2.2.1.

We can express ${\mathbf{M}}$ in a more explicit way using fundamental representations of ${\mathbf{G}}^{\mathrm{sc}}$ . Let $\alpha_1,\dots, \alpha_{n-1}$ denote the simple roots and $\varpi_1, \dots, \varpi_{n-1}$ the fundamental weights of ${\mathbf{G}}$ . Since ${\mathbf{G}}=\operatorname{GL}_n$ , the fundamental representations are simply $V_i=\bigwedge^ik^n$ for $1\leqslant i\leqslant n-1$ . We have the following representation of ${\mathbf{M}}^\times$ :

\begin{align*} \rho_\star\colon{\mathbf{M}}^\times=({\mathbf{T}}^{\mathrm{sc}}\times {\mathbf{G}}^{\mathrm{sc}})/{\mathbf{Z}}^{\mathrm{sc}} &\longrightarrow {\mathbb{A}}^{n-1}\oplus\bigoplus_{i=1}^{n-1}\operatorname{End}(V_i)\\ (z,g)&\longmapsto \bigg({\kern-1pt}\alpha_i(z),\varpi_i(z)\bigwedge^i g{\kern-1pt}\bigg),\end{align*}

and ${\mathbf{M}}$ is the normalization of the closure of $\rho_\star({\mathbf{M}}^\times)$ .

2.2.2.

We have the abelianization map of the reductive monoid

\begin{align*} \alpha_{\mathbf{M}}\colon {\mathbf{M}}\to {\mathbf{A}}_{\mathbf{M}}:= {\mathbf{M}}\mathbin{\!/\!\!/\!} ({\mathbf{G}}^{\mathrm{sc}}\times {\mathbf{G}}^{\mathrm{sc}})\simeq{\mathbb{A}}^{n-1},\end{align*}

which factors through the projection

\begin{align*} {\mathbb{A}}^{n-1}\oplus\bigoplus_{i=1}^{n-1}\operatorname{End}(V_i)\longrightarrow {\mathbb{A}}^{n-1}.\end{align*}

Note that ${\mathbf{A}}_{\mathbf{M}}$ is a commutative monoid and $\alpha_{\mathbf{M}}$ is a monoid homomorphism. We know that $\alpha_{\mathbf{M}}^{-1}({\mathbf{A}}_{\mathbf{M}}^\times)={\mathbf{M}}^\times$ and $\alpha_{\mathbf{M}}^{-1}(1)={\mathbf{G}}^{\mathrm{sc}}$ .

For any $x\in{\mathbf{M}}$ , we also define $x_i\in\operatorname{End}(V_i)$ to be its image under the obvious projection maps. By the general theory of reductive monoids, ${\mathbf{C}}_{\mathbf{M}}:={\mathbf{M}}\mathbin{\!/\!\!/\!}{\mathbf{G}}$ is isomorphic to affine space ${\mathbb{A}}^{2n-2}$ with coordinates given by

\begin{align*} \alpha_1,\ldots,\alpha_{n-1},\chi_1,\ldots,\chi_{n-1},\end{align*}

where $\alpha_i$ are treated as the coordinates on ${\mathbf{A}}_{\mathbf{M}}$ and $\chi_i$ sends x to $\operatorname{Tr}(x_i)$ . We denote by $\chi_{\mathbf{M}}$ the natural map ${\mathbf{M}}\to{\mathbf{C}}_{\mathbf{M}}$ . Let ${\mathbf{C}}\cong{\mathbb{A}}^{n-1}$ be the affine space with coordinates $\chi_i$ , then ${\mathbf{C}}_{\mathbf{M}}\simeq{\mathbf{A}}_{\mathbf{M}}\times{\mathbf{C}}$ . We also have a Chevalley–Steinberg-type isomorphism ( ${\mathbf{W}}$ is the Weyl group)

\begin{align*} {\mathbf{T}}_{\mathbf{M}}\mathbin{\!/\!\!/\!}{\mathbf{W}}\stackrel{\sim}{\longrightarrow}{\mathbf{C}}_{\mathbf{M}}.\end{align*}

2.2.3.

There is an extended discriminant function $\mathrm{Disc}_+$ on ${\mathbf{C}}_{\mathbf{M}}$ defined as follows: on the maximal torus ${\mathbf{T}}_{\mathbf{M}}$ , it has the formula

\begin{align*} \mathrm{Disc}_+=e^{(2\rho,0)}\prod_{\alpha\in\Phi}(1-e^{(0,\alpha)}),\end{align*}

where $\alpha$ ranges over the set of roots $\Phi$ of ${\mathbf{G}}$ . It extends to a regular function on $\bar{{\mathbf{T}}}_{\mathbf{M}}$ , and by its ${\mathbf{W}}$ -invariance, it descends to a function on ${\mathbf{C}}_{\mathbf{M}}$ and cuts out a reduced divisor on ${\mathbf{C}}_{\mathbf{M}}.$

2.2.4.

There is a natural section from ${\mathbf{A}}_{\mathbf{M}}$ to ${\mathbf{M}}$ denoted as $\delta_{\mathbf{M}}$ . On ${\mathbf{A}}_{\mathbf{M}}^\times\simeq{\mathbf{T}}^{\mathrm{ad}}$ , this section is given by $z\mapsto (z,z^{-1})$ . Similarly, the invariant map $\chi_{\mathbf{M}}$ also admits a section $\epsilon_{\mathbf{M}}$ , whose restriction to ${\mathbf{C}}_{\mathbf{M}}^\times\simeq{\mathbf{A}}_{\mathbf{M}}^\times\times{\mathbf{C}}$ is given by the formula

\begin{align*} (z,a)\mapsto \delta_{\mathbf{M}}(z)\epsilon(a),\end{align*}

where $\epsilon(a)\in{\mathbf{G}}^{\mathrm{sc}}$ is the companion matrix associated with $a=(a_1,\ldots,a_{n-1})$ :

\begin{align*} \epsilon(a)=\begin{pmatrix} a_1 & -a_2 & \cdots & (-1)^{n-2}a_{n-1} & (-1)^{n-1}\\[2pt] 1 & & & & \\ & 1 & & & \\ & & \ddots & & \\ & & & 1 & \end{pmatrix}\!.\end{align*}

See [Reference WangWan26, § 2.3.16 and Proposition 2.4.4] for details.

2.3.1.

The vector ${\mathbf{e}}$ and the co-vector ${\mathbf{e}}^\vee$ decompose $k^n$ into a direct sum $(k{\mathbf{e}}^\vee)^\perp\oplus k{\mathbf{e}}$ , where $(k{\mathbf{e}}^\vee)^\perp\simeq k^{n-1}$ . As a result, we can decompose $V_i=\bigwedge^i k^n$ ( $1\leqslant i\leqslant n-1$ ) as ${\mathbf{H}}$ -representations

\begin{align*} \bigwedge^i k^n=\bigwedge^i k^{n-1}\oplus\bigg(\bigwedge^{i-1}k^{n-1}\otimes k{\mathbf{e}}{\kern-1pt}\bigg),\end{align*}

and we denote

\begin{align*} V_i'&=\bigwedge^i k^{n-1},\\ V_i''&=\bigwedge^{i-1}k^{n-1}\otimes k{\mathbf{e}}.\end{align*}

In turn, for any $x_i\in\operatorname{End}(V_i)$ , its trace is the sum

\begin{align*} \operatorname{Tr}(x_i)=\operatorname{Tr}'(x_i)+\operatorname{Tr}''(x_i),\end{align*}

where $\operatorname{Tr}'$ (respectively $\operatorname{Tr}''$ ) is the induced trace on the direct summand $V_i'$ (respectively $V_i''$ ). Clearly, the functions

\begin{align*} \chi_i'\colon x &\longmapsto\operatorname{Tr}'(x_i),\\ \chi_i''\colon x&\longmapsto\operatorname{Tr}''(x_i)\end{align*}

on ${\mathbf{M}}$ are ${\mathbf{H}}$ -invariant, and we have $\chi_i=\chi_i'+\chi_i''$ .

2.3.2.

For a tuple $b=(b_1,\ldots,b_{n-1})\in{\mathbb{A}}^{n-1}$ , we define

\begin{align*} \beta(b):=\begin{pmatrix} 1 & & & & \\ &1& & & \\ & & \ddots & & \\ & & & 1 & \\[2pt] (-1)^{n-1}b_1 & (-1)^{n-2}b_2 &\cdots & -b_{n-1} & 1 \end{pmatrix}\!.\end{align*}

We can now state one of the main results of this section.

Theorem 2.1. The map

\begin{align*} \tilde{\chi}_{{\mathbf{M}},{\mathbf{H}}}\colon{\mathbf{M}}&\longrightarrow{\mathbf{A}}_{\mathbf{M}}\times{\mathbb{A}}^{2n-2}\\ x&\longmapsto (\alpha_{\mathbf{M}}(x),\chi_i'(x),\chi_i''(x)) \end{align*}

induces an isomorphism ${\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}\simeq {\mathbf{A}}_{\mathbf{M}}\times{\mathbb{A}}^{2n-2}\simeq{\mathbb{A}}^{3n-3}$ . Moreover, we have a section

\begin{align*} \epsilon_{{\mathbf{M}},{\mathbf{H}}}\colon {\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}&\longrightarrow {\mathbf{M}}\\ (z,a',a'')&\longmapsto \beta(a'')\epsilon_{\mathbf{M}}(z,a)\beta(a'')^{-1}, \end{align*}

where $a_i'$ (respectively $a_i''$ ) is the coordinate corresponding to $\chi_i'$ (respectively $\chi_i''$ ), $a'=(a_i')$ (respectively $a''=(a_i'')$ ), and $a=(a_i)$ is such that $a_i=a_i'+a_i''$ .

Proof. It is not hard to see (and well known) that the generic stabilizer of ${\mathbf{H}}$ acting on ${\mathbf{M}}^\times$ is trivial. Indeed, for a sufficiently general regular semisimple (in the usual sense) element $x\in{\mathbf{G}}^{\mathrm{sc}}$ , the vectors

\begin{align*} {\mathbf{e}},x{\mathbf{e}},\ldots,x^{n-1}{\mathbf{e}} \end{align*}

are linearly independent. If $h\in{\mathbf{H}}$ stabilizes x, then it fixes the above vectors as well, and so must be trivial. The same is true if $x\in{\mathbf{M}}^\times$ , because the ${\mathbf{H}}$ -action commutes with central translations. This implies that the map ${\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}\to {\mathbb{A}}^{3n-3}$ is dominant and generically quasi-finite after combining with the dimension counting:

\begin{align*} \dim{\mathbf{M}}-\dim{\mathbb{A}}^{3n-3}=(n^2+n-2)-(3n-3)=n^2-2n+1=\dim{{\mathbf{H}}}. \end{align*}

Therefore, if we can construct a section from ${\mathbb{A}}^{3n-3}$ to ${\mathbf{M}}$ , then we have an induced section ${\mathbb{A}}^{3n-3}\to{\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}$ , which will prove both claims since ${\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}$ is irreducible.

Note that the map $\epsilon_{{\mathbf{M}},{\mathbf{H}}}$ is well-defined on ${\mathbb{A}}^{3n-3}$ , and we show that $\tilde{\chi}_{{\mathbf{M}},{\mathbf{H}}}\circ\epsilon_{{\mathbf{M}},{\mathbf{H}}}$ is the identity map. To simplify notation, we let $A=\epsilon(a)$ and $B=\beta(a'')$ . We already know that the trace of $\epsilon_{\mathbf{M}}(z,a)=\delta_{\mathbf{M}}(z)A$ on $V_j$ is $a_j$ , and conjugation by B does not change it. So we only need to compute $\chi_j'$ (or $\chi_j''$ if we so choose). It is also easy to compute

\begin{align*} AB^{-1}=\begin{pmatrix} a_1' & -a_2' & \cdots & (-1)^{n-2}a_{n-1}' & (-1)^{n-1}\\[2pt] 1 & & & & \\ & 1 & & & \\ & & \ddots & & \\ & & & 1 & \end{pmatrix}\!. \end{align*}

The matrix coefficients of $AB^{-1}$ on $V_j$ , up to signs, are just its $j\times j$ minors, and the diagonal matrix coefficients correspond to minors such that if the ith row is chosen, then so is the ith column.

The action of $\delta_{\mathbf{M}}(z)$ on $V_j$ is described as follows: the highest weight of $V_j$ is $\varpi_j$ , a basis vector of which is ${\mathbf{e}}_{\varpi_j}={\mathbf{e}}_1\wedge\cdots\wedge{\mathbf{e}}_j$ , where ${\mathbf{e}}_i$ is the ith standard vector

\begin{align*} {\mathbf{e}}_i=\,^{\mathrm{t}}{}{(0,\ldots,1,\ldots,0)}, \end{align*}

with 1 in the ith row. Then $\delta_{\mathbf{M}}(z){\mathbf{e}}_{\varpi_j}={\mathbf{e}}_{\varpi_j}$ , and for any vector $e_\mu$ of weight $\mu=\varpi_j-\sum_ic_i\alpha_i$ ( $c_i\in{\mathbb{N}}$ ), we have

\begin{align*} \delta_{\mathbf{M}}(z)e_\mu=z_1^{c_1}\cdots z_{n-1}^{c_{n-1}}e_\mu, \end{align*}

where $z_i=\alpha_i(z)\in k$ . In addition, the action of (unipotent matrix) B on $V_j$ sends any pure wedge

\begin{align*} {\mathbf{e}}_{i_1}\wedge\cdots\wedge{\mathbf{e}}_{i_j}, \quad 1\leqslant i_1\lt\cdots\lt i_j\leqslant n, \end{align*}

to a sum of itself plus some wedges involving ${\mathbf{e}}_n={\mathbf{e}}$ , the latter of which is contained in the complement space $V_j''$ . This means that

\begin{align*} \chi_j'(B\delta_{\mathbf{M}}(z)AB^{-1})=\chi_j'(\delta_{\mathbf{M}}(z)AB^{-1}). \end{align*}

Since the action of $\delta_{\mathbf{M}}(z)$ is diagonal, it suffices to compute the diagonal matrix coefficients of $AB^{-1}$ and then scale them using $\delta_{\mathbf{M}}(z)$ . To this end, we need only consider the upper-left $(n-1)\times(n-1)$ block of $AB^{-1}$ .

For any $2\leqslant i\leqslant n-1$ , if the ith row of $AB^{-1}$ (without the last row and column) is chosen as part of a $j\times j$ minor, then for the minor to be non-vanishing (and contributing to $\chi_j'$ ), the $(i-1)$ th column must also be chosen, and then so is the $(i-1)$ th row. By induction, the only $j\times j$ minor contributing to $\chi_j'$ is the upper-left $j\times j$ block, corresponding to the highest weight $\varpi_j$ . This means that the scaling by $\delta_{\mathbf{M}}(z)$ is 1. As a result,

\begin{align*} \chi_j'\circ\epsilon_{{\mathbf{M}},{\mathbf{H}}}(z,a',a'')=\chi_j'(AB^{-1})=a_j'. \end{align*}

This finishes the proof.

2.5.1.

First, we have the following function on ${\mathbf{M}}^\times$ :

\begin{align*} \mathrm{Disc}_{{\mathbf{M}},{\mathbf{H}}}\colon (z,g)\longmapsto z^{2\rho}\det ({\mathbf{e}}^\vee g^{i+j-2}{\mathbf{e}})_{1\leqslant i,j\leqslant n},\end{align*}

where $\rho$ is the half-sum of all positive roots, viewed as a function on ${\mathbf{T}}^{\mathrm{sc}}$ written exponentially. One may check that this function is indeed well-defined: indeed, if $\zeta\in{\mathbf{Z}}^{\mathrm{sc}}$ , then $(z\zeta,\zeta^{-1}g)=(z,g)$ , but we also have $z^{2\rho}=(z\zeta)^{2\rho}$ and (we omitted the ranges $1\leqslant i,j\leqslant n$ below to avoid notation overload)

\begin{align*} \det({\mathbf{e}}^\vee(\zeta^{-1}g)^{i+j-2}{\mathbf{e}}) &=({\mathbf{e}}^\vee\zeta^{-1}{\mathbf{e}})^{n(n-1)}\det({\mathbf{e}}^\vee g^{i+j-2}{\mathbf{e}})\\ &=({\mathbf{e}}^\vee\zeta^{-n(n-1)}{\mathbf{e}})\det({\mathbf{e}}^\vee g^{i+j-2}{\mathbf{e}})\\ &=\det({\mathbf{e}}^\vee g^{i+j-2}{\mathbf{e}}),\end{align*}

because $\zeta^n=1$ .

2.5.2.

To extend this function to all of ${\mathbf{M}}$ , we make the following construction: recall that for any $x\in {\mathbf{M}}$ and any $1\leqslant i\leqslant n-1$ , we have the element $x_i\in\operatorname{End}(V_i)$ . Starting from ${\mathbf{f}}_1(x)=x_1{\mathbf{e}}$ , we define for $2\leqslant i\lt n$

\begin{align*} {\mathbf{f}}_i(x)=x_i({\mathbf{e}}\wedge {\mathbf{f}}_{i-1}(x))\in V_i,\end{align*}

and let ${\mathbf{f}}_n(x)={\mathbf{e}}\wedge{\mathbf{f}}_{n-1}(x)$ . Similarly, we have ${\mathbf{f}}_1^\vee(x)=({\mathbf{e}}^\vee)x_1$ ,

\begin{align*} {\mathbf{f}}_i^\vee(x)=({\mathbf{f}}_{i-1}^\vee(x)\wedge{\mathbf{e}}^\vee)x_i\in V_i^\vee,\end{align*}

and ${\mathbf{f}}_n^\vee(x)={\mathbf{f}}_{n-1}^\vee(x)\wedge{\mathbf{e}}^\vee$ .

Corollary 2.6. For any $x\in{\mathbf{M}}$ and any $1\lt i\leqslant n$ , we have an inclusion

\begin{align*} {\mathbf{f}}_i(x) \in {\mathbf{f}}_{i-1}(x)\wedge V_1\subset V_i, \end{align*}

where we use the convention $V_n=\bigwedge^nk^n$ . A similar result holds for ${\mathbf{f}}_i^\vee(x)$ .

For any i, since ${\mathbf{f}}_{i-1}\neq 0$ on S and is a pure tensor, the wedging map

\begin{align*} {\mathbf{f}}_{i-1}\wedge\colon V_1\longrightarrow V_i \end{align*}

is locally constant with $(i-1)$ -dimensional kernel. Since ${\mathbf{f}}_i(x_\eta)\in {\mathbf{f}}_{i-1}(x_\eta)\wedge V_1$ , we have

\begin{align*} {\mathbf{f}}_i(x)\in\bar{{\mathbf{f}}_{i-1}(x_\eta)\wedge V_1}, \end{align*}

and the fiber of the right-hand side over x is exactly ${\mathbf{f}}_{i-1}(x)\wedge V_1$ by local constancy. This proves the claim.

2.5.3.

The pairing $x\mapsto{\mathbf{f}}_n^\vee(x){\mathbf{f}}_n(x)$ defines a function on ${\mathbf{M}}$ , and one can check that if $x\in{\mathbf{M}}^\times$ , then we have

\begin{align*} \mathrm{Disc}_{{\mathbf{M}},{\mathbf{H}}}(x)= {\mathbf{f}}_n^\vee(x){\mathbf{f}}_n(x).\end{align*}

Since $k[{\mathbf{M}}]$ is a subring of $k[{\mathbf{M}}^\times]$ , we have the following lemma.

Definition 2.8. The extended ${\mathbf{H}}$ -discriminant function is defined as the product

\begin{align*} \mathrm{Disc}_{{\mathbf{M}},{\mathbf{H}}}^+:=\mathrm{Disc}_+\cdot\mathrm{Disc}_{{\mathbf{M}},{\mathbf{H}}}\in k[{\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}], \end{align*}

where $\mathrm{Disc}_+$ is as in § 2.2.3. The extended ${\mathbf{H}}$ -discriminant divisor ${\mathbf{D}}_{{\mathbf{M}},{\mathbf{H}}}^+$ is the principal divisor of ${\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}$ defined by $\mathrm{Disc}_{{\mathbf{M}},{\mathbf{H}}}^+$ .

Since ${\mathbf{f}}_n^\vee(x){\mathbf{f}}_n(x)\neq 0$ , the vector ${\mathbf{f}}_n(x)\neq 0$ either. Therefore, ${\mathbf{f}}_i(x)\neq 0$ for any i. Suppose $h\in {\mathbf{H}}_x$ , then $\bigwedge^i h$ fixes ${\mathbf{f}}_i(x)$ for any i. When $i=1$ , this means that ${\mathbf{e}}_1':= x_1({\mathbf{e}})$ is an eigenvector of h with eigenvalue 1. By Corollary 2.6 and induction, we can find linearly independent vectors

\begin{align*} {\mathbf{e}}_0'={\mathbf{e}},{\mathbf{e}}_1',\ldots,{\mathbf{e}}_{n-1}', \end{align*}

such that $h({\mathbf{e}}_i')\in{\mathbf{e}}_i'+ \sum_{j=0}^{i-1}k{\mathbf{e}}_j'$ . This shows that ${\mathbf{H}}_x$ is unipotent, as claimed.

2.6.1.

The commutative monoid ${\mathbf{A}}_{\mathbf{M}}$ admits an involution $\iota_{\mathbf{A}}$ sending a simple root $\alpha$ to $-w_0(\alpha)$ , where $w_0$ is the longest element in the Weyl group with respect to the Borel (this is valid for the universal monoid associated with a general split semisimple and simply connected group, but we will not need it). Explicitly for ${\mathbf{G}}$ (in other words, $\operatorname{GL}_n$ ), this is induced by flipping the Dynkin diagram of ${\mathbf{G}}$ .

The involution $\iota_{\mathbf{A}}$ can be upgraded to an anti-involution $\iota$ on ${\mathbf{M}}$ as follows: given a fundamental representation $V_i$ of ${\mathbf{G}}^{\mathrm{sc}}$ , its contragredient representation is again a fundamental representation, denoted by $V_i^\vee$ . Any $x_i\in\operatorname{End}(V_i)$ then has the adjoint element $x_i^\vee\in\operatorname{End}(V_i^\vee)$ . There is a unique $j=n-i$ such that $\varpi_j=-w_0(\varpi_i)$ and $V_j\cong V_i^\vee$ as ${\mathbf{G}}^{\mathrm{sc}}$ -representations, and we fix one such isomorphism $\phi_{i}$ . Then $x_i^\vee$ may be viewed as an element $y_j\in\operatorname{End}(V_j)$ , and $y_j$ is independent of the choice of $\phi_i$ by Schur’s lemma. This defines an anti-involution

\begin{align*} \iota_\star\colon {\mathbf{A}}_{\mathbf{M}}\oplus\bigoplus_{i=1}^{n-1}\operatorname{End}(V_i)&\longrightarrow {\mathbf{A}}_{\mathbf{M}}\oplus\bigoplus_{i=1}^{n-1}\operatorname{End}(V_i)\\ (z, x_i)&\longmapsto (\iota_{\mathbf{A}}(z), y_i).\end{align*}

It is easy to see that $\iota_\star$ preserves ${\mathbf{M}}^\times$ : for $(z,g)\in{\mathbf{M}}^\times$ , $\iota_\star$ is given by the formula

\begin{align*} (z,g)&\longmapsto (\iota_{\mathbf{A}}(z),g^{-1}).\end{align*}

Therefore, $\iota_\star$ induces an anti-involution $\iota$ on ${\mathbf{M}}$ as desired.

Example 2.13. When $n=2$ , we have ${\mathbf{M}}=\operatorname{Mat}_2$ , and $\iota$ is the map

\begin{align*} \bigg(\begin{matrix} a & b\\ c & d \end{matrix}\bigg) \longmapsto \bigg(\begin{matrix} d & -b\\ -c & a \end{matrix}\bigg). \end{align*}

When $n\geqslant 3$ , however, it is impossible to lift $\iota_{\mathbf{A}}$ to the monoid $\operatorname{Mat}_n$ , unlike the case of ${\mathbf{M}}$ .

2.6.2.

There is also an involution $\tau$ on ${\mathbf{M}}$ specific to the ${\mathbf{G}}=\operatorname{GL}_n$ case. Indeed, the group $\operatorname{Out}({\mathbf{G}}^{\mathrm{sc}})$ of outer automorphisms has order 2, and it induces an involution $\tilde{\tau}$ on ${\mathbf{M}}$ using the standard pinning of ${\mathbf{G}}^{\mathrm{sc}}$ . Let $\dot{w}_0$ be the following element of $\operatorname{SL}_n$ lifting $w_0$ :

(2.1)

and define

\begin{align*} \tau:= \operatorname{Ad}_{\dot{w}_0}\circ\tilde{\tau}.\end{align*}

One can verify that on ${\mathbf{M}}^\times$ , $\tau$ is given by the formula

\begin{align*} (z,g)\longmapsto (\iota_{\mathbf{A}}(z),\,^{\mathrm{t}}{}{g}^{-1}),\end{align*}

so it is indeed an involution. Note that $\tau$ is compatible with the involution on ${\mathbf{G}}$ (respectively ${\mathbf{H}}$ ) given by transpose-inverse.

2.6.3.

For any k-scheme X and an étale double cover $\vartheta\colon X'\to X$ , we consider the Weil restriction

\begin{align*} {\mathbf{M}}':=\vartheta_*({\mathbf{M}}\times X'),\end{align*}

viewed as a reductive monoid scheme with ${\mathbf{H}}':=\vartheta_*({\mathbf{H}}\times X')$ -action over X. For convenience, we let ${\mathbf{G}}':= \vartheta_*({\mathbf{G}}\times X')$ . The anti-involution $\iota$ on ${\mathbf{M}}$ induces an anti-involution on ${\mathbf{M}}'$ , still denoted by $\iota$ , and the latter clearly commutes with the unique non-trivial element $\sigma\in\operatorname{Gal}(X'/X)$ . Let ${\mathfrak{M}}$ be the monoidal symmetric space

\begin{align*} {\mathfrak{M}}:= ({\mathbf{M}}')^{\iota\sigma}.\end{align*}

The group ${\mathbf{H}}$ can be identified with the fixed-point subgroup $H:= ({\mathbf{H}}')^\sigma$ , and H acts on ${\mathfrak{M}}$ .

2.6.4.

On the other hand, we have the unitary form of ${\mathbf{M}}$ defined as

\begin{align*} {\mathfrak{M}}':= ({\mathbf{M}}')^{\tau\sigma},\end{align*}

as well as unitary groups

\begin{align*} G'&:=\mathrm{U}_n=({\mathbf{G}}')^{\tau\sigma},\\ H'&:=\mathrm{U}_{n-1}=({\mathbf{H}}')^{\tau\sigma}.\end{align*}

Clearly, H’ acts on ${\mathfrak{M}}'$ by the adjoint action.

2.6.5.

The abelianization map $\alpha_{{\mathbf{M}}'}$ is easily seen to be equivariant under both $\iota\sigma$ and $\tau\sigma$ , and these two Galois actions coincide when descended to ${\mathbf{A}}_{{\mathbf{M}}'}$ . Therefore, we may define the common abelianization of ${\mathfrak{M}}$ and ${\mathfrak{M}}'$ to be the invariant space ${\mathbf{A}}_{{\mathbf{M}}'}^{\iota\sigma}={\mathbf{A}}_{{\mathbf{M}}'}^{\tau\sigma}$ . We denote it simply by ${\mathfrak{A}}_{\mathfrak{M}}$ without any prime (’) symbol.

2.6.6.

Now we study the invariant theory of the H-action on ${\mathfrak{M}}$ and that of the H’-action on ${\mathfrak{M}}'$ using the results from the ${\mathbf{H}}$ -action on ${\mathbf{M}}$ .

First, we note that both $\iota\sigma$ and $\tau\sigma$ map ${\mathbf{H}}'$ -orbits to ${\mathbf{H}}'$ -orbits, and so they descend to involutions on ${\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}':=\vartheta_*({\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}\times X')$ . Moreover, note that the action of $\iota\sigma$ coincides with that of $\tau\sigma$ , because on ${\mathbf{M}}^\times$ , $\iota$ and $\tau$ only differ by a transposition on the ${\mathbf{G}}^{\mathrm{sc}}$ -factor, which does not affect the traces on $V_i'$ and $V_i''$ . This allows us to define

\begin{align*} {\mathfrak{C}}_{{\mathfrak{M}},H}:= ({\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}')^{\iota\sigma}=({\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}')^{\tau\sigma}.\end{align*}

As a result, we have the following commutative diagram.

Lemma 2.14. We have natural isomorphisms

\begin{align*} {\mathfrak{M}}\mathbin{\!/\!\!/\!} H\stackrel{\sim}{\longrightarrow}{\mathfrak{C}}_{{\mathfrak{M}},H}\stackrel{\sim}{\longleftarrow} {\mathfrak{M}}'\mathbin{\!/\!\!/\!} H'. \end{align*}

2.6.7.

Define the twisted forms of $\chi_{{\mathbf{M}},{\mathbf{H}}}$ :

\begin{align*} \chi_{{\mathfrak{M}},H}\colon {\mathfrak{M}}&\longrightarrow{\mathfrak{C}}_{{\mathfrak{M}},H},\\ \chi_{{\mathfrak{M}},H}'\colon {\mathfrak{M}}'&\longrightarrow{\mathfrak{C}}_{{\mathfrak{M}},H},\end{align*}

then their fibers can be similarly studied using fibers of $\chi_{{\mathbf{M}},{\mathbf{H}}}$ . The extended ${\mathbf{H}}$ -discriminant divisor ${\mathbf{D}}_{{\mathbf{M}},{\mathbf{H}}}^+$ also similarly induces a principal extended H-discriminant divisor ${\mathfrak{D}}_{{\mathfrak{M}},H}^+$ on ${\mathfrak{C}}_{{\mathfrak{M}},H}$ , whose complement is the H-srs locus ${\mathfrak{C}}_{{\mathfrak{M}},H}^{\mathrm{srs}}$ . These facts are straightforward to check, and we leave the details to the reader.

2.6.8.

For future convenience, we also describe the scheme ${\mathfrak{C}}_{{\mathfrak{M}},H}$ in a more explicit manner. We will only consider the case when $X=\operatorname{Spec} R$ and $X'=\operatorname{Spec} R'$ are affine, and the general case is similar.

The scheme ${\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}'$ in this case is just an affine space ${\mathbb{A}}_{R'}^{3n-3}$ , so it suffices to describe the set ${\mathfrak{C}}_{{\mathfrak{M}},H}(R)$ as a subset of $(R')^{3n-3}$ . On the direct factor $(R')^{n-1}$ corresponding to the abelianization, the involution $\iota\sigma=\tau\sigma$ takes $(z_i)$ to the point $(\sigma z_{n-i})$ . On the remaining direct factor, if we write $(a_i',a_i'')$ corresponding to coordinates $(\chi_i',\chi_i'')$ , then the involution is the map

\begin{align*} (a_i',a_i'')\longmapsto (\sigma a_{n-i}'',\sigma a_{n-i}').\end{align*}

To see this, one only needs to notice that if we identify $V_i^\vee$ with $V_{n-i}$ using the standard volume form, then its direct summand $(V_i')^\vee$ is identified with $V_{n-i-1}'\otimes k{\mathbf{e}}$ , which is exactly $V_{n-i}''$ . As a result, ${\mathfrak{C}}_{{\mathfrak{M}},H}(R)$ is the set

\begin{align*} \{{(z_i,a_i',a_i'')\mid z_i=\sigma z_{n-i}, a_i'=\sigma a_{n-i}'', a_i''=\sigma a_{n-i}'.}\},\end{align*}

where the $z_i$ part belongs to ${\mathfrak{A}}_{\mathfrak{M}}$ .

2.7.1.

We first consider the split case and begin by reformulating $[{{\mathbf{M}}/{\mathbf{H}}}]$ in a coordinate-free way. We consider the following classifying stack (S is a k-scheme):

\begin{align*} {\underline{\mathscr{M}}}(S)=\left\{{({\mathcal{E}},x,e,e^\vee)\,\left\lvert\,\! \begin{array}{l} {\mathcal{E}}\in\operatorname{Bun}_{{\mathbf{G}}}(S)\\ x\in {\mathbf{M}}\times^{{\mathbf{G}}}{\mathcal{E}}\\ e\colon{\mathcal{O}}_S\to V_{\mathcal{E}}, e^\vee\colon V_{\mathcal{E}}\to {\mathcal{O}}_S \end{array}\right.}\!\!\!\right\}\!,\end{align*}

where $V_{\mathcal{E}}$ means the vector bundle associated with the ${\mathbf{G}}$ -torsor ${\mathcal{E}}$ and $V_{\mathcal{E}}^\vee$ its vector dual. It maps to $[{{\mathbf{M}}/{\mathbf{G}}}]$ by forgetting the vector e and co-vector $e^\vee$ , and this forgetful map is representable (by section spaces of vector bundles). As a result, ${\underline{\mathscr{M}}}$ is an algebraic stack of finite type.

Inside ${\underline{\mathscr{M}}}$ we have the closed substack ${\underline{\mathscr{M}}}_1$ consisting of tuples such that $e^\vee e=1$ .

Lemma 2.15. We have a natural isomorphism

\begin{align*} [{{\mathbf{M}}/{\mathbf{H}}}]\stackrel{\sim}{\longrightarrow}{\underline{\mathscr{M}}}_1. \end{align*}

The inverse functor is similarly straightforward: if $e^\vee e=1$ , then together e and $e^\vee$ define a direct summand of $V_{\mathcal{E}}$ that is a trivial line bundle, which is equivalent to an ${\mathbf{H}}$ -reduction ${\mathcal{E}}_{\mathbf{H}}$ of ${\mathcal{E}}$ . The rest of the verifications are left to the reader.

2.7.2.

There is an obvious fibration $b_0\colon {\underline{\mathscr{M}}}\to {\mathbb{A}}^1$ sending $({\mathcal{E}},x,e,e^\vee)$ to $e^\vee e$ , and for any $s\in{\mathbb{A}}^1$ , we let ${\underline{\mathscr{M}}}_s=b_0^{-1}(s)$ be the preimage of s. The following result is obvious.

Lemma 2.16. We have an isomorphism

\begin{align*} {\underline{\mathscr{M}}}_1\times {\mathbb{G}}_{\mathrm{m}} &\longrightarrow b_0^{-1}({\mathbb{G}}_{\mathrm{m}})\\ (({\mathcal{E}},x,e,e^\vee),s) &\longmapsto ({\mathcal{E}},x,e,se^\vee). \end{align*}

2.7.3.

We review a little bit of linear algebra before we proceed. Given a vector space V and a vector $e\in V$ , we may define, for any $i\in{\mathbb{N}}$ , a natural linear map

\begin{align*} e\wedge\colon \bigwedge^i V\longrightarrow \bigwedge^{i+1}V\end{align*}

by wedging e on the left. On the other hand, given a co-vector $e^\vee \in V^\vee $ , we may also define a contraction (linear) map

\begin{align*} e^\vee \colon\bigwedge^{i+1}V&\longrightarrow \bigwedge^iV\\ e_0\wedge\cdots\wedge e_i&\longmapsto \sum_{j=0}^i(-1)^j(e^\vee e_j) (e_0\wedge\cdots\wedge\hat{e}_j\wedge\cdots \wedge e_i),\end{align*}

where $\hat{e}_j$ means the $e_j$ -term is deleted.

2.7.4.

Let ${\mathbf{C}}_{\underline{\mathscr{M}}}:= {\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}\times{\mathbb{A}}^1$ , and we now define a map

\begin{align*} {\underline{\mathscr{M}}}\longrightarrow{\mathbf{C}}_{\underline{\mathscr{M}}}\end{align*}

extending the map $[{{\mathbf{M}}/{\mathbf{H}}}]\to {\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}$ . Given $({\mathcal{E}},x,e,e^\vee )\in{\underline{\mathscr{M}}}$ , we have invariants $a_i=\chi_i(x)$ as before. For $1\leqslant i\leqslant n-1$ , recall the element $y_i\in \operatorname{End}(V_i)$ induced by any element $y\in{\mathbf{M}}$ . By twisting by ${\mathcal{E}}$ , we have $x_i\in\operatorname{End}(V_{{\mathcal{E}},i})$ associated with x, where $V_{{\mathcal{E}},i}=\bigwedge^iV_{\mathcal{E}}$ . Define $x_i^{e,e^\vee }$ to be the composition map

\begin{align*} V_{{\mathcal{E}},i-1}\stackrel{e\wedge}{\longrightarrow}V_{{\mathcal{E}},i}\stackrel{x_i}{\longrightarrow} V_{{\mathcal{E}},i}\stackrel{e^\vee }{\longrightarrow}V_{{\mathcal{E}},i-1},\end{align*}

and let $\chi_i^{e,e^\vee }(x)$ be the trace of $x_i^{e,e^\vee }$ . Let

\begin{align*} b_i= \begin{cases} e^\vee e, & i=0,\\ \chi_i^{e,e^\vee }(x), & 1\leqslant i\leqslant n-1. \end{cases}\end{align*}

When $V=k^n$ , $e={\mathbf{e}}$ , $e^\vee ={\mathbf{e}}^\vee $ , it is easy to see that $b_0=1$ and for $i\geqslant 1$ , $b_i=a_i''$ . We already have $a_i=a_i'+a_i''$ , so $\alpha_i$ (viewed as functions on ${\mathbf{A}}_{\mathbf{M}}$ ), $\chi_i$ , and $\chi_i^{e,e^\vee }$ give an alternative set of coordinates of ${\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}$ . Define

\begin{align*} \chi_{\underline{\mathscr{M}}}\colon {\underline{\mathscr{M}}}&\longrightarrow {\mathbf{C}}_{\underline{\mathscr{M}}}\\ ({\mathcal{E}},x,e,e^\vee )&\longmapsto (\alpha_{\mathbf{M}}(x),\chi_i(x),\chi_i^{e,e^\vee }(x),b_0).\end{align*}

The restriction of $\chi_{\underline{\mathscr{M}}}$ to ${\underline{\mathscr{M}}}_1$ is the same as the one induced by $\chi_{{\mathbf{M}},{\mathbf{H}}}$ .

2.7.5.

There is a section from ${\mathbf{C}}_{\underline{\mathscr{M}}}$ to ${\underline{\mathscr{M}}}$ extending the one on ${\mathbf{C}}_{{\mathbf{M}},{\mathbf{H}}}$ , defined as follows: given $z\in{\mathbf{A}}_{\mathbf{M}}$ , $a=(a_i)$ , $b=(b_i)$ (including $b_0$ ), we let ${\mathcal{E}}={\mathcal{E}}_0$ be the trivial ${\mathbf{G}}$ -torsor (so that $V_{\mathcal{E}}=k^n$ ), $x_{z,a}=\epsilon_{\mathbf{M}}(z,a)$ , $e={\mathbf{e}}$ , and

\begin{align*} e^\vee ={\mathbf{e}}_b^\vee := (-1)^{n-1}b_1{\mathbf{e}}_1^\vee+(-1)^{n-2}b_2{\mathbf{e}}_2^\vee+\cdots-b_{n-1}{\mathbf{e}}_{n-1}^\vee+b_0{\mathbf{e}}^\vee.\end{align*}

We define the map

\begin{align*} \epsilon_{\underline{\mathscr{M}}}\colon{\mathbf{C}}_{\underline{\mathscr{M}}}&\longrightarrow {\underline{\mathscr{M}}}\\ (z,a,b)&\longmapsto ({\mathcal{E}}_0,x_{z,a},{\mathbf{e}},{\mathbf{e}}_b^\vee ).\end{align*}

Proof. The proof is, in essence, similar to that for $\epsilon_{{\mathbf{M}},{\mathbf{H}}}$ , and clearly we only need to consider the $b_j$ -coordinates for $j\geqslant 1$ . We choose the basis of $V_j$ induced by the standard basis of $k^n$ . Since wedging with ${\mathbf{e}}$ kills any pure tensor with factor ${\mathbf{e}}$ , we only need to consider the effect of $x_j^{e,e^\vee }$ on basis elements of the form

\begin{align*} {\mathbf{e}}_{i_1\cdots i_{j-1}}={\mathbf{e}}_{i_1}\wedge\cdots\wedge {\mathbf{e}}_{i_{j-1}},\quad 1\leqslant i_1\lt\cdots\lt i_{j-1}\leqslant n-1. \end{align*}

As before, let $A=\epsilon(a)\in\operatorname{SL}_n$ be the companion matrix associated with a, then A sends ${\mathbf{e}}_i$ to $(-1)^{i-1}a_i{\mathbf{e}}_1+{\mathbf{e}}_{i+1}$ (recall ${\mathbf{e}}_n={\mathbf{e}}$ ) and ${\mathbf{e}}$ to $(-1)^{n-1}{\mathbf{e}}_1$ . This means that ${\mathbf{e}}\wedge{\mathbf{e}}_{i_1\cdots i_{j-1}}$ is sent to

\begin{align*} & (-1)^{n-1}{\mathbf{e}}_1\wedge ((-1)^{i_1-1}a_{i_1}{\mathbf{e}}_1+{\mathbf{e}}_{i_1+1})\wedge \cdots\wedge ((-1)^{i_{j-1}-1}a_{i_{j-1}}{\mathbf{e}}_1+{\mathbf{e}}_{i_{j-1}+1})\\&\quad = (-1)^{n-1}{\mathbf{e}}_1\wedge {\mathbf{e}}_{i_1+1}\wedge \cdots\wedge {\mathbf{e}}_{i_{j-1}+1}. \end{align*}

Scaling the result by $\delta_{\mathbf{M}}(z)$ , we see that $x_{z,a}$ sends ${\mathbf{e}}\wedge{\mathbf{e}}_{i_1\cdots i_{j-1}}$ to

\begin{align*} ((z_2\cdots z_{i_1})(z_3\cdots z_{i_2})\cdots(z_{j}\cdots z_{i_{j-1}})(-1)^{n-1}){\mathbf{e}}_1\wedge {\mathbf{e}}_{i_1+1}\wedge \cdots\wedge {\mathbf{e}}_{i_{j-1}+1}, \end{align*}

where the product $z_{l+1}\cdots z_{i_l}$ is 1 if $i_l\lt l+1$ , or equivalently, $i_l=l$ . To compute the contribution of ${\mathbf{e}}_{i_1\cdots i_{j-1}}$ to the trace $\chi_j^{e,e^\vee }$ , we need to compute the contraction with ${\mathbf{e}}_b^\vee $ and then project to the basis ${\mathbf{e}}_{i_1\cdots i_{j-1}}$ . Therefore, for ${\mathbf{e}}_{i_1\cdots i_{j-1}}$ to have non-zero contribution to $b_j$ , we must have

\begin{align*} (1,i_1+1,\ldots,i_{j-2}+1)=(i_1,i_2,\ldots,i_{j-1})=(1,2,\ldots,j-1). \end{align*}

Moreover, only the term involving ${\mathbf{e}}_{j}^\vee $ in ${\mathbf{e}}_b^\vee $ can contribute to $b_j$ . The trace $\chi_j^{e,e^\vee }$ can then be computed to be the coefficient in front of the pure tensor

\begin{align*} (-1)^{n-j}b_j{\mathbf{e}}_j^\vee ((-1)^{n-1}{\mathbf{e}}_1\wedge {\mathbf{e}}_{2}\wedge \cdots\wedge {\mathbf{e}}_{j}) = b_j{\mathbf{e}}_1\wedge {\mathbf{e}}_{2}\wedge \cdots\wedge {\mathbf{e}}_{j-1}, \end{align*}

which is $b_j$ . This finishes the proof.

2.7.6.

By replacing ${\mathbf{e}}$ with e (respectively ${\mathbf{e}}^\vee $ with $e^\vee $ ), we can define the analogue of ${\mathbf{f}}_i(x)$ (respectively ${\mathbf{f}}_i^\vee (x)$ ) for any $m=({\mathcal{E}},x,e,e^\vee )$ :

\begin{align*} {\mathbf{f}}_i(m)&= \begin{cases} x_1 e, & i=1,\\ x_i(e\wedge {\mathbf{f}}_{i-1}(m)), & 1\lt i\lt n,\\ e\wedge {\mathbf{f}}_{n-1}(m), & i=n; \end{cases}\\ {\mathbf{f}}_i^\vee (m)&= \begin{cases} e^\vee x_1, & i=1,\\ ({\mathbf{f}}_{i-1}^\vee (m)\wedge e^\vee )x_i, & 1\lt i\lt n,\\ {\mathbf{f}}_{n-1}^\vee (m)\wedge e^\vee, & i=n. \end{cases}\end{align*}

Let ${\underline{\mathscr{M}}}^\times\subset{\underline{\mathscr{M}}}$ be the preimage of $[{{\mathbf{M}}^\times/{\mathbf{G}}}]$ , then clearly it is an open dense substack. Similar to Corollary 2.6, we have the following.

On the other hand, we claim that ${\underline{\mathscr{M}}}^{\times,{\mathrm{srs}}}$ is equal to $\chi_{{\underline{\mathscr{M}}}}^{-1}(\chi_{{\underline{\mathscr{M}}}}({\underline{\mathscr{M}}}^{\times,{\mathrm{srs}}}))$ and also isomorphic to $\chi_{{\underline{\mathscr{M}}}}({\underline{\mathscr{M}}}^{\times,{\mathrm{srs}}})$ . By ${\mathbf{Z}}_{\mathbf{M}}$ -equivariance, we only need to prove this for the closed substack of ${\underline{\mathscr{M}}}^{\times,{\mathrm{srs}}}$ lying over $[{{\mathbf{G}}^{\mathrm{sc}}/{\mathbf{G}}}]$ , or equivalently, the abelianization is 1. By embedding ${\mathbf{G}}^{\mathrm{sc}}=\operatorname{SL}_n$ into $\operatorname{Mat}_n=\mathfrak{gl}_n$ , the claim then follows from the well-known result from the Lie algebra case [Reference YunYun11].

Finally, given a map ${\underline{\mathscr{M}}}\to S$ , where S is affine, its restriction to ${\underline{\mathscr{M}}}^{\times,{\mathrm{srs}}}\cup b_0^{-1}({\mathbb{G}}_{\mathrm{m}})$ uniquely factors through the image of the latter in ${\mathbf{C}}_{{\underline{\mathscr{M}}}}$ , whose complement has codimension 2. Since S is affine and ${\mathbf{C}}_{\underline{\mathscr{M}}}$ is normal, the factorization then extends to the whole ${\mathbf{C}}_{\underline{\mathscr{M}}}$ .

By definition, x is semisimple, so we may choose the trivialization of ${\mathcal{E}}$ in such a way that x is identified with a point in $\bar{{\mathbf{T}}}_{\mathbf{M}}(R)$ . Since x is also ${\mathbf{G}}$ -regular, it is contained in the big-cell locus (e.g. see [Reference WangWan26, § 2.4.15] for a quick summary). By possibly conjugating by the Weyl group, we may assume that x can be written as

\begin{align*} x=e_{I,\Delta}zt, \end{align*}

where $e_{I,\Delta}$ is an idempotent in $\bar{{\mathbf{T}}}_{\mathbf{M}}$ , $t\in{\mathbf{T}}^{\mathrm{sc}}(R)$ , and $z\in{\mathbf{Z}}_{\mathbf{M}}\simeq{\mathbf{T}}^{\mathrm{sc}}(R)$ . Since the stabilizer stays constant along ${\mathbf{Z}}_{\mathbf{M}}$ -orbits in ${\mathbf{M}}\times{\mathbb{A}}^n\times{\mathbb{A}}^n$ (where ${\mathbf{Z}}_{\mathbf{M}}$ acts on ${\mathbb{A}}^n\times{\mathbb{A}}^n$ trivially), we may assume that $z=1$ . Some basic properties of $e_{I,\Delta}$ can be found in [Reference WangWan26, § 2.4.17], and since ${\mathbf{G}}=\operatorname{GL}_n$ , an explicit description is as follows: the standard basis of the standard representation $V_1$ has weights

\begin{align*} \varpi_1,\varpi_1-\alpha_1,\ldots,\varpi_1-\alpha_1-\cdots -\alpha_{n-1}. \end{align*}

For each $1\lt i\leqslant n-1$ , the highest weight of $V_i$ is

\begin{align*} \varpi_i&=\sum_{j=1}^{i}\bigg(\varpi_1-\sum_{l=1}^{j-1}\alpha_l\bigg)\\ &=i\varpi_1-\sum_{j=1}^{i}(i-j)\alpha_j, \end{align*}

and so lower weights in $V_i$ can be written as

\begin{align*} \mu:=\varpi_i-\sum_{j=i}^{n-1}c_j\alpha_j, \end{align*}

such that if $c_j\neq 0$ , then $c_l\neq 0$ for all $i\leqslant l\lt j$ . The eigenvalue $\mu(e_{I,\Delta})$ is 1 if $c_j=0$ outside I, and 0 otherwise. Therefore, we may assume without loss of generality that

\begin{align*} I=\{1,\ldots,s\},\quad 0\leqslant s\leqslant n-1. \end{align*}

Let $t=\operatorname{diag}(t_1,\ldots,t_n)$ , and suppose $e=\,^{\mathrm{t}}{}{(}e_1,\ldots,e_n)\in R^n$ , then ${\mathbf{f}}_n(m)$ is the determinant of the matrix

Since ${\mathbf{f}}_n(m)$ is invertible, we must have $e_i\in R^\times$ for all i. The stabilizer of x in ${\mathbf{G}}$ is diagonal, so $e_i\in R^\times$ implies that if $\operatorname{Ad}_g(x)=x$ and $ge=e$ , then $g=1$ . This proves the claim that ${\underline{\mathscr{M}}}^{\mathrm{srs}}$ is a sheaf.

Still assuming $z=1$ , the invariants $b_i$ can be similarly computed as entries in the row vector

\begin{align*} (e_1e_1^\vee,\ldots,e_ne_n^\vee){\mathbf{F}}, \end{align*}

where ${\mathbf{F}}$ is the matrix

in which $\Sigma_{i,l}$ is the summation

\begin{align*} \sum_{\substack{1\leqslant j_1\lt\cdots\lt j_l\leqslant\max\{l,s+1\}\\i\neq j_1,\ldots, j_l}}t_{j_1}\cdots t_{j_l}. \end{align*}

We claim that ${\mathbf{F}}$ is invertible: indeed, if $s\lt l\lt n$ , then $\Sigma_{l+1,l}$ is simply the product

\begin{align*} t_1\cdots t_{l}\in R^\times, \end{align*}

so we only need to show that the determinant of the upper-right block of ${\mathbf{F}}$ is invertible. Using elementary column operations and induction, we see that the said determinant is, up to a sign, equal to that of the Vandermonde matrix

\begin{align*} \begin{pmatrix} t_1^{s+1} & \cdots & t_1\\ \vdots & & \vdots \\[2pt] t_{s+1}^{s+1} & \cdots & t_{s+1} \end{pmatrix}, \end{align*}

which is invertible because the upper-right corner of ${\mathbf{f}}_n(m)$ is invertible.

If m and m’ are two points in ${\underline{\mathscr{M}}}^{\mathrm{srs}}$ with the same invariants, then by ${\mathbf{G}}$ -conjugation, we may assume that ${\mathcal{E}}={\mathcal{E}}'$ and $x=x'$ . Since ${\mathbf{F}}$ is invertible, we have $e_ie_i^\vee=e_i'e_i^{\vee\prime}$ for all i. Therefore, using that $e_i\in R^\times$ , we may find a scalar in $R^\times$ that simultaneously sends e to e’ and $e^\vee$ to $e^{\vee\prime}$ . This shows that $\chi_{{\underline{\mathscr{M}}}}$ is a bijection on R-points for any Artinian algebra R, hence it must be an isomorphism.

2.7.7.

The Galois twisted form of ${\underline{\mathscr{M}}}$ is similarly formulated. We already have the monoidal symmetric space ${\mathfrak{M}}$ and groups $G\simeq{\mathbf{G}}$ and $H\simeq{\mathbf{H}}$ over X, induced by a given étale double cover $\vartheta\colon X'\to X$ .

We define the unitary form ${\mathscr{M}}'$ to be the stack classifying tuples $({\mathcal{E}}',x',e')$ , where ${\mathcal{E}}'$ is a $G'=\mathrm{U}_n$ -torsor over X, $x'\in{\mathfrak{M}}'\times^{G'}{\mathcal{E}}'(S)$ , and $e'\colon {\mathcal{O}}_{S'}\to V'_{{\mathcal{E}}'}$ is a vector. Here, $S'=X'\times_X S$ and $V'_{{\mathcal{E}}'}$ is the Hermitian vector bundle on S’ associated with ${\mathcal{E}}'$ . Through the Hermitian form, we may identify $V'_{{\mathcal{E}}'}$ with the dual of $\sigma^*V'_{{\mathcal{E}}'}$ , and so e’ induces a co-vector

\begin{align*} e^{\prime\vee}\colon V'_{{\mathcal{E}}'}\simeq(\sigma^*V'_{{\mathcal{E}}'})^\vee \to\sigma^*{\mathcal{O}}_{S'}\simeq {\mathcal{O}}_{S'}.\end{align*}

2.7.8.

Both ${\mathscr{M}}$ and ${\mathscr{M}}'$ map to the same space ${\mathfrak{C}}_{\mathscr{M}}$ of invariants

\begin{align*} {\mathscr{M}}\longrightarrow {\mathfrak{C}}_{\mathscr{M}}\longleftarrow {\mathscr{M}}'.\end{align*}

Here, ${\mathfrak{C}}_{\mathscr{M}}$ is isomorphic to ${\mathfrak{C}}_{{\mathfrak{M}},H}\times{\mathbb{A}}^1$ , where ${\mathbb{A}}^1$ corresponds to the invariant $b_0$ . More concretely, if $X=\operatorname{Spec}{R}$ and $X'=\operatorname{Spec}{R'}$ , then ${\mathfrak{C}}_{\mathscr{M}}(R)$ is the subset of $(R')^{3n-2}$

\begin{align*} \{{(z_1,\ldots,z_{n-1},a_1,\ldots,a_{n-1},b_1,\ldots,b_{n-1},b_0) \mid z_i=\sigma z_{n-i},a_i=\sigma a_{n-i}, b_i=\sigma (b_0a_{n-i}-b_{n-i})}\},\end{align*}

where we use the convention $a_n:= 1$ and $b_n:= 0$ , so that $b_0=\sigma (b_0a_n-b_n)$ is equivalent to saying $b_0\in R$ .

2.7.9.

The pure tensors ${\mathbf{f}}_i$ and ${\mathbf{f}}_i^\vee $ are not stable under the Galois action if $0\lt i\lt n$ , but both ${\mathbf{f}}_n$ and ${\mathbf{f}}_n^\vee $ have twisted forms, which we denote by $f_n(x)$ and $f_n^\vee (x)$ for $({\mathcal{E}},x,e,e^\vee )\in {\mathscr{M}}$ , and $f_n'(x')$ for $({\mathcal{E}}',x',e')$ for $({\mathcal{E}}',x',e')\in{\mathscr{M}}'$ . In the unitary case, the co-tensor $f_n^{\prime\vee}(x)$ is the dual of $\sigma^*f_n'(x')$ . Additionally, ${\mathbf{f}}_i$ and ${\mathbf{f}}_i^\vee $ are still well-defined as tensors over S’, and if $i=n$ , they coincide with the pullbacks of $f_n$ and $f_n^\vee $ , respectively.

3.1.1.

Recall we have surjective morphisms of stacks

\begin{align*} {\mathscr{M}}\longrightarrow {\mathfrak{C}}_{\mathscr{M}}\longleftarrow {\mathscr{M}}'.\end{align*}

We define ${\mathfrak{C}}_{\mathscr{M}}^\heartsuit({\mathcal{O}}_v)$ to be the subset of ${\mathfrak{C}}_{\mathscr{M}}({\mathcal{O}}_v)$ consisting of points whose restriction to $F_v$ is invertible and strongly regular semisimple.

Similarly as in [Reference WangWan26, § 4], the image of $a\in{\mathfrak{C}}_{\mathscr{M}}({\mathcal{O}}_v)$ that is invertible over $F_v$ induces a point in ${\mathfrak{A}}_{\mathfrak{M}}({\mathcal{O}}_v)\cap{\mathfrak{A}}_{\mathfrak{M}}^\times(F_v)$ , and so we have a canonically attached dominant cocharacter $\lambda_v\in\check{\mathbb{X}}(T^{\mathrm{ad}})$ , called the boundary divisor of a.

Definition 3.1 Let $a\in{\mathfrak{C}}_{\mathscr{M}}^\heartsuit({\mathcal{O}}_v)$ . The affine Jacquet–Rallis fiber ${\mathcal{M}}_{v}(a)$ is the k-stack (for the étale topology) associated with the prestack sending a k-algebra R to the groupoid

\begin{align*} \{{m\in {\mathscr{M}}({\mathcal{O}}_v\hat{\otimes}_k R)\mid \chi_{\mathscr{M}}(m)=a\in{\mathfrak{C}}_{\mathscr{M}}({\mathcal{O}}_v\hat{\otimes}_k R)}\}. \end{align*}

Similarly, we define ${\mathcal{M}}_v'(a)$ by replacing ${\mathscr{M}}$ with ${\mathscr{M}}'$ . For any k-field embedding $\bar{v}\colon k_v\to \bar{k}$ , we also define ${\mathcal{M}}_{\bar{v}}(a)$ (respectively ${\mathcal{M}}_{\bar{v}}'(a)$ ) to be the analogue of ${\mathcal{M}}_v(a)$ (respectively ${\mathcal{M}}_v'(a)$ ) by replacing ${\mathcal{O}}_v\hat{\otimes}_k R$ by ${\mathcal{O}}_{\bar{v}}\hat{\otimes}_{\bar{k}}R$ for any $\bar{k}$ -algebra R.

Clearly, we have the canonical isomorphism

\begin{align*} {\mathcal{M}}_v(a)\otimes_k\bar{k}\simeq\prod_{\bar{v}\colon k_v\to \bar{k}}{\mathcal{M}}_{\bar{v}}(a),\end{align*}

where $\operatorname{Gal}(k_v/k)$ acts on the direct factors of the right-hand side simply transitively by permutations, and similarly for ${\mathcal{M}}_v'(a)$ .

3.1.2.

Before we study the properties of ${\mathcal{M}}_v(a)$ and ${\mathcal{M}}_v'(a)$ , we need to also recall the usual notion of multiplicative affine Springer fibers (MASFs), following [Reference WangWan26, § 4].

Note that over ${\mathcal{O}}_v$ , the unitary group G’ becomes quasi-split, and so the formulations in [Reference WangWan26, § 4] apply directly to the unitary side. For the symmetric-space side, it is not directly addressed in [Reference WangWan26], but the basic definitions still make sense using the monoidal symmetric space ${\mathfrak{M}}$ , and the geometric results still hold because ${\mathfrak{M}}$ becomes split after base change to $\bar{k}$ , and so we only have to worry about rationality questions.

Let $\bar{a}\in{\mathfrak{C}}_{\mathfrak{M}}({\mathcal{O}}_v)$ be the image of a, which is generically invertible and regular semisimple, and we restrict ourselves to only the MASF associated with $\bar{a}$ . The definition of MASFs in [Reference WangWan26, 4] relies on an $F_v$ -point $\gamma\in{\mathfrak{M}}(F_v)$ (respectively $\gamma'\in{\mathfrak{M}}'(F_v)$ ) lying over $\bar{a}$ . In the unitary case, $\gamma'$ always exists due to a well-known result by Steinberg (one can also see [Reference WangWan26, Theorem 2.4.24] for a slightly stronger result based on Steinberg’s argument).

In the symmetric case, since a is generically strongly regular semisimple, it lifts to a unique $F_v$ -point of ${\mathscr{M}}$ , which in particular gives a regular semisimple element $[\gamma]\in[{{\mathfrak{M}}/G}](F_v)$ . Since $G=\operatorname{GL}_n$ , any G-bundle over $F_v$ is trivial, and so if we choose a trivialization of the G-bundle associated with $[\gamma]$ , then it induces the desired $\gamma$ .

3.1.3.

A subtlety here is that a also induces a point $[\gamma'']\in[{{\mathfrak{M}}'/G'}](F_v)$ , whose associated G’-bundle may or may not be trivial. By local Tate–Nakayama duality, it has a canonical obstruction element

\begin{align*} \operatorname{Ob}_v(a)=\operatorname{Ob}{[\gamma'']}\in\operatorname{H}^1(F_v,G')\simeq\pi_0(Z_{\check{{\mathbf{G}}}}^{\Gamma_v})^*,\end{align*}

where superscript $*$ means Pontryagin dual, and $\Gamma_v$ is the Galois group of $F_v$ , which acts on $Z_{\check{{\mathbf{G}}}}={\mathbb{G}}_{\mathrm{m}}$ through $\operatorname{Gal}(k_v'/k_v)$ . This obstruction is non-trivial only if v’ is non-split over v, in which case the Frobenius element acts on ${\mathbb{G}}_{\mathrm{m}}$ by inversion, and we have

\begin{align*} \pi_0(Z_{\check{{\mathbf{G}}}}^{\Gamma_v})=\pi_0({\mathbb{G}}_{\mathrm{m}}^{\operatorname{Gal}(k_v'/k_v)})=\{\pm 1\}\simeq {\mathbb{Z}}/2{\mathbb{Z}}.\end{align*}

We will study the obstruction $\operatorname{Ob}_v(a)$ in more detail in § 3.2.

3.1.4.

Using $\gamma$ (respectively $\gamma'$ ), we can define MASF ${\mathcal{M}}_v^{\mathrm{Hit}}(a)$ (respectively ${\mathcal{M}}_v^{\prime{\mathrm{Hit}}}(a)$ ) the same way as in [Reference WangWan26, § 4]. Concretely, ${\mathcal{M}}_v^{\mathrm{Hit}}(a)(R)$ is the groupoid classifying tuples $(E,\phi,\iota)$ , where E is a G-bundle over $X_{v,R}=\operatorname{Spec}{{\mathcal{O}}_v\hat{\otimes}_k R}$ , $\phi$ is a section of ${\mathfrak{M}}\times^GE$ over $X_{v,R}$ such that $(E,\phi)$ lies over a, and $\iota$ is an isomorphism over $X_{v,R}^\bullet$ between $(E,\phi)$ and $(E_0,\gamma)$ ( $E_0$ means the trivial G-torsor). ${\mathcal{M}}_v^{\prime{\mathrm{Hit}}}(a)$ is described similarly. An explicit description of the k-points of MASF (in the unitary case) involving Cartan double cosets and the boundary divisor of $\gamma'$ may be found in [Reference WangWan26, §§ 4.1.2 and 4.1.12].

In the symmetric case, we have a well-defined forgetful map

(3.1) \begin{align} {\mathcal{M}}_v(a)\longrightarrow {\mathcal{M}}_v^{\mathrm{Hit}}(a)\end{align}

by forgetting the vector and the co-vector. Note that for any $m\in{\mathcal{M}}_v(a)$ , the rational isomorphism $\iota$ of its image under this map is induced by the fact that ${\mathscr{M}}^{\mathrm{srs}}\to{\mathfrak{C}}_{\mathscr{M}}^{\mathrm{srs}}$ is an isomorphism.

In the unitary case, due to the obstruction $\operatorname{Ob}_v(a)$ , we may not have a forgetful map

(3.2) \begin{align} {\mathcal{M}}_v'(a)\longrightarrow {\mathcal{M}}_v^{\prime{\mathrm{Hit}}}(a).\end{align}

Nevertheless, we may slightly modify the definition of ${\mathcal{M}}_v^{\prime{\mathrm{Hit}}}(a)$ by replacing the distinguished point $(E_0,\gamma')$ by $[\gamma'']$ . The resulting functor ${\mathcal{M}}_v^{\prime\prime{\mathrm{Hit}}}(a)$ is isomorphic to ${\mathcal{M}}_v^{\prime{\mathrm{Hit}}}(a)$ after base changing to $\bar{k}$ , so its geometric properties are still the same, and only the Galois structure changes. We then have a forgetful map

(3.3) \begin{align} {\mathcal{M}}_v'(a)\longrightarrow {\mathcal{M}}_v^{\prime\prime{\mathrm{Hit}}}(a).\end{align}

We first prove that both functors in (3.1) are sheaves. Since, for any k-algebra R, we have the inclusion map ${\mathcal{O}}_v\hat{\otimes}_k R\subset F_v\hat{\otimes}_k R$ , automorphisms of any point in ${\mathcal{M}}_v(a)(R)$ (a tuple defined over ${\mathcal{O}}_v\hat{\otimes}_k R$ ) are determined after restricting to $F_v\hat{\otimes}_k R$ , which must be trivial because a is strongly regular semisimple. Therefore, ${\mathcal{M}}_v(a)$ is a sheaf. It is also known (and easily seen) that ${\mathcal{M}}_v^{\mathrm{Hit}}(a)$ is a sheaf. So (3.1) is a morphism of sheaves.

Next, we show injectivity. Suppose m and m’ are two points in ${\mathcal{M}}_v(a)(R)$ with isomorphic image in ${\mathcal{M}}_v^{\mathrm{Hit}}(a)$ . Without loss of generality, we may even assume that their images are identical, denoted by $(E,\phi,\iota)$ . Since a is strongly regular semisimple over $F_v$ , m and m’, viewed as the G-bundle E together with some additional data over ${\mathcal{O}}_v\hat{\otimes}_k R$ , are isomorphic when restricted to $F_v\hat{\otimes}_k R$ . In particular, the vector (respectively co-vector) parts of both m and m’ are isomorphic, via $\iota$ , to the same vector (respectively co-vector) in $(F_v\hat{\otimes}_kR)^n$ , the latter being the (trivial) vector bundle induced by $[\gamma]$ over $F_v\hat{\otimes}_k R$ . Since the map ${\mathcal{O}}_v\hat{\otimes}_kR\to F_v\hat{\otimes}_kR$ is injective, this means the vector and co-vector parts of m and m’ remain, respectively, isomorphic over ${\mathcal{O}}_v\hat{\otimes}_kR$ . This proves injectivity.

3.1.5.

In order to show the representability of ${\mathcal{M}}_v(a)$ and ${\mathcal{M}}_v'(a)$ , we need an alternative description of these functors using lattices. In fact, we are going to prove that both functors are representable by a projective scheme over k, and by Galois descent, it suffices to prove the same claim over $\bar{k}$ . Therefore, we may assume that v’ is split over v and ${\mathcal{M}}_v(a)\cong {\mathcal{M}}_v'(a)$ . We may also assume $k_v=k$ for simplicity.

By splitness, we now have a section from ${\mathfrak{C}}_{\mathscr{M}}$ to ${\mathscr{M}}$ , sending a to a distinguished point $m=(V,\gamma,e,e^\vee)\in{\mathscr{M}}({\mathcal{O}}_v)$ , where V is the trivial vector bundle of rank n, $\gamma\in{\mathfrak{M}}({\mathcal{O}}_v)$ , and e (respectively $e^\vee$ ) is a vector (respectively co-vector) in $V({\mathcal{O}}_v)$ . For any local k-algebra R, an R-point of ${\mathcal{M}}_v(a)$ can then be identified with an element $g\in G(R(\!(\pi_v)\!))/G(R[\![\pi_v]\!])=\operatorname{GL}_n(R(\!(\pi_v)\!))/\operatorname{GL}_n(R[\![\pi_v]\!])$ such that $\operatorname{Ad}_g^{-1}(\gamma)\in {\mathfrak{M}}(R[\![\pi_v]\!])$ , and both $g^{-1}e$ and $e^\vee g$ are integral (namely, have $R[\![\pi_v]\!]$ entries).

For each $i\geqslant 0$ , let $V_i=\bigwedge^i V$ as before, then g induces a full-rank $R[\![\pi_v]\!]$ -lattice $\Lambda=gV(R[\![\pi_v]\!])$ in $V(R(\!(\pi_v)\!))$ , hence also lattices

\begin{align*} \Lambda_i:=\bigwedge^i\Lambda_i=\bigg(\bigwedge^i g\bigg)V_i(R[\![\pi_v]\!]).\end{align*}

Let $\gamma_i\in\operatorname{End}(V_i)$ ( $1\leqslant i\leqslant n-1$ ) be the elements induced by $\gamma$ , then we must have

\begin{align*} \gamma_i\Lambda_i\subset\Lambda_i,\end{align*}

because $\operatorname{Ad}_g^{-1}(\gamma)$ is integral. An immediate consequence is the following.

Lemma 3.3. Suppose v’ is split over v, then the functor ${\mathcal{M}}_v(a)\cong{\mathcal{M}}_v'(a)$ is isomorphic to the functor whose R-points are

(3.4) \begin{align} \{{R[\![\pi_v]\!]\text{-lattices }\Lambda\in V(R(\!(\pi_v)\!)) \mid \gamma_i\Lambda_i\subset\Lambda_i, e\in\Lambda, e^\vee\in\Lambda^\vee}\}. \end{align}

3.1.6.

Recall that, to $m=(V,\gamma,e,e^\vee)$ , we can associate pure tensors $f_i(m)$ and $f_i^\vee(m)$ , defined inductively (cf. § 2.7). These tensors are integral because m is. Similarly, the tuple $m'=(V,\operatorname{Ad}_g^{-1}(\gamma), g^{-1}e,e^\vee g)$ is also integral, so its associated tensors $f_i(m')$ and $f_i^\vee(m')$ are again integral. It is not hard to see that

\begin{align*} f_i(m)&=\bigg(\bigwedge^i g\bigg)f_i(m')\in \Lambda_i,\\ f_i^\vee(m)&=f_i^\vee(m')\bigg(\bigwedge^i g^{-1}\bigg)\in \Lambda_i^\vee.\end{align*}

Lemma 3.4. There exists some integer N depending only on a, such that for any $\Lambda\in{\mathcal{M}}_v(a)(R)$ , we have

\begin{align*} \pi_v^{N}V(R[\![\pi_v]\!])\subset\Lambda\subset\pi_v^{-N}V(R[\![\pi_v]\!]). \end{align*}

Proof. First, we have $F_v$ -linearly independent vectors e and $f_1(m)=\gamma_1e$ in $\Lambda$ . Since $\gamma_1\Lambda\subset\Lambda$ , the vector $\gamma_1^2e$ is contained in $\Lambda$ as well. Since a is generically invertible, $f_2(m)$ is equal to $c_2(\gamma_1e\wedge\gamma_1^2e)$ for some $c_2\in F_v^\times$ . Such $c_2$ depends on the abelianization part of a. By induction, and the fact that $f_n(m)\in F_v^\times$ because a is generically strongly regular semisimple, we obtain linearly independent vectors in $\Lambda$ :

\begin{align*} e,\gamma_1e,\ldots,\gamma_1^{n-1}e. \end{align*}

They generate a sublattice of $\Lambda$ , and thus $\Lambda$ must be bounded below. Similarly, $\Lambda^\vee$ must also be bounded below, which implies that $\Lambda$ is bounded above. Both bounds depend only on $\gamma$ or a as claimed.

3.2.1.

We first describe $\operatorname{Ob}_v(a)$ . We assume that v’ is non-split over v, and let $F_v'=F_v\otimes_{k_v}k_v'$ . Let $\sigma$ be the non-trivial element in $\operatorname{Gal}(F_v'/F_v)$ , which is identified with the Frobenius element of $k_v$ . Let $\sigma$ (respectively $g\mapsto \bar{g}$ ) be the Frobenius action on $G'(F_v')=\operatorname{GL}_n(F_v')$ induced by G’ (respectively the split $F_v$ -group $\operatorname{GL}_n$ ), then

\begin{align*} \sigma(g)=\,^{\mathrm{t}}{}{\bar{g}}^{-1}\end{align*}

for all $g\in G'(F_v')$ . Since $\operatorname{GL}_n$ -torsors are trivial over $F_v'$ , any G’-torsor over $F_v$ is induced by a cocycle

\begin{align*} \sigma\longmapsto g\in G'(F_v'),\end{align*}

where g is such that $g\sigma(g)=1$ , or, in other words, $g=\,^{\mathrm{t}}{}{\bar{g}}$ is Hermitian.

Let $A'=G'/(G')^{\mathrm{der}}$ . It is a non-split one-dimensional torus whose $F_v$ -points are norm-1 elements in $(F_v')^\times$ . Then the cocycle given by $g\in G'(F_v')$ is a coboundary if and only if $\det{g}\in A'(F_v')=(F_v')^\times$ has even valuation.

3.2.2.

Consider $a\in{\mathfrak{C}}_{\mathscr{M}}^\heartsuit({\mathcal{O}}_v)$ and its unique $F_v$ -lift $m'\in{\mathscr{M}}'(F_v)$ . The image of m’ in $[{\mathfrak{M}}'/G'](F_v)$ we denote by $[\gamma'']$ as in § 3.1, and we have seen that there exists some $\gamma'\in{\mathfrak{M}}'(F_v)$ lying over the image $\bar{a}\in{\mathfrak{C}}_{\mathfrak{M}}({\mathcal{O}}_v)$ of a.

The canonical point $[\gamma'']$ is a G’-torsor ${\mathcal{E}}'$ over $F_v$ together with a G’-equivariant map

\begin{align*} \phi'\colon {\mathcal{E}}'\longrightarrow {\mathfrak{M}}',\end{align*}

whose scheme-theoretic image is a closed $F_v$ -subvariety $O_{[\gamma'']}$ in ${\mathfrak{M}}'_{F_v}$ . Let ${\mathfrak{J}}_{\bar{a}}'$ be the regular centralizer of the G’-action at $\bar{a}$ , then $\phi'$ induces a $\chi_{{\mathfrak{M}}'}^*{\mathfrak{J}}_{\bar{a}}'$ -torsor

\begin{align*} \phi'_{\gamma'}\colon {\mathcal{E}}'\longrightarrow O_{[\gamma'']},\end{align*}

because $\bar{a}$ is generically regular semisimple.

We know that the homogeneous G’-space $O_{[\gamma'']}$ contains at least one $F_v$ -point, namely $\gamma'$ , and so the fiber of $\phi'_{\gamma'}$ over $\gamma'$ is a ${\mathfrak{J}}_{\bar{a}}'$ -torsor $\mu_{{\mathfrak{J}}'}$ over $F_v$ . Using the canonical isomorphism between the centralizer $G'_{\gamma'}=\operatorname{C}_{G'}(\gamma')$ and ${\mathfrak{J}}_{\bar{a}}'$ , we view $\mu_{{\mathfrak{J}}'}$ as a $G'_{\gamma'}$ -torsor, which then induces a G’-torsor $\mu'$ .

3.2.3.

The Langlands dual group $\check{{\mathbf{J}}}$ of ${\mathfrak{J}}_{\bar{a}}'$ may be identified with the maximal torus $\check{{\mathbf{T}}}\subset\check{{\mathbf{G}}}$ and such identification is well-defined up to $\check{{\mathbf{G}}}$ -conjugacy. In particular, we have a canonical embedding of tori

\begin{align*} Z_{\check{{\mathbf{G}}}}\longrightarrow \check{{\mathbf{J}}}.\end{align*}

The $\Gamma_v$ -action on $\check{{\mathbf{J}}}$ may be identified with some W-twisted $\Gamma_v$ -action on $\check{{\mathbf{T}}}$ , and such identification depends on $\gamma'$ . Since W acts on $Z_{\check{{\mathbf{G}}}}$ trivially, we further have a canonical embedding

\begin{align*} Z_{\check{{\mathbf{G}}}}^{\Gamma_v}\longrightarrow \check{{\mathbf{J}}}^{\Gamma_v}.\end{align*}

This leads to a canonical commutative diagram.

Lemma 3.6 shows that $\operatorname{Ob}_v(a)$ is equal to $\mu'$ for any choice of $\gamma'$ .

3.2.4.

To further explicate $\mu'$ , we need to use the information contained in m’. Over $F_v'$ , we can extend m’ to an ${\mathcal{O}}_v'$ -point $\tilde{m}$ in ${\mathscr{M}}'$ . Explicitly, $\tilde{m}$ is a tuple $(V,\gamma,e,e^\vee)$ , where V is the free ${\mathcal{O}}_v'$ -module $({\mathcal{O}}_v')^n$ , $\gamma\in {\mathfrak{M}}'({\mathcal{O}}_v')$ , $e\in V({\mathcal{O}}_v')$ , and $e^\vee\in V^\vee({\mathcal{O}}_v')$ . Since both $\tilde{m}$ and $\sigma(\tilde{m})$ lie over a, there exists a unique $h\in G'(F_v')$ such that

\begin{align*} \sigma(\gamma,e,e^\vee) =(\operatorname{Ad}_h^{-1}(\gamma),h^{-1}e,e^\vee h).\end{align*}

This implies that $\sigma(h)^{-1}h^{-1}$ lies in the centralizer of tuple $(\gamma,e,e^\vee)$ (the vector bundle V is always trivial), hence it must be trivial because a is generically strongly regular semisimple. In other words,

\begin{align*} h=\sigma(h)^{-1}=\,^{\mathrm{t}}{}{\bar{h}}\end{align*}

is Hermitian, hence it induces a G’-torsor over $F_v$ . It is a non-trivial torsor if and only if $\operatorname{Ob}_v(a)$ is, hence it must be equal to the latter. Recall the discriminant function (2.23) $\mathrm{Disc}=\mathrm{Disc}_{\underline{\mathscr{M}}}={\mathbf{f}}_n^\vee{\mathbf{f}}_n=f_n^\vee f_n$ on ${\mathfrak{C}}_{\mathscr{M}}$ . We claim that $\operatorname{Ob}_v(a)$ may be described by the valuation of $\mathrm{Disc}(a)$ .

Proof. Let $\gamma_1$ be the image of $\gamma$ in $\operatorname{End}(V)({\mathcal{O}}_v')$ , and for $1\leqslant i\leqslant n-1$ , let $z_i\in {\mathcal{O}}_v'-\{0\}$ be the coordinate of the abelianization of $\gamma$ corresponding to the ith simple root. Then it is straightforward to compute

\begin{align*} f_n(\tilde{m}) &=\prod_{i=1}^{n-1}z_i^{{(n-i)(n-i-1)}/{2}}(e\wedge \gamma_1 e\wedge\cdots\wedge \gamma_1^{n-1} e),\\ f_n^\vee(\tilde{m}) &=\prod_{i=1}^{n-1}z_i^{{(n-i)(n-i-1)}/{2}}(e^\vee\gamma_1^{n-1}\wedge\cdots\wedge e^\vee\gamma_1 \wedge e^\vee), \end{align*}

and so

\begin{align*} \operatorname{val}_v(\mathrm{Disc}(a))\equiv \operatorname{val}_v(\langle e\wedge \gamma_1 e\wedge\cdots\wedge \gamma_1^{n-1} e,e^\vee\gamma_1^{n-1}\wedge\cdots\wedge e^\vee\gamma_1 \wedge e^\vee \rangle)\bmod 2. \end{align*}

If we regard e as a column vector and $e^\vee$ as a row vector, then the condition $\sigma(e,e^\vee)=(h^{-1}e,e^\vee h)$ is equivalent to saying

\begin{align*} e^\vee =\,^{\mathrm{t}}{}{\bar{e}}h^{-1}=\,^{\mathrm{t}}{}{\bar{e}}\,^{\mathrm{t}}{}{\bar{h}}^{-1}. \end{align*}

Therefore, we have

\begin{align*} e^\vee\gamma_1^{n-1}\wedge\cdots\wedge e^\vee\gamma_1 \wedge e^\vee &= \,^{\mathrm{t}}{}{\bar{e}}h^{-1}\gamma_1^{n-1}\wedge\cdots\wedge \,^{\mathrm{t}}{}{\bar{e}}h^{-1}\gamma_1 \wedge \,^{\mathrm{t}}{}{\bar{e}}h^{-1}\\ &= (\,^{\mathrm{t}}{}{\bar{e}}h^{-1}\gamma_1^{n-1}h\wedge\cdots\wedge \,^{\mathrm{t}}{}{\bar{e}}h^{-1}\gamma_1h \wedge \,^{\mathrm{t}}{}{\bar{e}})\det{h^{-1}}. \end{align*}

Since we also have $\sigma(\gamma)=\operatorname{Ad}_h^{-1}(\gamma)$ , we have

\begin{align*} h^{-1}\gamma_1h=\sigma(\gamma)_1=\bigg(\prod_{i=1}^{n-1}\bar{z}_i^{n-i}\bigg)^{\!{2}/{n}}\,^{\mathrm{t}}{}{\bar{\gamma_1}}^{-1}. \end{align*}

This implies that

\begin{align*} \,^{\mathrm{t}}{}{\bar{e}}h^{-1}\gamma_1^{n-1}h\wedge\cdots\wedge \,^{\mathrm{t}}{}{\bar{e}}h^{-1}\gamma_1h \wedge \,^{\mathrm{t}}{}{\bar{e}} &= (\,^{\mathrm{t}}{}{\bar{e}}\,^{\mathrm{t}}{}{\bar{\gamma_1}}^{1-n}\wedge\cdots\wedge \,^{\mathrm{t}}{}{\bar{e}}\,^{\mathrm{t}}{}{\bar{\gamma_1}}^{-1} \wedge \,^{\mathrm{t}}{}{\bar{e}})\bigg(\prod_{i=1}^{n-1}\bar{z}_i^{n-i}\bigg)^{\!n-1}\!, \end{align*}

and so

\begin{align*} & \operatorname{val}_v(e^\vee\gamma_1^{n-1}\wedge\cdots\wedge e^\vee\gamma_1 \wedge e^\vee)\\ &\quad =\operatorname{val}_v(\,^{\mathrm{t}}{}{\bar{e}}\,^{\mathrm{t}}{}{\bar{\gamma_1}}^{1-n}\wedge\cdots\wedge \,^{\mathrm{t}}{}{\bar{e}}\,^{\mathrm{t}}{}{\bar{\gamma_1}}^{-1} \wedge \,^{\mathrm{t}}{}{\bar{e}})+(n-1)\sum_{i=1}^{n-1}(n-i)\operatorname{val}_v(z_i)\\ &\quad = \operatorname{val}_v(\gamma_1^{-(n-1)} e\wedge \cdots\wedge\gamma_1^{-1} e\wedge e) +(n-1)\sum_{i=1}^{n-1}(n-i)\operatorname{val}_v(z_i)\\ &\quad =\operatorname{val}_v(e\wedge \gamma_1 e\wedge\cdots\wedge \gamma_1^{n-1} e) +(n-1)\bigg[\sum_{i=1}^{n-1}(n-i)\operatorname{val}_v(z_i)-\operatorname{val}_v(\det(\gamma_1))\bigg]. \end{align*}

As a result, it suffices to prove that

\begin{align*} (n-1)\bigg[\sum_{i=1}^{n-1}(n-i)\operatorname{val}_v(z_i)-\operatorname{val}_v(\det(\gamma_1))\bigg] \end{align*}

is even. Indeed, write $\gamma$ as (z,x) for some ramified element $z\in T^{\mathrm{sc}}(k_v'(\!(\pi_v^{1/n})\!))$ and $x\in G^{\mathrm{sc}}(k_v'(\!(\pi_v^{1/n})\!))$ , then we have $z_i=\alpha_i(z)$ and

\begin{align*} \det{\gamma_1}=\varpi_1(z)^n=\prod_{i=1}^{n-1}z_i^{n-i}, \end{align*}

which clearly shows that

\begin{align*} \sum_{i=1}^{n-1}(n-i)\operatorname{val}_v(z_i)-\operatorname{val}_v(\det(\gamma_1))=0. \end{align*}

This finishes the proof.

Corollary 3.8. When either v’ is split over v or $\operatorname{val}_v(\mathrm{Disc}(a))$ is even, we have a k-isomorphism

\begin{align*} {\mathcal{M}}_v^{\prime{\mathrm{Hit}}}(a)\cong{\mathcal{M}}_v^{\prime\prime{\mathrm{Hit}}}(a). \end{align*}

Consequently, the forgetful map (3.3) induces a forgetful map (3.2).

3.2.5.

With $\operatorname{Ob}_v(a)$ completely described, we can now look at the Frobenius action on (3.4) induced by ${\mathcal{M}}_v'(a)$ . We will also describe the similar action induced by ${\mathcal{M}}_v(a)$ . For this purpose, it suffices to let $R=k_v'$ in (3.4). We let $\Lambda_0=V({\mathcal{O}}_v')=({\mathcal{O}}_v')^n$ be the standard lattice in $V(F_v')$ . As before, we let $\tilde{m}=(V,\gamma,e,e^\vee)$ be the image of a under the section over ${\mathcal{O}}_v'$ , and $h\in G'(F_v')$ is the canonical Hermitian element induced by the $F_v$ -structure on $\tilde{m}$ .

The usual $\sigma$ -action sending $(\gamma,e,e^\vee)$ to $\sigma(\gamma,e,e^\vee)$ corresponds to sending $g\Lambda_0$ to $\,^{\mathrm{t}}{}{\bar{g}}^{-1}\Lambda_0$ , while $\operatorname{Ad}_h^{-1}$ sends $g\Lambda_0$ to $hg\Lambda_0$ . Since $(\gamma,e,e^\vee)$ is fixed by the $\sigma$ -action induced by $\tilde{m}$ , the $\sigma$ -action on (3.4) induced by that of ${\mathcal{M}}_v'(a)$ is

\begin{align*} g\Lambda_0\longmapsto h^{-1}\,^{\mathrm{t}}{}{\bar{g}}^{-1}\Lambda_0.\end{align*}

If $g\Lambda_0$ is a fixed point of this action, then we must have

\begin{align*} \,^{\mathrm{t}}{}{\bar{g}}hg\in G'({\mathcal{O}}_v'),\end{align*}

which is possible if and only if either v’ is split over v or $\det{h}$ has even valuation, or equivalently, $\operatorname{Ob}_v(a)$ is trivial.

3.2.6.

Suppose we define a bilinear form

\begin{align*} ({-},{-})\colon V(F_v')\times V(F_v')&\longrightarrow F_v'\\ (x,y)&\longmapsto \,^{\mathrm{t}}{}{\bar{x}}h^{-1}y,\end{align*}

then

(3.5) \begin{align} {\mathcal{M}}_v'(a)(k_v)= \left\{{\kern-1pt}{{\mathcal{O}}_v'\text{-lattices }\Lambda\in V(F_v') \,\left\vert\, \! \begin{array}{l} \Lambda\text{ is self-dual with respect to }({-},{-})\\ \gamma_i\Lambda_i\subset\Lambda_i, e\in\Lambda, e^\vee\in\Lambda^\vee \end{array}\right. }\!\!\!\right\}{\kern-1pt}.\end{align}

It is non-empty if and only if $\operatorname{Ob}_v(a)$ is trivial, or equivalently, either v’ is split over v or $\mathrm{Disc}(a)$ has even valuation.

3.2.7.

Similarly, there is a canonical element $s\in\operatorname{GL}_n(F_v')$ , with $s\bar{s}=1$ induced by $\tilde{m}$ and the Frobenius structure on ${\mathcal{M}}_v(a)$ , and

(3.6) \begin{align} {\mathcal{M}}_v(a)(k_v)= \left\{{\kern-1pt}{{\mathcal{O}}_v'\text{-lattices }\Lambda\in V(F_v') \,\left\vert\, \!\begin{array}{l} s\Lambda=\bar{\Lambda}\\ \gamma_i\Lambda_i\subset\Lambda_i, e\in\Lambda, e^\vee\in\Lambda^\vee \end{array}\right. }\!\!\!\!\right\}{\kern-1pt}.\end{align}

3.2.8.

There is an involution on the set ${\mathcal{M}}_v(a)(k_v)$ defined as follows: write $\gamma=(z,x)$ , where $z\in T^{\mathrm{sc}}(k_v'(\!(\pi_v^{1/n})\!))$ and $x\in G^{\mathrm{sc}}(k_v'(\!(\pi_v^{1/n})\!))$ . We claim $\gamma^*=(z,\,^{\mathrm{t}}{}{x})$ has the same image in ${\mathfrak{C}}_{\mathfrak{M}}$ as $\gamma$ : indeed, its abelianization is the same as that of $\gamma$ , and clearly $\varpi_i(z)\bigwedge^i\,^{\mathrm{t}}{}{x}$ has the same trace as $\varpi_i(z)\bigwedge^i x$ . Since $\gamma^*_i=\,^{\mathrm{t}}{}{\gamma_i}$ for each $1\leqslant i\leqslant n-1$ , $\gamma^*\in{\mathfrak{M}}({\mathcal{O}}_v')$ as well. Let

\begin{align*} \tilde{m}^*=(V,\gamma^*,\,^{\mathrm{t}}{}{e}^\vee,\,^{\mathrm{t}}{}{e})\in {\mathscr{M}}({\mathcal{O}}_v').\end{align*}

Note that for any $e\in V$ , the transpose of the wedging map $e\wedge$ is precisely the contraction by $\,^{\mathrm{t}}{}{e}$ , and as a result, $\tilde{m}^*$ has the same image of $\tilde{m}$ in ${\mathfrak{C}}_{\mathscr{M}}$ . This shows that there exists a unique element ${\mathfrak{d}}\in \operatorname{GL}_n(F_v')$ such that

\begin{align*} (\gamma^*,\,^{\mathrm{t}}{}{e}^\vee,\,^{\mathrm{t}}{}{e})=(\operatorname{Ad}_{{\mathfrak{d}}}(\gamma),{\mathfrak{d}} e,e^\vee {\mathfrak{d}}^{-1}).\end{align*}