N_P(F)K$ .

Fix the Haar measure on K such that ${\mathrm{vol}}(K)=1$ . In particular, we emphasise that this convention applies to the case where G is a torus. For $P\in\mathscr{F}^G(M_0)$ , fix a Haar measure on $N_P(F)$ such that

\begin{align*} \int_{N_P(F)} \exp (2\rho_{\overline{P}}(H_{\overline{P}}(n))) \,dn =1, \end{align*}

where $\rho_{\overline{P}}$ is the half of the sum of roots (with multiplicity) associated to $\overline{P}$ . We equip ${\mathfrak{n}}_P(F)$ with the corresponding Haar measure via the exponential map. For $M\in\mathscr{L}^G(M_0)$ , there are compatible Haar measures on G(F) and M(F) in the sense of [Reference ArthurArt91, (1.1), p. 12] such that, for all $P\in\mathscr{P}^G(M)$ and all continuous and compactly supported functions f on G(F), we have the equality

\begin{align*} \int_{G(F)} f(x) \,dx = \int_{M(F)\times N_P(F)\times K} f(mnk) \,dk \, dn \, dm. \end{align*}

We shall choose such measures.

Let V be an F-linear space of finite dimension. If F is non-archimedean, denote by ${\mathcal {C}}_c^\infty(V)={\mathcal {S}}(V)$ the space of locally constant, compactly supported and complex-valued functions on V. If F is archimedean, denote by ${\mathcal {S}}(V)$ the space of Schwartz functions on V. For $f\in{\mathcal {S}}(V)$ , denote by ${\mathrm{Supp}}(f)$ its support.

Fix a continuous and nontrivial unitary character $\psi: F\rightarrow{\mathbb {C}}^\times$ . Let $\langle\cdot,\cdot\rangle$ be a non-degenerate symmetric bilinear form on ${\mathfrak{g}}(F)$ , which is invariant under conjugation. Let ${\mathfrak{v}}$ be an F-linear subspace of ${\mathfrak{g}}(F)$ , on which the restriction of $\langle\cdot,\cdot\rangle$ is non-degenerate; examples for such ${\mathfrak{v}}$ include ${\mathfrak{m}}(F)$ with $M\in\mathscr{L}^G(M_0)$ . It is equipped with the unique self-dual Haar measure with respect to $\psi(\langle\cdot,\cdot\rangle)$ . For $f\in{\mathcal {S}}({\mathfrak{v}})$ , define its Fourier transform $\hat{f}\in{\mathcal {S}}({\mathfrak{v}})$ by

\begin{align*} \forall X\in {\mathfrak{v}}, \hat{f}(X):=\int_{{\mathfrak{v}}} f(Y) \psi(\langle X,Y \rangle) \, dY. \end{align*}

Let $M\in\mathscr{L}^G(M_0)$ and $Q\in\mathscr{F}^G(M)$ . For $x\in G(F)$ , we define

\begin{align*} v_P(\lambda,x):=e^{-\lambda(H_P(x))}, \forall \lambda\in i{\mathfrak{a}}_M^\ast, P\in\mathscr{P}^G(M). \end{align*}

By [Reference ArthurArt81, p. 40–41], this is a (G,M)-family in the sense of [Reference ArthurArt81, p. 36]. By [Reference ArthurArt81, Lemma 6.2], we obtain Arthur’s weight function

(2.1) \begin{equation} v_M^Q(x):=\lim_{\lambda\rightarrow 0} \sum_{\{P\in\mathscr{P}^G(M): P\subseteq Q\}} v_P(\lambda,x) \theta_P^Q(\lambda)^{-1}, \forall x\in G(F),\end{equation}

where $\theta_P^Q(\lambda)$ is defined in [Reference ArthurArt81, p. 15].

Given an F-linear space of finite dimension equipped with a non-degenerate symmetric bilinear form $q(\cdot,\cdot)$ and a Haar measure, we denote by $\gamma_\psi(q)$ the Weil constant given in [Reference WeilWei64, § 14, Théorème 2].

2.2.1.

Let F be a local field of characteristic zero or a number field. A symmetric pair in the sense of [Reference Aizenbud and GourevitchAG09, Definition 7.1.1] is a triple $(G,H,\theta)$ , where $H\subseteq G$ are a pair of reductive groups and $\theta$ is an involution of G such that $H=G^\theta$ . Let ${\mathfrak{s}}$ be the tangent space at the neutral element of the symmetric space $S:=G/H$ . We shall always view ${\mathfrak{s}}$ as a subspace of ${\mathfrak{g}}$ . Thus,

\begin{align*} {\mathfrak{s}}=\{X\in{\mathfrak{g}}:(d\theta)(X)=-X\}, \end{align*}

on which H acts by the restriction of ${\mathrm{Ad}}$ . By [Reference Aizenbud and GourevitchAG09, Lemma 7.1.9], there exists a G-invariant $\theta$ -invariant non-degenerate symmetric bilinear form on ${\mathfrak{g}}$ .

An element $X\in{\mathfrak{s}}$ is said to be semi-simple if ${\mathrm{Ad}}(H)(X)$ is Zariski closed in ${\mathfrak{s}}$ . If F is a local field of characteristic zero, $X\in{\mathfrak{s}}(F)$ is semi-simple if and only if ${\mathrm{Ad}}(H(F))(X)$ is closed in ${\mathfrak{s}}(F)$ in the analytic topology by [Reference Rader and RallisRR96, Fact A, p. 108–109]. We say that an element $X\in{\mathfrak{s}}$ is regular if $H_X$ has minimal dimension. Denote by ${\mathfrak{s}}_{\mathrm{rs}}$ the subset of ${\mathfrak{s}}$ consisting of regular semi-simple elements in ${\mathfrak{s}}$ .

2.2.2.

Now suppose that F is a local field of characteristic zero. A Cartan subspace of ${\mathfrak{s}}$ is defined as a maximal abelian subspace ${\mathfrak{c}}\subseteq{\mathfrak{s}}$ defined over F consisting of semi-simple elements. Denote by $\mathscr{T}^{\mathfrak{s}}$ the set of Cartan subspaces of ${\mathfrak{s}}$ . Fix a (finite) set of representatives $\mathscr{T}_0^{\mathfrak{s}}$ for H(F)-conjugacy classes in $\mathscr{T}^{\mathfrak{s}}$ . Let ${\mathfrak{c}}\in\mathscr{T}^{\mathfrak{s}}$ . We write ${\mathfrak{c}}_{\mathrm{reg}}:={\mathfrak{c}}\cap{\mathfrak{s}}_{\mathrm{rs}}$ . Denote by $T_{\mathfrak{c}}$ the centraliser of ${\mathfrak{c}}$ in H. Define the Weyl group

(2.2) \begin{equation} W(H,{\mathfrak{c}}):={\mathrm{Norm}}_{H(F)}({\mathfrak{c}})/T_{\mathfrak{c}}(F).\end{equation}

For ${\mathfrak{c}}_1, {\mathfrak{c}}_2\in\mathscr{T}^{\mathfrak{s}}$ , denote by $W(H,{\mathfrak{c}}_1,{\mathfrak{c}}_2)$ the set of isomorphisms from ${\mathfrak{c}}_1$ onto ${\mathfrak{c}}_2$ induced by ${\mathrm{Ad}}(x)$ for some $x\in H(F)$ . In particular, $W(H,{\mathfrak{c}},{\mathfrak{c}})$ is nothing but $W(H,{\mathfrak{c}})$ (viewed as a set). For $X\in {\mathfrak{c}}(F)$ , define the Weyl discriminant factor

\begin{align*} |D^{\mathfrak{s}}(X)|_F:=|\det({\mathrm{ad}}(X)|_{{\mathfrak{h}}/{\mathfrak{t}}_{\mathfrak{c}}\oplus{\mathfrak{s}}/{\mathfrak{c}}})|_F^{1/2}. \end{align*}

Let $\langle\cdot,\cdot\rangle$ be a G-invariant $\theta$ -invariant non-degenerate symmetric bilinear form on ${\mathfrak{g}}$ . For any F-linear subspace ${\mathfrak{v}}$ of ${\mathfrak{g}}(F)$ such that the restriction of $\langle \cdot,\cdot\rangle$ on ${\mathfrak{v}}$ is non-degenerate, denote by $\gamma_\psi({\mathfrak{v}})$ the Weil constant associated to ${\mathfrak{v}}$ (see the end of § 2.1.2). Let ${\mathfrak{c}}\in\mathscr{T}^{\mathfrak{s}}$ . For $X,Y\in({\mathfrak{c}}\cap{\mathfrak{s}}_{\mathrm{rs}})(F)$ , we define a bilinear form $q_{X,Y}$ on ${\mathfrak{h}}(F)/{\mathfrak{t}}_{{\mathfrak{c}}}(F)$ by

\begin{align*} q_{X,Y}(Z,Z'):=\langle [Z,X],[Y,Z'] \rangle. \end{align*}

It is non-degenerate and symmetric and we have $q_{X,Y}=q_{Y,X}$ . Write

(2.3) \begin{equation}\gamma_\psi(X,Y):=\gamma_\psi(q_{X,Y}).\end{equation}

2.3.1.

We refer to [Reference LiLi22, Reference LiLi23] for more details of the following facts about the first symmetric pair.

2.3.2.

Let F be a local field of characteristic zero or a number field. Let $G=G_n:=GL_{2n}$ and denote by $H=H_n:=GL_n\times GL_n$ its subgroup via diagonal embedding. In fact, H is the subgroup of fixed points of the involution ${\mathrm{Ad}}(\omega_0)$ on G, where

\begin{align*} \omega_0:=\bigg(\begin{matrix}1_n & \\ & -1_n\end{matrix}\bigg)\in G(F). \end{align*}

We also write ${\mathfrak{s}}_n:={\mathfrak{s}}$ . We shall embed G into ${\mathfrak{g}}$ in the standard way. For an F-subvariety ${\mathfrak{v}}$ of ${\mathfrak{g}}$ , we write ${\mathfrak{v}}^\times:={\mathfrak{v}}\cap G$ . From [Reference ZhangZha15, Proposition 4.2], we know ${\mathfrak{s}}_{\mathrm{rs}}\subseteq{\mathfrak{s}}^\times$ in this case. Let $\langle\cdot,\cdot\rangle$ be the non-degenerate symmetric bilinear form on ${\mathfrak{g}}(F)$ defined by

(2.4) \begin{equation} \langle X,Y\rangle:={\mathrm{tr}}(XY),\forall X,Y\in{\mathfrak{g}}(F),\end{equation}

which is invariant under the adjoint action of G(F) and ${\mathrm{Ad}}(\omega_0)$ .

Let $M_0$ be the group of diagonal matrices in G. Set

\begin{align*} \omega:=\bigg(\begin{matrix}0 & 1_n \\ 1_n & 0\end{matrix}\bigg)\in G(F). \end{align*}

For $P\in\mathscr{F}^G(M_0)$ , we say that P is ‘ $\omega$ -stable’ if $\omega\in P$ . Denote by $\mathscr{F}^{G,\omega}(M_0)$ the subset of $\mathscr{F}^G(M_0)$ consisting of $\omega$ -stable parabolic subgroups. For $M\in\mathscr{L}^G(M_0)$ , we say that M is ‘ $\omega$ -stable’ if $M=M_P$ for some $P\in\mathscr{F}^{G,\omega}(M_0)$ . Denote by $\mathscr{L}^{G,\omega}(M_0)$ the subset in $\mathscr{L}^G(M_0)$ consisting of $\omega$ -stable Levi subgroups. Let $A_n$ be the group of diagonal matrices in $GL_n$ . There is a bijection (cf. [Reference LiLi22, Proposition 5.2]) between $\mathscr{L}^{GL_n}(A_n)$ and $\mathscr{L}^{G,\omega}(M_0)$ induced by

(2.5) \begin{equation} M_n\mapsto M=\bigg(\begin{matrix}{\mathfrak{m}}_n & {\mathfrak{m}}_n \\ {\mathfrak{m}}_n & {\mathfrak{m}}_n\end{matrix}\bigg)^{\!\times}.\end{equation}

Given $M\in\mathscr{L}^{G,\omega}(M_0)$ , we shall always write $M_n$ for the preimage of M under this bijection. Note that if $M\in\mathscr{L}^{G,\omega}(M_0)$ , for $L\in\mathscr{L}^G(M)$ and $Q\in\mathscr{F}^G(M)$ , we have $L\in\mathscr{L}^{G,\omega}(M_0)$ and $Q\in\mathscr{F}^{G,\omega}(M_0)$ . There is also a bijection (cf. [Reference LiLi22, Proposition 5.2]) between $\mathscr{F}^{GL_n}(A_n)$ and $\mathscr{F}^{G,\omega}(M_0)$ induced by

\begin{align*} P_n\mapsto P=\bigg(\begin{matrix}{\mathfrak{p}}_n & {\mathfrak{p}}_n \\ {\mathfrak{p}}_n & {\mathfrak{p}}_n\end{matrix}\bigg)^{\!\times}. \end{align*}

Given $P\in\mathscr{F}^{G,\omega}(M_0)$ , we shall always write $P_n$ for the preimage of P under this bijection. For ${\mathfrak{c}}\in\mathscr{T}^{\mathfrak{s}}$ and $L\in\mathscr{L}^{G,\omega}(M_0)$ , define

(2.6) \begin{equation} W(H,{\mathfrak{c}},{\mathfrak{l}}\cap{\mathfrak{s}}):=\bigsqcup_{{\mathfrak{c}}_2\in\mathscr{T}_0^{{\mathfrak{l}}\cap{\mathfrak{s}}}} W(H,{\mathfrak{c}},{\mathfrak{c}}_2).\end{equation}

Let $M\in\mathscr{L}^{G,\omega}(M_0)$ . We say that an element $X\in({\mathfrak{m}}\cap{\mathfrak{s}}_{\mathrm{rs}})(F)$ (respectively a Cartan subspace ${\mathfrak{c}}\subseteq{\mathfrak{m}}\cap{\mathfrak{s}}$ ) is M-elliptic if $A_M$ is the maximal F-split torus in the torus $H_X$ (respectively in $T_{\mathfrak{c}}$ ). Denote by $({\mathfrak{m}}\cap{\mathfrak{s}}_{\mathrm{rs}})(F)_\mathrm{ell}$ the set of M-elliptic elements in $({\mathfrak{m}}\cap{\mathfrak{s}}_{\mathrm{rs}})(F)$ . In particular, ${\mathfrak{s}}_{\mathrm{rs}}(F)_\mathrm{ell}$ is the set of G-elliptic elements in ${\mathfrak{s}}_{\mathrm{rs}}(F)$ . Write $M_H:=M\cap H$ . Denote by $\Gamma_\mathrm{ell}(({\mathfrak{m}}\cap{\mathfrak{s}}_{\mathrm{rs}})(F))$ the set of $M_H(F)$ -conjugacy classes in $({\mathfrak{m}}\cap{\mathfrak{s}}_{\mathrm{rs}})(F)_\mathrm{ell}$ . Denote by $\mathscr{T}_\mathrm{ell}^{{\mathfrak{m}}\cap{\mathfrak{s}}}$ the subset of $\mathscr{T}_0^{{\mathfrak{m}}\cap{\mathfrak{s}}}$ consisting of representatives that are M-elliptic Cartan subspaces. For $X\in{\mathfrak{s}}_{\mathrm{rs}}(F)$ , $X\in{\mathfrak{s}}_{\mathrm{rs}}(F)_\mathrm{ell}$ if and only if $\chi_X(\lambda)=p(\lambda^2)$ for some irreducible polynomial $p(\lambda)\in F[\lambda]$ of degree n.

2.3.3.

Now suppose that F is a local field of characteristic zero. Let $P\in\mathscr{F}^{G,\omega}(M_0)$ . Then

\begin{align*} {\mathfrak{m}}_P=\bigg(\begin{matrix}{\mathfrak{m}}_n & {\mathfrak{m}}_n \\ {\mathfrak{m}}_n & {\mathfrak{m}}_n\end{matrix}\bigg) {\kern-.5pt}\quad\text{and}\quad {\mathfrak{n}}_P=\bigg(\begin{matrix}{\mathfrak{n}}_n & {\mathfrak{n}}_n \\ {\mathfrak{n}}_n & {\mathfrak{n}}_n\end{matrix}\bigg), \end{align*}

where we denote $M_n:=M_{P_n}$ and $N_n:=N_{P_n}$ . We shall choose the same Haar measure for any of the four copies in ${\mathfrak{m}}_P(F)$ or ${\mathfrak{n}}_P(F)$ under these identifications. For $f\in{\mathcal {S}}({\mathfrak{s}}(F))$ , we define a function $f_P^\eta\in{\mathcal {S}}(({\mathfrak{m}}_P\cap{\mathfrak{s}})(F))$ by

\begin{align*} f_P^\eta(Z):=\int_{K_H\times({\mathfrak{n}}_P\cap{\mathfrak{s}})(F)} f({\mathrm{Ad}}(k^{-1})(Z+U)) \eta(\det(k)) \, dU \, dk \end{align*}

for all $Z\in({\mathfrak{m}}_P\cap{\mathfrak{s}})(F)$ . Recall that $(\hat{f})_P^\eta=\widehat{f_P^\eta}$ , and we shall denote it by $\hat{f}_P^\eta$ without confusion.

Let $M\in\mathscr{L}^{G,\omega}(M_0)$ and $Q\in\mathscr{F}^G(M)$ . For $f\in {\mathcal {C}}_c^\infty({\mathfrak{s}}(F))$ and $X\in ({\mathfrak{m}}\cap{\mathfrak{s}}_{\mathrm{rs}})(F)$ , define the weighted orbital integral

(2.7) \begin{equation} J_M^Q(\eta, X, f):=|D^{\mathfrak{s}}(X)|_F^{1/2} \int_{H_X(F)\backslash H(F)} f({\mathrm{Ad}}(x^{-1})(X)) \eta(\det(x)) v_M^Q(x) \, dx.\end{equation}

In particular, if $M=G$ , then it is reduced to the usual normalised relative orbital integral twisted by $\eta\circ\det$ . For $X=({\kern-1pt}\begin{smallmatrix}0 & A \\[1pt] B & 0\end{smallmatrix}{\kern-1pt})\in{\mathfrak{s}}_{\mathrm{rs}}(F)$ , we define a transfer factor (see [Reference ZhangZha15, Definition 5.7])

(2.8) \begin{equation} \kappa(X):=\eta(\det(A)),\end{equation}

which satisfies

\begin{align*} \kappa({\mathrm{Ad}}(x^{-1})(X))=\eta(\det(x))\kappa(X) \end{align*}

for all $x\in H(F)$ . By [Reference LiLi23, Proposition 4.1(3) and (4)], the function $\kappa(\cdot)J_M^Q(\eta, \cdot, f)$ is constant on ${\mathrm{Ad}}(M_H(F))(X)$ .

2.3.4.

Now suppose additionally that F is non-archimedean. Let $M\in\mathscr{L}^{G,\omega}(M_0)$ and $X\in ({\mathfrak{m}}\cap{\mathfrak{s}}_{\mathrm{rs}})(F)$ . In [Reference LiLi23, § 8.1], we deduce from $J_M^G(\eta,X,\cdot)$ an $(H,\eta)$ -invariant distribution $I_M^G(\eta, X, \cdot)$ on ${\mathfrak{s}}(F)$ inductively such that their respective Fourier transforms $\hat{J}_M^G(\eta,X,\cdot)$ and $\hat{I}_M^G(\eta,X,\cdot)$ satisfy

\begin{equation*} \hat{I}_M^G(\eta, X, f)=\hat{J}_M^G(\eta, X, f)-\sum_{L\in\mathscr{L}^G(M), L\neq G} \hat{I}_M^{L,G,w}(\eta, X, f).\end{equation*}

Here $\hat{I}_M^{L,G,w}(\eta, X, \cdot)$ denotes the $(H,\eta)$ -invariant distribution on ${\mathfrak{s}}(F)$ defined by the weighted induction (see [Reference LiLi23, (7.1.10)] for more details) of the $(L_H, \eta)$ -invariant distribution $\hat{I}_M^L(\eta, X, \cdot)$ on $({\mathfrak{l}}\cap{\mathfrak{s}})(F)$ , which is a product analogue of $\hat{I}_M^G(\eta,X,\cdot)$ in lower ranks. By [Reference LiLi23, Propositions 7.2 and 8.1], there are unique locally constant functions $\hat{j}_M^G(\eta,X,\cdot)$ and $\hat{i}_M^G(\eta,X,\cdot)$ on ${\mathfrak{s}}_{\mathrm{rs}}(F)$ representing $\hat{J}_M^G(\eta,X,\cdot)$ and $\hat{I}_M^G(\eta,X,\cdot)$ , respectively. That is to say, for all $f\in{\mathcal {C}}_c^\infty({\mathfrak{s}}(F))$ , we have

\begin{align*} \hat{J}_M^G(\eta, X, f):=J_M^G(\eta, X, \hat{f})=\int_{{\mathfrak{s}}(F)} f(U) \hat{j}_M^G(\eta, X, U) |D^{\mathfrak{s}}(U)|_F^{-1/2} \, dU \end{align*}

and

\begin{align*} \hat{I}_M^G(\eta, X, f):=I_M^G(\eta, X, \hat{f})=\int_{{\mathfrak{s}}(F)} f(U) \hat{i}_M^G(\eta, X, U) |D^{\mathfrak{s}}(U)|_F^{-1/2} \, dU. \end{align*}

2.4.1.

We refer to [Reference LiLi21, Reference LiLi23] for more details of the following facts about the second symmetric pair.

2.4.2.

Let F be a local field of characteristic zero or a number field. Let E be a quadratic extension of F. Let ${\mathfrak{g}}'$ be a central simple algebra over F with a fixed F-algebra embedding $E\hookrightarrow{\mathfrak{g}}'(F)$ . Let ${\mathfrak{h}}':={\mathrm{Cent}}_{{\mathfrak{g}}'}(E)$ be the centraliser of E in ${\mathfrak{g}}'$ . Then ${\mathfrak{h}}'(F)$ is a central simple algebra over E by the double centraliser theorem. Denote by $G':={{\mathfrak{g}}'}^\times$ (respectively $H':={{\mathfrak{h}}'}^\times$ ) the group of invertible elements in ${\mathfrak{g}}'$ (respectively ${\mathfrak{h}}'$ ). Let $\alpha\in E\backslash F$ be such that $\alpha^2\in F$ , so $E=F(\alpha)$ . In fact, H′ is the subgroup of fixed points of the involution ${\mathrm{Ad}}(\alpha)$ on G′. Denote by ${\mathfrak{s}}'$ the corresponding tangent space of $G'/H'$ at the neutral element. For an F-subvariety ${\mathfrak{v}}'\subseteq{\mathfrak{g}}'$ , we write ${{\mathfrak{v}}'}^\times:={\mathfrak{v}}'\cap G'$ . By the base change to $\overline{F}$ , we see ${\mathfrak{s}}'_{\mathrm{rs}}\subseteq{{\mathfrak{s}}'}^\times$ in this case. Let $\langle\cdot,\cdot\rangle$ be the non-degenerate symmetric bilinear form on ${\mathfrak{g}}'(F)$ defined by

(2.9) \begin{equation} \langle X,Y\rangle:={\mathrm{Trd}}(XY),\forall X,Y\in{\mathfrak{g}}'(F),\end{equation}

which is invariant under the adjoint action of G′(F) and ${\mathrm{Ad}}(\alpha)$ .

By the Wedderburn–Artin theorem, G′ is isomorphic to $GL_{r,D}$ , where $r:=\mathrm{rk}_F(G')$ for some central division algebra D over F such that $r\deg(D)$ is even. By the Noether–Skolem theorem, up to conjugation by G′(F), the embedding $H'\hookrightarrow G'$ is isomorphic to one of the two cases below (see [Reference ChommauxCho19, §§ 2.1 and 3.1] and [Reference LiLi21, § 3.4]).

Case I. If there is an embedding $E\rightarrow D$ as F-algebras, then

\begin{align*} (G',H')\simeq(GL_{r,D},{\mathrm{Res}}_{E/F} GL_{r,D'}), \end{align*}

where $D':={\mathrm{Cent}}_D(E)$ is a central division algebra over E of degree ${\deg(D)}/{2}$ . Let $M'_0\simeq ({\mathrm{Res}}_{E/F} {\mathbb {G}}_{m,D'})^r$ (respectively $M'_{\widetilde{0}}\simeq ({\mathbb {G}}_{m,D})^r$ ) be the subgroup of diagonal elements in H′ (respectively G′). For $M'\in\mathscr{L}^{H'}(M'_0)$ , denote by $\widetilde{M'}$ the unique (see the discussion after [Reference LiLi21, Proposition 3.7]) element in $\mathscr{L}^{G'}(M'_{\widetilde{0}})$ such that $\widetilde{M'}\cap H'=M'$ . The map $M'\mapsto\widetilde{M'}$ induces a bijection between $\mathscr{L}^{H'}(M'_0)$ and $\mathscr{L}^{G'}(M'_{\widetilde{0}})$ , and we can identify $A_{M'}$ with $A_{\widetilde{M'}}$ .

Case II. If there is no embedding $E\rightarrow D$ as F-algebras, then

\begin{align*} (G',H')\simeq(GL_{r,D},{\mathrm{Res}}_{E/F} GL_{{r}/{2},D\otimes_F E}), \end{align*}

where $D\otimes_F E$ is a central division algebra over E of degree $\deg(D)$ . Let $M'_0\simeq ({\mathrm{Res}}_{E/F} {\mathbb {G}}_{m,D\otimes_FE})^{{r}/{2}}$ (respectively $M'_{\widetilde{0}}\simeq ({\mathbb{G}}_{m,D})^r$ ) be the subgroup of diagonal elements in H′ (respectively G′). Denote by $\mathscr{L}^{G'}(M'_{\widetilde{0}}, M'_0)$ the subset of elements in $\mathscr{L}^{G'}(M'_{\widetilde{0}})$ containing $M'_0$ . For $M'\in\mathscr{L}^{H'}(M'_0)$ , denote by $\widetilde{M'}$ the unique (see the discussion after [Reference LiLi21, Proposition 3.7]) element in $\mathscr{L}^{G'}(M'_{\widetilde{0}}, M'_0)$ such that $\widetilde{M'}\cap H'=M'$ . The map $M'\mapsto\widetilde{M'}$ induces a bijection between $\mathscr{L}^{H'}(M'_0)$ and $\mathscr{L}^{G'}(M'_{\widetilde{0}}, M'_0)$ , and we can identify $A_{M'}$ with $A_{\widetilde{M'}}$ .

If $\mathrm{rk}_F(G')=r$ , we also write $G'_r:=G'$ , $H'_r:=H'$ and ${\mathfrak{s}}'_r:={\mathfrak{s}}'$ . Note that $\mathrm{rk}_F(H'_r)=r$ in Case I (respectively ${r}/{2}$ in Case II). For $P'\in\mathscr{F}^{H'}(M'_0)$ , denote by $\widetilde{P'}$ the unique element in $\mathscr{F}^{G'}(\widetilde{M'_0})$ such that $\widetilde{P'}\cap H'=P'$ . The map $P'\mapsto\widetilde{P'}$ induces a bijection between $\mathscr{F}^{H'}(M'_0)$ and $\mathscr{F}^{G'}(\widetilde{M'_0})$ in both Case I and Case II. Let $\tau\in D^\times$ in Case I (respectively $\tau\in GL_2(D)$ in Case II) be an element such that ${\mathrm{Ad}}(\alpha)(\tau)=-\tau$ . Let $P'\in\mathscr{F}^{H'}(M'_0)$ . By [Reference LiLi21, Proposition 3.12], we have

\begin{align*} {\mathfrak{m}}_{\widetilde{P'}}\cap{\mathfrak{s}}'={\mathfrak{m}}_{P'}\tau=\tau{\mathfrak{m}}_{P'} \end{align*}

and

\begin{align*}{\mathfrak{n}}_{\widetilde{P'}}\cap{\mathfrak{s}}'={\mathfrak{n}}_{P'}\tau=\tau{\mathfrak{n}}_{P'}.\end{align*}

For ${\mathfrak{c}}'\in\mathscr{T}^{{\mathfrak{s}}'}$ and $L'\in\mathscr{L}^{H'}(M'_0)$ , define

(2.10) \begin{equation} W(H',{\mathfrak{c}}',\widetilde{{\mathfrak{l}}'}\cap{\mathfrak{s}}'):=\bigsqcup_{{\mathfrak{c}}'_2\in\mathscr{T}_0^{\widetilde{{\mathfrak{l}}'}\cap{\mathfrak{s}}'}} W(H',{\mathfrak{c}}',{\mathfrak{c}}'_2).\end{equation}

Let $M'\in\mathscr{L}^{H'}(M'_0)$ . We say that an element $Y\in(\widetilde{{\mathfrak{m}}'}\cap{\mathfrak{s}}'_{\mathrm{rs}})(F)$ (respectively a Cartan subspace ${\mathfrak{c}}'\subseteq\widetilde{{\mathfrak{m}}'}\cap{\mathfrak{s}}'$ ) is M′-elliptic if $A_{M'}$ is the maximal F-split torus in the torus $H'_Y$ (respectively in $T_{{\mathfrak{c}}'}$ ). Denote by $(\widetilde{{\mathfrak{m}}'}\cap{\mathfrak{s}}'_{\mathrm{rs}})(F)_\mathrm{ell}$ the set of M′-elliptic elements in $(\widetilde{{\mathfrak{m}}'}\cap{\mathfrak{s}}'_{\mathrm{rs}})(F)$ . In particular, ${\mathfrak{s}}'_{\mathrm{rs}}(F)_\mathrm{ell}$ is the set of H′-elliptic elements in ${\mathfrak{s}}'_{\mathrm{rs}}(F)$ . Denote by $\Gamma_\mathrm{ell}((\widetilde{{\mathfrak{m}}'}\cap{\mathfrak{s}}'_{\mathrm{rs}})(F))$ the set of M′(F)-conjugacy classes in $(\widetilde{{\mathfrak{m}}'}\cap{\mathfrak{s}}'_{\mathrm{rs}})(F)_\mathrm{ell}$ . Denote by $\mathscr{T}_\mathrm{ell}^{\widetilde{{\mathfrak{m}}'}\cap{\mathfrak{s}}'}$ the subset of $\mathscr{T}_0^{\widetilde{{\mathfrak{m}}'}\cap{\mathfrak{s}}'}$ consisting of representatives that are M′-elliptic Cartan subspaces.

Proof. For $Y\in{\mathfrak{s}}'_{\mathrm{rs}}(F)$ , by the base change to an algebraic closure of F and reduction to the case of § 2.3, we obtain (cf. the proof of [Reference LiLi21, Proposition 3.3])

\begin{align*} {\mathrm{Prd}}_Y(\lambda)=p(\lambda^2), \end{align*}

where $p\in F[\lambda]$ denotes the reduced characteristic polynomial of $Y^2\in {\mathfrak{h}}'(F)$ (viewed as a central simple algebra over E). Moreover, $Y^2$ is a regular semi-simple element in ${\mathfrak{h}}'(F)$ in the usual sense. Then every element in the centraliser of $Y^2$ in ${\mathfrak{h}}'(F)$ can be written as a polynomial of $Y^2$ and thus commutes with Y. It follows that $H'_{Y^2}(F)\subset H'_Y(F)$ . Since it is clear that $H'_Y\subset H'_{Y^2}$ , we deduce that the tori $H'_Y$ and $H'_{Y^2}$ have the same F-split rank.

By definition of H′-elliptic elements, $Y\in{\mathfrak{s}}'_{\mathrm{rs}}(F)_\mathrm{ell}$ if and only if the F-split rank of $H'_Y$ is 1. In this case, the F-split rank of $H'_{Y^2}$ is 1. Since $F[\lambda]/(p)\simeq F[Y^2]$ is included in the centraliser of $Y^2$ in ${\mathfrak{h}}'(F)$ , this implies that p is irreducible in $F[\lambda]$ .

By [Reference LiLi21, Lemma 9.1], if $Y\notin{\mathfrak{s}}'_{\mathrm{rs}}(F)_\mathrm{ell}$ , then Y is H′(F)-conjugate to an element

\begin{align*} Z={\mathrm{diag}}(Z_1, \ldots, Z_\ell)\in(\widetilde{{\mathfrak{m}}'}\cap{\mathfrak{s}}')(F)\simeq\bigoplus_{i=1}^\ell {\mathfrak{s}}'_{r_i}(F) \end{align*}

for some $M'\in\mathscr{L}^{H'}(M'_0), M'\subsetneq H'$ , where $\ell\geqslant2$ and $1\leqslant r_i\lt r$ for all $1\leqslant i\leqslant\ell$ verify $\sum\nolimits_{i=1}^\ell r_i=r$ . Thus,

\begin{align*}{\mathrm{Prd}}_Y={\mathrm{Prd}}_Z=\prod\limits_{i=1}^\ell {\mathrm{Prd}}_{Z_i}, \end{align*}

with ${\mathrm{Prd}}_{Z_i}(\lambda)=p_i(\lambda^2)$ , where $p_i\in F[\lambda]$ is the reduced characteristic polynomial of $Z_i^2\in{\mathfrak{h}}'_{r_i}(F)$ . It follows that $p=\prod\nolimits_{i=1}^\ell p_i$ is not irreducible in $F[\lambda]$ .

2.4.3.

Now suppose that F is a local field of characteristic zero. Let $P'\in\mathscr{F}^{H'}(M'_0)$ . We shall choose the same Haar measures on $({\mathfrak{m}}_{\widetilde{P'}}\cap{\mathfrak{s}}')(F)$ and $({\mathfrak{n}}_{\widetilde{P'}}\cap{\mathfrak{s}}')(F)$ as those on ${\mathfrak{m}}_{P'}(F)$ and ${\mathfrak{n}}_{P'}(F)$ , respectively, using the above identifications induced by $\tau$ . Such Haar measures depend on the choice of $\tau$ . For $f'\in{\mathcal {C}}_c^\infty({\mathfrak{s}}'(F))$ , we define a function $f'_{P'}\in{\mathcal {C}}_c^\infty(({\mathfrak{m}}_{\widetilde{P'}}\cap{\mathfrak{s}}')(F))$ by

\begin{align*} f'_{P'}(Z):=\int_{K_{H'}\times({\mathfrak{n}}_{\widetilde{P'}}\cap{\mathfrak{s}}')(F)} f'({\mathrm{Ad}}(k^{-1})(Z+U)) \, dU \, dk \end{align*}

for all $Z\in({\mathfrak{m}}_{\widetilde{P'}}\cap{\mathfrak{s}}')(F)$ . Recall that $(\hat{f'})_{P'}=\widehat{f'_{P'}}$ , and we shall denote it by $\hat{f}'_{P'}$ without confusion.

Let $M'\in\mathscr{L}^{H'}(M'_0)$ and $Q'\in\mathscr{F}^{H'}(M')$ . For $f'\in {\mathcal {C}}_c^\infty({\mathfrak{s}}'(F))$ and $Y\in(\widetilde{{\mathfrak{m}}'}\cap{\mathfrak{s}}'_{\mathrm{rs}})(F)$ , define the weighted orbital integral

(2.11) \begin{equation} J_{M'}^{Q'}(Y, f'):=|D^{{\mathfrak{s}}'}(Y)|_F^{1/2} \int_{H'_Y(F)\backslash H'(F)} f'({\mathrm{Ad}}(x^{-1})(Y)) v_{M'}^{Q'}(x) \, dx.\end{equation}

In particular, if $M'=H'$ , then it is reduced to the usual normalised relative orbital integral.

2.4.4.

Now suppose additionally that F is non-archimedean. Let $M'\in\mathscr{L}^{H'}(M'_0)$ and $Y\in(\widetilde{{\mathfrak{m}}'}\cap{\mathfrak{s}}'_{\mathrm{rs}})(F)$ . In [Reference LiLi23, § 8.2], we deduce from $J_{M'}^{H'}(Y,\cdot)$ an H′-invariant distribution $I_{M'}^{H'}(Y, \cdot)$ on ${\mathfrak{s}}'(F)$ inductively such that their respective Fourier transforms $\hat{J}_{M'}^{H'}(Y,\cdot)$ and $\hat{I}_{M'}^{H'}(Y,\cdot)$ satisfy

\begin{equation*} \hat{I}_{M'}^{H'}(Y, f')=\hat{J}_{M'}^{H'}(Y, f')-\sum_{L'\in\mathscr{L}^{H'}(M'), L'\neq H'} \hat{I}_{M'}^{L',H',w}(Y, f').\end{equation*}

Here, $\hat{I}_{M'}^{L',H',w}(Y, \cdot)$ denotes the H′-invariant distribution on ${\mathfrak{s}}'(F)$ defined by the weighted induction (see [Reference LiLi23, (7.2.4)] for more details) of the L′-invariant distribution $\hat{I}_{M'}^{L'}(Y, \cdot)$ on $(\widetilde{{\mathfrak{l}}'}\cap{\mathfrak{s}}')(F)$ , which is a product analogue of $\hat{I}_{M'}^{H'}(Y,\cdot)$ in lower ranks. By [Reference LiLi23, Propositions 7.10 and 8.6], there are unique locally constant functions $\hat{j}_{M'}^{H'}(Y,\cdot)$ and $\hat{i}_{M'}^{H'}(Y,\cdot)$ on ${\mathfrak{s}}'_{\mathrm{rs}}(F)$ representing $\hat{J}_{M'}^{H'}(Y,\cdot)$ and $\hat{I}_{M'}^{H'}(Y,\cdot)$ , respectively. That is to say, for all $f'\in{\mathcal {C}}_c^\infty({\mathfrak{s}}'(F))$ , we have

\begin{align*} \hat{J}_{M'}^{H'}(Y, f'):=J_{M'}^{H'}(Y, \hat{f'})=\int_{{\mathfrak{s}}'(F)} f'(V) \hat{j}_{M'}^{H'}(Y, V) |D^{{\mathfrak{s}}'}(V)|_F^{-1/2} \, dV \end{align*}

and

\begin{align*} \hat{I}_{M'}^{H'}(Y, f'):=I_{M'}^{H'}(Y, \hat{f'})=\int_{{\mathfrak{s}}'(F)} f'(V) \hat{i}_{M'}^{H'}(Y, V) |D^{{\mathfrak{s}}'}(V)|_F^{-1/2} \, dV. \end{align*}

3.1.1 Matching orbits

Denote by ${\mathbf {A}}^n$ the affine space over F of dimension n. By [Reference LiLi22, Proposition 3.3], the map

\begin{align*} {\mathfrak{s}}\rightarrow{\mathbf {A}}^n,X\mapsto\chi_X \end{align*}

defines a categorical quotient ${\mathfrak{s}}//H$ over F, where $\chi_X$ denotes the characteristic polynomial of $X\in{\mathfrak{g}}$ . By [Reference LiLi21, Proposition 3.5], the map

\begin{align*} {\mathfrak{s}}'\rightarrow{\mathbf {A}}^n, Y\mapsto{\mathrm{Prd}}_Y \end{align*}

defines a categorical quotient ${\mathfrak{s}}'//{H'}$ over F, where ${\mathrm{Prd}}_Y$ denotes the reduced characteristic polynomial of $Y\in{\mathfrak{g}}'$ . Therefore, we can identify

\begin{align*} {\mathfrak{s}}//H\simeq{\mathbf {A}}^n\simeq{\mathfrak{s}}'//{H'}. \end{align*}

By [Reference LiLi21, Proposition 3.3], it induces an injection from the set of H′(F)-orbits in ${\mathfrak{s}}'_{\mathrm{rs}}(F)$ into the set of H(F)-orbits in ${\mathfrak{s}}_{\mathrm{rs}}(F)$ .

3.1.2 Matching Levi subgroups

There is an injection $M'\mapsto M$ from $\mathscr{L}^{H'}(M'_0)$ into $\mathscr{L}^{G,\omega}(M_0)$ such that the restriction of $G'_{\overline{F}}\simeq G_{\overline{F}}$ induces an isomorphism $\widetilde{M'}_{\overline{F}}\simeq M_{\overline{F}}$ , where $\widetilde{M'}\in\mathscr{L}^{G'}(M'_{\widetilde{0}})$ is associated to M′ as in 2.4.2. Let $M'\in\mathscr{L}^{H'}(M'_0)$ . We shall always denote by M the image of M′ under this injection, and say that M′ and M are matching Levi subgroups. There are compatible isomorphisms

(3.1) \begin{equation} (M,M_H, {\mathfrak{m}}\cap{\mathfrak{s}})\simeq \prod_{i=1}^\ell (G_{n_i},H_{n_i}, {\mathfrak{s}}_{n_i})\end{equation}

and

(3.2) \begin{equation} (\widetilde{M'},M', \widetilde{{\mathfrak{m}}'}\cap{\mathfrak{s}}')\simeq \prod_{i=1}^\ell (G'_{r_i},H'_{r_i},{\mathfrak{s}}'_{r_i})\end{equation}

such that the restriction of $G'_{\overline{F}}\simeq G_{\overline{F}}$ induces an isomorphism

\begin{align*} (G_{n_i},H_{n_i}, {\mathfrak{s}}_{n_i})_{\overline{F}}\simeq (G'_{r_i},H'_{r_i},{\mathfrak{s}}'_{r_i})_{\overline{F}} \end{align*}

for each $1\leqslant i\leqslant \ell$ . Denote

(3.3) \begin{equation} e_{M'}:=((-1)^{r_i})_{1\leqslant i\leqslant\ell}\in\{\pm1\}^\ell.\end{equation}

For $L\in\mathscr{L}^G(M)\subseteq\mathscr{L}^{G,\omega}(M_0)$ , it matches a unique element in $\mathscr{L}^{H'}(M')$ to be denoted by L′. The map $L\mapsto L'$ induces a bijection between $\mathscr{L}^G(M)$ and $\mathscr{L}^{H'}(M')$ . For $Q\in\mathscr{F}^G(M)\subseteq\mathscr{F}^{G,\omega}(M_0)$ , we shall always denote by Q′ the unique element in $\mathscr{F}^{H'}(M')$ such that the restriction of $G'_{\overline{F}}\simeq G_{\overline{F}}$ induces an isomorphism $\widetilde{Q'}_{\overline{F}}\simeq Q_{\overline{F}}$ , where $\widetilde{Q'}\in\mathscr{F}^{G'}(\widetilde{M'_0})$ is associated to Q′ as in § 2.4.2. The map $Q\mapsto Q'$ induces a bijection between $\mathscr{F}^G(M)$ and $\mathscr{F}^{H'}(M')$ .

3.1.3 M-matching orbits

Fix a pair of matching Levi subgroups $M'\in\mathscr{L}^{H'}(M'_0)$ and $M\in\mathscr{L}^{G,\omega}(M_0)$ as in § 3.1.2. Let $X\in({\mathfrak{m}}\cap{\mathfrak{s}}_{\mathrm{rs}})(F)$ and $Y\in(\widetilde{{\mathfrak{m}}'}\cap{\mathfrak{s}}'_{\mathrm{rs}})(F)$ . Write

(3.4) \begin{equation} X={\mathrm{diag}}(X_1,\ldots,X_\ell)\in\bigoplus_{i=1}^\ell {\mathfrak{s}}_{n_i}(F)\end{equation}

and

\begin{align*} Y={\mathrm{diag}}(Y_1,\ldots,Y_\ell)\in\bigoplus_{i=1}^\ell {\mathfrak{s}}'_{r_i}(F). \end{align*}

3.1.4 Criterion via the Kottwitz sign

Now suppose that F is a local field of characteristic zero. Denote by e(G′) the Kottwitz sign of G′ in the sense of [Reference KottwitzKot83].

For $X=({\kern-1pt}\begin{smallmatrix}0 & A \\[1pt] B & 0\end{smallmatrix}{\kern-1pt})\in{\mathfrak{s}}_{\mathrm{rs}}(F)$ , we define its sign by

\begin{align*} \eta(X):=\eta(\det(AB)). \end{align*}

Let $M'\in\mathscr{L}^{H'}(M'_0)$ and $M\in\mathscr{L}^{G,\omega}(M_0)$ be a pair of matching Levi subgroups as in § 3.1.2. Let $X\in({\mathfrak{m}}\cap{\mathfrak{s}}_{\mathrm{rs}})(F)$ as in (3.4). Denote

(3.5) \begin{equation} \eta_M(X):=(\eta(X_i))_{1\leqslant i\leqslant\ell}\in\{\pm1\}^\ell.\end{equation}

3.2.1 Classification of $E\hookrightarrow{\mathfrak{g}}'(F)$

Recall that $E=F(\alpha)$ with $\alpha^2\in F$ . Set

\begin{align*} \alpha_0:=\bigg(\begin{matrix}0 & \alpha^2 1_n \\ 1_n & 0\end{matrix}\bigg)\in G(F). \end{align*}

There is an F-algebra embedding $i: E\hookrightarrow{\mathfrak{g}}(F)$ defined by $i(\alpha)=\alpha_0$ . Denote

\begin{align*} H_0:=G_{\alpha_0}\simeq{\mathrm{Res}}_{E/F}GL_{n,E}. \end{align*}

We have

\begin{align*} {\mathfrak{h}}_0=\bigg\{\bigg(\begin{matrix}A & \alpha^2 C \\ C & A\end{matrix}\bigg): A,C\in{\mathfrak{g}}{\mathfrak{l}}_n\bigg\}. \end{align*}

Denote by ${\mathfrak{s}}_0$ the corresponding tangent space of $G/H_0$ at the neutral element. Then

\begin{align*} {\mathfrak{s}}_0=\bigg\{\bigg(\begin{matrix}A & -\alpha^2 C \\ C & -A\end{matrix}\bigg): A,C\in{\mathfrak{g}}{\mathfrak{l}}_n\bigg\}. \end{align*}

Recall that $\omega_0=({\kern-1pt}\begin{smallmatrix}1_n & 0 \\[1pt] 0 & -1_n\end{smallmatrix}{\kern-1pt})\in{\mathfrak{s}}_0(F)$ . Then

\begin{align*} \omega_0^2=1_{2n} \text{ and } {\mathfrak{s}}_0={\mathfrak{h}}_0\omega_0=\omega_0{\mathfrak{h}}_0. \end{align*}

The action ${\mathrm{Ad}}(\omega_0)$ induces the Galois involution on $H_0$ and ${\mathfrak{h}}_0$ . Consider pairs of the form $({\mathfrak{g}}',j)$ , where ${\mathfrak{g}}'$ is a central simple algebra over F of degree 2n and $j: E\hookrightarrow{\mathfrak{g}}'(F)$ is an F-algebra embedding. Denote

\begin{align*} \alpha':=j(\alpha)\in G'(F). \end{align*}

By Proposition 3.9(ii), there is an isomorphism $\varphi: {\mathfrak{g}}'_{\overline{F}}\rightarrow{\mathfrak{g}}_{\overline{F}}$ such that

\begin{align*} \varphi(\alpha')=\alpha_0 \end{align*}

and

(3.6) \begin{equation} \varphi\circ\sigma\circ\varphi^{-1}\circ\sigma^{-1}={\mathrm{Ad}}(u_\sigma)\end{equation}

for all $\sigma\in\Gamma$ , where $u_\sigma$ is a Galois 1-cocycle with values in $H_0/Z_G$ . We remark that $\sigma$ and $\sigma^{-1}$ on the left-hand side of (3.6) denote the Galois actions, respectively, on ${\mathfrak{g}}'$ and ${\mathfrak{g}}$ .

3.2.2 Link to the split case

We shall fix a pair $({\mathfrak{g}}',j)$ from § 3.2.1 until the end of § 3.2. Set

\begin{align*} \alpha_1:=\bigg(\begin{matrix}\alpha1_n & \\ & -\alpha1_n\end{matrix}\bigg)\in G(E). \end{align*}

Recall that $H=G_{\omega_0}=G_{\alpha_1}$ and that $\omega=({\kern-1pt}\begin{smallmatrix}0& 1_n \\[1pt] 1_n & 0\end{smallmatrix}{\kern-1pt})\in{\mathfrak{s}}(F)$ . We have

\begin{align*} \omega^2=1_{2n}\quad \text{and}\quad {\mathfrak{s}}={\mathfrak{h}}\omega=\omega{\mathfrak{h}}. \end{align*}

The action ${\mathrm{Ad}}(\omega)$ induces an involution on H and ${\mathfrak{h}}$ . Let $y\in G$ be such that

(3.7) \begin{equation} {\mathrm{Ad}}(y)\circ\varphi(\alpha')=\alpha_1,\end{equation}

i.e. ${\mathrm{Ad}}(y)(\alpha_0)=\alpha_1$ . Recall that $H'=G'_{\alpha'}$ . The morphism ${\mathrm{Ad}}(y)\circ\varphi$ induces isomorphisms $H'_{\overline{F}}\rightarrow H_{\overline{F}}$ and ${\mathfrak{s}}'_{\overline{F}}\rightarrow{\mathfrak{s}}_{\overline{F}}$ . For all $\sigma\in\Gamma$ , we have

\begin{align*} {\mathrm{Ad}}(\sigma(y))\circ\sigma\circ\varphi(\alpha')=\sigma(\alpha_1)=\varepsilon_\sigma\alpha_1, \end{align*}

where $\varepsilon_\sigma$ denotes the quadratic character of $\Gamma$ associated to $E/F$ . Since

\begin{align*} \sigma\circ\varphi(\alpha')={\mathrm{Ad}}(u_\sigma^{-1})\circ\varphi\circ\sigma(\alpha')={\mathrm{Ad}}(u_\sigma^{-1})\circ\varphi(\alpha')={\mathrm{Ad}}(u_\sigma^{-1})\circ{\mathrm{Ad}}(y^{-1})(\alpha_1), \end{align*}

we obtain

\begin{align*}{\mathrm{Ad}}(yu_\sigma\sigma(y)^{-1})(\alpha_1)=\varepsilon_\sigma\alpha_1.\end{align*}

For all $\sigma\in\Gamma$ , define $w_\sigma:=1_{2n}$ if $\sigma\in{\mathrm{Gal}}(\overline{F}/E)$ and $w_\sigma:=\omega$ otherwise. Then ${\mathrm{Ad}}(w_\sigma)(\alpha_1)=\varepsilon_\sigma\alpha_1$ . We deduce that

(3.8) \begin{equation} yu_\sigma\sigma(y)^{-1}w_\sigma\in H.\end{equation}

The elements y verifying (3.7) form an H-torsor of the form Hv, where

(3.9) \begin{equation}v:=\bigg(\begin{matrix}1_n & \alpha1_n \\ 1_n & -\alpha1_n\end{matrix}\bigg)\in G(E).\end{equation}

We easily check that $\sigma(v)=w_\sigma v$ for all $\sigma\in\Gamma$ and that $\omega v=v\omega_0$ .

3.2.3 Isomorphisms over $\overline{F}$ of centralisers

Let $Y\in{\mathfrak{s}}'(F)$ be a semi-simple element. There exists $y\in Hv$ and a semi-simple element $X\in{\mathfrak{s}}(F)$ such that

\begin{align*} {\mathrm{Ad}}(y)\circ\varphi(Y)=X. \end{align*}

Then ${\mathrm{Ad}}(y)\circ\varphi$ induces isomorphisms $H'_Y\rightarrow H_X$ and ${\mathfrak{s}}'_{Y}\rightarrow {\mathfrak{s}}_X$ over $\overline{F}$ . For all $\sigma\in\Gamma$ , we see that

(3.10) \begin{align}({\mathrm{Ad}}(y)\circ\varphi)\circ\sigma\circ({\mathrm{Ad}}(y)\circ\varphi)^{-1}\circ\sigma^{-1}&={\mathrm{Ad}}(y)\circ(\varphi\circ\sigma\circ\varphi^{-1}\circ\sigma^{-1})\circ(\sigma\circ{\mathrm{Ad}}(y^{-1})\circ\sigma^{-1})\nonumber\\&={\mathrm{Ad}}(yu_\sigma\sigma(y)^{-1}).\end{align}

Since

\begin{align*} ({\mathrm{Ad}}(y)\circ\varphi)\circ\sigma\circ({\mathrm{Ad}}(y)\circ\varphi)^{-1}\circ\sigma^{-1}(X)&=({\mathrm{Ad}}(y)\circ\varphi)\circ\sigma\circ({\mathrm{Ad}}(y)\circ\varphi)^{-1}(X) \\& =({\mathrm{Ad}}(y)\circ\varphi)\circ\sigma(Y)=({\mathrm{Ad}}(y)\circ\varphi)(Y)=X,\end{align*}

we obtain

(3.11) \begin{equation} yu_\sigma\sigma(y)^{-1}\in G_X.\end{equation}

Combining this with (3.8), we have

(3.12) \begin{equation} yu_\sigma\sigma(y)^{-1}\in Hw_\sigma\cap G_X.\end{equation}

Since $w_\sigma$ normalises H (respectively ${\mathfrak{s}}$ ), we see that $yu_\sigma\sigma(y)^{-1}$ normalises $H_X$ (respectively ${\mathfrak{s}}_X$ ).

3.2.4 Case of invertible elements: inner twists

We first consider the particular case where $Y\in{{\mathfrak{s}}'}^\times(F)$ is a semi-simple element. In this case, by [Reference Jacquet and RallisJR96, Proposition 2.1], up to conjugation by H(F), we may suppose that

\begin{align*} X=\bigg(\begin{matrix}0 & 1_n \\ A & 0\end{matrix}\bigg), \end{align*}

where $A\in GL_n(F)$ is semi-simple in the classical sense. By [Reference Jacquet and RallisJR96, Proposition 2.2] or direct calculation, for such X, there is an isomorphism of representations

\begin{align*} (H_X, {\mathfrak{s}}_X)\simeq(GL_{m,A}, {\mathfrak{g}}{\mathfrak{l}}_{m,A}). \end{align*}

3.2.5 Case of regular elements: isomorphisms over F

We now consider an even more special case. Let $X\in{\mathfrak{s}}_{\mathrm{rs}}(F)$ and $Y\in{\mathfrak{s}}'_{\mathrm{rs}}(F)$ be such that $X\leftrightarrow Y$ . Then there exists $y\in Hv$ such that ${\mathrm{Ad}}(y)\circ\varphi(Y)=X$ .

Suppose that F is a local field of characteristic zero. With regard to Lemma 3.12, we shall choose compatible Haar measures on $H'_Y(F)\rightarrow H_X(F)$ and ${\mathfrak{s}}'_Y(F)\simeq{\mathfrak{s}}_X(F)$ .

3.2.6 General case

We return to the general case of a semi-simple element $Y\in{\mathfrak{s}}'(F)$ . We identify ${\mathfrak{s}}'$ with the $(-1)$ -eigenspace of the involution ${\mathrm{Ad}}(\alpha)$ on ${\mathfrak{g}}'={\mathfrak{g}}{\mathfrak{l}}_{r,D}$ , where $\alpha$ is viewed as an element of ${\mathfrak{g}}'$ via the diagonal embedding. By [Reference LiLi23, Proposition 3.25], up to H′(F)-conjugation, we may suppose that

\begin{align*} Y=\bigg(\begin{matrix}B & \\ & 0_{r-s}\end{matrix}\bigg), \end{align*}

where $1\leqslant s\leqslant r$ is an integer in Case I (respectively an even integer in Case II) and $B\in{{\mathfrak{s}}'_s}^\times(F)$ is a semi-simple element with respect to the $H'_s$ -action. For such Y, there is an isomorphism of representations

\begin{align*}(H'_Y, {\mathfrak{s}}'_Y)\simeq(H'_{s,B},{\mathfrak{s}}'_{s,B})\times(H'_{r-s}, {\mathfrak{s}}'_{r-s}). \end{align*}

By [Reference Jacquet and RallisJR96, Proposition 2.1], up to conjugation by H(F), we may suppose that

\begin{align*} X=\left(\begin{array}{c@{\quad}c@{\quad}c@{\quad}c}0 & 0 & 1_{m} & 0 \\0 & 0 & 0 & 0 \\A & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\\end{array}\right)\!, \end{align*}

where $1\leqslant m\leqslant n$ is an integer and $A\in GL_m(F)$ is semi-simple in the classical sense. By [Reference Jacquet and RallisJR96, Proposition 2.2], for such X there is an isomorphism of representations

\begin{align*} (H_X, {\mathfrak{s}}_X)\simeq(GL_{m,A}, {\mathfrak{g}}{\mathfrak{l}}_{m,A})\times(H_{n-m}, {\mathfrak{s}}_{n-m}). \end{align*}

Since ${\mathrm{Ad}}(y)\circ\varphi(Y)=X$ , we have $2m=s\deg(D)$ . By Lemma 3.10, $H'_{s,B}$ is an inner form of $GL_{m,A}$ . This generalises [Reference ZhangZha15, Remark 5.5] and is useful for the smooth transfer.

3.3.1 The Galois 1-cocycle associated to X.

Let

\begin{align*} X=\bigg(\begin{matrix}0 & 1_n \\ A & 0\end{matrix}\bigg)\in{\mathfrak{s}}_{\mathrm{rs}}(F), \end{align*}

where $A\in GL_n(F)$ is regular semi-simple in the classical sense (see [Reference ZhangZha15, Proposition 4.2]). Then $GL_{n,A}$ is a maximal F-torus in $GL_n$ . Denote

\begin{align*} X_0:={\mathrm{Ad}}(v^{-1})(X)\in{\mathfrak{s}}_0(E),\end{align*}

where v is defined by (3.9). Since $H_X=\{(x,x)\in H: x\in GL_{n,A}\}$ , we see that ${\mathrm{Ad}}(v^{-1})$ acts trivially on $H_X$ and thus

\begin{align*} H_{0,X_0}={\mathrm{Ad}}(v^{-1})(H_X)=H_X\simeq GL_{n,A}. \end{align*}

Denote

\begin{align*} H_A:={\mathrm{Cent}}_H({\mathrm{diag}}(A,A))=GL_{n,A}\times GL_{n,A}, \end{align*}

which is a maximal F-torus in H, and

\begin{align*} T_X:={\mathrm{Cent}}_{H_0}({\mathrm{diag}}(A,A))={\mathrm{Ad}}(v^{-1})(H_A). \end{align*}

Proof. For all $\sigma\in\Gamma$ , we have

\begin{align*} \sigma({\mathrm{Ad}}(v^{-1})(V))={\mathrm{Ad}}(\sigma(v)^{-1})(V)={\mathrm{Ad}}(v^{-1} w_\sigma^{-1})(V)={\mathrm{Ad}}(v^{-1})(V). \end{align*}

By [Reference BorelBor91, § AG, Theorem 14.4], ${\mathrm{Ad}}(v^{-1})(V)$ is defined over F. The other assertions are obvious.

By Lemma 3.13, $T_X$ is a maximal F-torus in $H_0$ . Note that $H_X=H_A^{\omega}$ and $H_{0,X_0}=T_X^{\omega_0}$ , where the superscripts denote fixed points by the corresponding element. We also observe that

\begin{align*} T_X\simeq{\mathrm{Res}}_{E/F} (GL_{n,A})_E \end{align*}

and that the inclusion $H_{0,X_0}\subseteq T_X$ is isomorphic to the inclusion $GL_{n,A}\subseteq{\mathrm{Res}}_{E/F} (GL_{n,A})_E$ . For simplicity, we also write

(3.13) \begin{equation} T:=T_X\quad \text{and}\quad R:=H_{0,X_0},\end{equation}

which are maximal F-tori of $H_0$ and $H_0^{\omega_0}$ , respectively.

Lemma 3.14. There exists a unique Galois 1-cocycle $t_\sigma$ with values in $T/R$ such that, for all $\sigma\in\Gamma$ ,

\begin{align*} {\mathrm{Ad}}(t_\sigma)\circ\sigma(X_0)=X_0. \end{align*}

Proof. The uniqueness is obvious from the definition of R. It suffices to consider the existence. The cocycle condition is also automatic. It is thus enough to check the equality. If $\sigma\in{\mathrm{Gal}}(\overline{F}/E)$ , then $\sigma(X_0)=X_0$ , so it suffices to take $t_\sigma:=1$ . Now suppose that $\sigma\notin{\mathrm{Gal}}(\overline{F}/E)$ . Since

\begin{align*} \sigma(X_0)={\mathrm{Ad}}(\sigma(v)^{-1})\circ\sigma(X)={\mathrm{Ad}}(v^{-1}\omega^{-1})(X), \end{align*}

it suffices to find an element $t_\sigma\in T$ such that $vt_\sigma v^{-1}\omega^{-1}\in G_X$ . In fact, we can take

(3.14) \begin{equation} t_\sigma:={\mathrm{Ad}}(v^{-1})\bigg(\begin{matrix}1 & \\ & A\end{matrix}\bigg).\end{equation}

Since $A\in GL_n(F)$ is regular semi-simple, its characteristic polynomial $\chi_A$ is separable and factorised into a product $\chi_A=\prod_{i\in I}\chi_i$ of pairwise distinct monic irreducible polynomials over F. Then

\begin{align*} GL_{n,A}\simeq \prod_{i\in I} {\mathrm{Res}}_{F_i/F} {\mathbb {G}}_{m,F_i}, \end{align*}

where $F_i=F[\lambda]/(\chi_i(\lambda))$ . Using the inflation-restriction exact sequence, Shapiro’s lemma and Hilbert’s Theorem 90 (see e.g. [Reference Knus, Merkurjev, Rost and TignolKMR⁺98, § 29.A]), we can identify

\begin{equation*} H^1(F,T/R)\simeq H^1({\mathrm{Gal}}(E/F), T(E)/R(E)).\end{equation*}

We shall compute it in the following lemma.

Proof. For all $\sigma\in\Gamma$ and $h':={\mathrm{Ad}}(v^{-1})(h)\in T$ , where $h\in H_A$ , we see that

\begin{align*} \sigma(h')={\mathrm{Ad}}(\sigma(v)^{-1})\circ\sigma(h)={\mathrm{Ad}}(v^{-1})\circ{\mathrm{Ad}}(w_\sigma)\circ\sigma(h). \end{align*}

Therefore, we can regard T as the subgroup $H_A$ of G equipped with the Galois action ${\mathrm{Ad}}(w_\sigma)\circ\sigma$ . Then the inclusion $R(E)\subseteq T(E)$ is isomorphic to the diagonal inclusion $H_X(E)\subseteq H_A(E)$ . Let $\sigma$ be the nontrivial element in ${\mathrm{Gal}}(E/F)$ and $s_\sigma:=(B,C)\in H_A(E)/H_X(E)$ . By a straightforward calculation, we observe that:

1. (i) $s_\sigma$ defines a 1-cocycle if and only if $s_\sigma{\mathrm{Ad}}(w_\sigma)\circ\sigma(s_\sigma)\in H_X(E)$ , i.e. $BC^\sigma=CB^\sigma$ ;

2. (ii) $s_\sigma$ defines a 1-coboundary if and only if $s_\sigma=(B,C)^{-1}{\mathrm{Ad}}(w_\sigma)\circ\sigma(B,C)$ for some $(B,C)\in H_A(E)/H_X(E)$ , i.e. $s_\sigma=(C^\sigma B^{-1},B^\sigma C^{-1})\in H_A(E)/H_X(E)$ .

Then $H^1({\mathrm{Gal}}(E/F),T(E)/R(E))$ is the quotient of

\begin{align*} &\{(B,C)\in H_A(E)/H_X(E): BC^\sigma=CB^\sigma\}=\{(1,CB^{-1})\in H_A(E)/H_X(E): (CB^{-1})^\sigma=CB^{-1}\} \\&\quad=\{(1,B)\in H_A(E)/H_X(E): B^\sigma=B\}=\{(1,B)\in H_A(E)/H_X(E): B\in GL_{n,A}(F)\} \\\end{align*}

by

\begin{align*} &\{(C^\sigma B^{-1},B^\sigma C^{-1})\in H_A(E)/H_X(E): B,C\in GL_{n,A}(E)\}\\&\quad =\{(B^{-\sigma},B)\in H_A(E)/H_X(E): B\in GL_{n,A}(E)\}\\&\quad =\{(1,BB^\sigma)\in H_A(E)/H_X(E): B\in GL_{n,A}(E)\}.\end{align*}

That is to say, it is the quotient of $GL_{n,A}(F)$ by the group of norms of $GL_{n,A}(E)$ . Under the identification of T and $H_A$ with the twisted Galois action, we see from (3.14) that the Galois 1-cocycle $t_\sigma$ in Lemma 3.14 corresponds exactly to the class of A.

Denote $E_i:=F_i\otimes_F E$ for $i\in I$ . Let $I_0$ be the subset of I consisting of i such that $E_i$ is a field. By Lemma 3.15, we have

(3.15) \begin{equation} H^1(F, T/R)\simeq\prod_{i\in I} F_i^\times/N_{E_i/F_i}(E_i^\times)=\prod_{i\in I_0} F_i^\times/N_{E_i/F_i}(E_i^\times).\end{equation}

If F is a local field of characteristic zero, then $H^1(F,T/R)=({\mathbb {Z}}/2{\mathbb {Z}})^{I_0}$ .

3.3.2 Criterion via Galois 1-cocycles

Note that $Z_G\subseteq R$ .

Lemma 3.16. Let $t\in H^1(F,T/R)$ be the class of the Galois 1-cocycle $t_\sigma$ in Lemma 3.14 associated to X. Let $u\in H^1(F,H_0/Z_G)$ be the class of the Galois 1-cocycle $u_\sigma$ in § 3.2.1 associated to ${\mathfrak{g}}'$ . Then the following conditions are equivalent:

1. (i) there exists $Y\in{\mathfrak{s}}'(F)$ and $h\in H_0$ such that ${\mathrm{Ad}}(h)\circ\varphi(Y)=X_0$ ;

2. (ii) there exists an element of $H^1(F,T/Z_G)$ which has images $t\in H^1(F,T/R)$ and $u\in H^1(F,H_0/Z_G)$ under the following natural maps.

Proof of Lemma 3.16. Given any $h\in H_0$ , let $Y\in{\mathfrak{s}}'$ be the unique element such that ${\mathrm{Ad}}(h)\circ\varphi(Y)=X_0$ . For all $\sigma\in\Gamma$ , we have

\begin{align*} {\mathrm{Ad}}(\sigma(h))\circ\sigma\circ\varphi(Y)=\sigma(X_0)={\mathrm{Ad}}(t_\sigma^{-1})(X_0) \end{align*}

and

\begin{align*} \sigma\circ\varphi(Y)={\mathrm{Ad}}(u_\sigma^{-1})\circ\varphi\circ\sigma(Y). \end{align*}

But $Y\in{\mathfrak{s}}'(F)$ if and only if

\begin{align*} \varphi\circ\sigma(Y)=\varphi(Y)={\mathrm{Ad}}(h^{-1})(X_0) \end{align*}

for all $\sigma\in\Gamma$ . We have shown that condition (i) is equivalent to the following: there exists $h\in H_0$ such that

(3.16) \begin{equation} {\mathrm{Ad}}(\sigma(h)u_\sigma^{-1}h^{-1})(X_0)={\mathrm{Ad}}(t_\sigma^{-1})(X_0)\end{equation}

for all $\sigma\in\Gamma$ . But (3.16) means exactly

\begin{align*} hu_\sigma\sigma(h)^{-1}\in Rt_\sigma. \end{align*}

If such an h exists, then $hu_\sigma\sigma(h)^{-1}\in Rt_\sigma\subseteq T$ defines a Galois 1-cocycle with values in $T/Z_G$ which satisfies condition (ii). Conversely, any Galois 1-cocycle with values in $T/Z_G$ having image u must be of the form $hu_\sigma\sigma(h)^{-1}$ , where $h\in H_0$ . If it also has image t, then by replacing h with t′h, where $t'\in T$ , we may suppose that $hu_\sigma\sigma(h)^{-1}\in Rt_\sigma$ .

3.3.3 Criterion via the Kottwitz sign revisited

Note that $H_0^{\omega_0}=H^\omega\simeq GL_n$ . We shall abuse notation and denote by $R_{\mathrm{der}}$ (respectively $T_{\mathrm{der}}$ ) the preimage of R (respectively T) in $H_{0,{\mathrm{der}}}^{\omega_0}=H_{0,{\mathrm{sc}}}^{\omega_0}$ (respectively $H_{0,{\mathrm{der}}}=H_{0,{\mathrm{sc}}}$ ). We shall use the index ‘ ${\mathrm{ab}}$ ’ to denote the abelianised Galois cohomology defined in [Reference BorovoiBor98, Definition 2.2] and [Reference LabesseLab99, 1.6 and 1.8]. By [Reference LabesseLab99, Proposition 1.6.7], we have abelianisation maps $H^1(F,H_0^{\omega_0}/Z_G)\rightarrow H^1_{\mathrm{ab}}(F,H_0^{\omega_0}/Z_G)$ and $H^1(F,H_0/Z_G)\rightarrow H^1_{\mathrm{ab}}(F,H_0/Z_G)$ , which are surjective. If F is a non-archimedean local field of characteristic zero, they are also injective by Kneser’s theorem (see [Reference LabesseLab99]). By the short exact sequence of complexes

\begin{align*} 1\rightarrow[1\rightarrow Z_G]\rightarrow[R_{\mathrm{der}}\rightarrow R]\rightarrow[R_{\mathrm{der}}\rightarrow R/Z_G]\rightarrow 1, \end{align*}

we have

\begin{align*}H^i_{\mathrm{ab}}(F,H_0^{\omega_0}/Z_G)=H^i_{\mathrm{ab}}(F,Z_G\rightarrow H_0^{\omega_0})=H^i_{\mathrm{ab}}(F,R_{\mathrm{der}}\rightarrow R/Z_G), \end{align*}

where $i=1, 2$ . Similarly, we see that

\begin{align*} H^1_{\mathrm{ab}}(F,H_0/Z_G)=H^1_{\mathrm{ab}}(F,T_{\mathrm{der}}\rightarrow T/Z_G). \end{align*}

By the short exact sequence of complexes

\begin{align*} 1\rightarrow[R_{\mathrm{der}}\rightarrow R/Z_G]\rightarrow[T_{\mathrm{der}}\rightarrow T/Z_G]\rightarrow [T_{\mathrm{der}}/R_{\mathrm{der}}\rightarrow T/R]\rightarrow 1, \end{align*}

we have

(3.17) \begin{equation} H^1_{\mathrm{ab}}(F, H_0^{\omega_0}/Z_G\rightarrow H_0/Z_G)=H^1_{\mathrm{ab}}(F,T_{\mathrm{der}}/R_{\mathrm{der}}\rightarrow T/R).\end{equation}

From the theory of Galois hypercohomology (see [Reference Kottwitz and ShelstadKS99, Appendix A.1]), we obtain the following commutative diagram with exact columns and rows.

(3.18)

Let $t\in H^1(F,T/R)$ and $u\in H^1(F,H_0/Z_G)$ be as in Lemma 3.16. Denote by $\varepsilon_X$ the image of t under the homomorphism

\begin{align*} H^1(F,T/R)\rightarrow H^1_{\mathrm{ab}}(F, H_0^{\omega_0}/Z_G\rightarrow H_0/Z_G). \end{align*}

Denote by $\varepsilon'$ the image of u under the morphism

\begin{align*} H^1(F,H_0/Z_G)\rightarrow H^1_{\mathrm{ab}}(F,H_0/Z_G)\rightarrow H^1_{\mathrm{ab}}(F, H_0^{\omega_0}/Z_G\rightarrow H_0/Z_G). \end{align*}

Proposition 3.19. There is a natural identification

\begin{align*} H^1_{\mathrm{ab}}(F, H_0^{\omega_0}/Z_G\rightarrow H_0/Z_G)\simeq H^1(F,C_{H_0}/C_{H_0^{\omega_0}}) \end{align*}

where $C_{H_0}$ (respectively $C_{H_0^{\omega_0}}$ ) denotes the cocentre of $H_0$ (respectively $H_0^{\omega_0}$ ).

Remark 3.20. We have $C_{H_0}{\kern-.5pt}={\kern-.2pt}{\mathrm{Res}}_{E/F}(C_{H_0^\omega})_E$ , where $C_{H_0^\omega}{\kern-.5pt}\simeq{\kern-.3pt} C_{GL_n}{\kern-.5pt}\simeq{\kern-.3pt} {\mathbb {G}}_m$ . Let ${\mathbb{G}}_m^\ast{\kern-.2pt}:={\kern-.2pt}{\mathrm{Res}}_{E/F} {\mathbb {G}}_{m,E}/{\mathbb {G}}_m$ . It is a well-known exercise in Galois cohomology (also included in the proof of Lemma 3.15) that

\begin{align*} H^1(F,{\mathbb {G}}_m^\ast)\simeq H^1({\mathrm{Gal}}(E/F),{\mathbb {G}}_m^\ast(E))\simeq F^\times/N_{E/F}(E^\times). \end{align*}

Proof of Proposition 3.19. By (3.17) and the short exact sequence

\begin{align*} 1\rightarrow[R_{\mathrm{der}}\rightarrow T_{\mathrm{der}}] \rightarrow[R\rightarrow T] \rightarrow[R/R_{\mathrm{der}}\rightarrow T/T_{\mathrm{der}}] \rightarrow1, \end{align*}

we have

(3.19) \begin{equation} H^1_{\mathrm{ab}}(F, H_0^{\omega_0}/Z_G\rightarrow H_0/Z_G)=H^1(F, (T/T_{\mathrm{der}})/(R/R_{\mathrm{der}})).\end{equation}

But $T/T_{\mathrm{der}}=C_{H_0}$ and $R/R_{\mathrm{der}}=C_{H_0^{\omega_0}}$ .