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# Multiplicative Higgs bundles and involutions

Guillermo Gallego  Facultad de Ciencias Matemáticas, UCM, Plaza Ciencias 3,
28040 Madrid, SPAIN<EMAIL_ADDRESS>and  Oscar García-Prada  Instituto de
Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Nicolás Cabrera, 13–15, 28049 Madrid,
SPAIN<EMAIL_ADDRESS>
###### Abstract.

In this paper we generalize the theory of multiplicative $G$-Higgs bundles
over a curve to pairs $(G,\theta)$, where $G$ is a reductive algebraic group
and $\theta$ is an involution of $G$. This generalization involves the notion
of a multiplicative Higgs bundle taking values in a symmetric variety
associated to $\theta$, or in an equivariant embedding of it. We also study
how these objects appear as fixed points of involutions of the moduli space of
multiplicative $G$-Higgs bundles, induced by the involution $\theta$.

###### 2020 Mathematics Subject Classification:

Primary 14D23; Secondary 14H60, 14M17

The first author’s research is supported by the UCM under the contract
CT63/19-CT64/19. This work of the second author is partially supported by the
Spanish Ministry of Science and Innovation, through the “Severo Ochoa
Programme for Centres of Excellence in R&D (CEX2019-000904-S)” and grant No.
PID2019-109339GB-C31.

##  1\. Introduction

###  1.1. Multiplicative Higgs bundles

Let $k$ be an algebraically closed field of characteristic $0$, $G$ a
reductive algebraic group over $k$ and $X$ a smooth algebraic curve over $k$.
A _multiplicative $G$-Higgs bundle_ on $X$ is a pair $(E,\varphi)$ where $E$
is a principal $G$-bundle over $X$ and $\varphi$ is a section of the adjoint
group bundle $E(G)$ over $X\setminus|D|$ the complement of a finite subset
$|D|\subset X$. This is indeed a “multiplicative” version of an (ordinary,
twisted) _$G$ -Higgs bundle_ on $X$, which is a pair $(E,\varphi)$ with $E$ a
principal $G$-bundle over $X$ and $\varphi$ a section of the adjoint Lie
algebra bundle $E(\mathfrak{g})$ twisted by some line bundle $L$ on $X$. As in
the ordinary case, one wants to prescribe the singularities of $\varphi$. In
the multiplicative case, instead of fixing the twisting line bundle $L$, one
controls the singularity at each point $x\in|D|$ by fixing a dominant
cocharacter $\lambda\in\mathrm{X}_{*}(T)_{+}$, for a given choice $T\subset
B\subset G$ of maximal torus $T$ and Borel subgroup $B$.

Just like for ordinary Higgs bundles [hitchin], one can define a Hitchin map
in the multiplicative case, which is based on the “multiplicative version” of
the Chevalley restriction map $G\rightarrow T/W$, for $W=N_{G}(T)/T$ the Weyl
group of $T$. This leads to the _multiplicative Hitchin fibration_. A general
reference reviewing and explaining the several perspectives around
multiplicative Higgs bundles and the multiplicative Hitchin fibration is
presented by Elliot and Pestun [elliot-pestun].

Multiplicative Higgs bundles were originally introduced in algebraic geometry
by Hurtubise and Markman [hurtubise-markman], although they had made previous
appearances in the physics literature (see [elliot-pestun] for references).
Hurtubise and Markman considered the particular case where $X$ is an elliptic
curve and, in this case, they constructed an algebraic symplectic form on the
moduli space of _simple_ multiplicative Higgs bundles and proved that the
multiplicative Hitchin fibration defines an algebraically completely
integrable system. As explained in Elliot–Pestun [elliot-pestun], these
results can be generalized to the case where $X$ is Calabi–Yau, that is, if
$X$ is an elliptic curve, $\mathbb{A}^{1}$, or $\mathbb{G}_{m}$.

The multiplicative Hitchin fibration was also studied independently by Frenkel
and Ngô [frenkel-ngo] in the context of geometrization of trace formulas. In
particular, they suggest the use of Vinberg’s enveloping monoid to describe
the moduli stack of multiplicative Higgs bundles and construct the
multiplicative Hitchin fibration. That paper was the starting point for a
programme to study the multiplicative Hitchin fibration from the point of view
of Vinberg’s theory of reductive monoids [vinberg]. This has been continued in
several papers by Bouthier and J. Chi [bouthier_Springer, bouthier_fibration,
bouthier-chi, chi], which study analogues of affine Springer fibres, and in
the thesis of G. Wang [wang], dedicated to the Fundamental Lemma of
Langlands–Shelstad for the groups, in the spirit of Ngô’s work on the
Fundamental Lemma for the Lie algebras [ngo_lemme].

When the base field is $k=\mathbb{C}$ the field of complex numbers,
multiplicative $G$-Higgs bundles on $X$ have also been related to singular
$K$-monopoles on the $3$-manifold $X^{\operatorname{an}}\times S^{1}$, for
$K\subset G$ a maximal compact subgroup, in the works of Charbonneau–Hurtubise
[charbonneau-hurtubise] and Smith [smith]. The correspondence is given by
constructing a multiplicative Higgs bundle as the _scattering map_ of a
monopole. In the other direction, by “gluing” from a multiplicative Higgs
bundle one constructs a holomorphic bundle on the $4$-manifold
$X^{\operatorname{an}}\times S^{1}\times S^{1}$, solves the Hermitian
Yang–Mills equations on it and reduces to $X^{\operatorname{an}}\times S^{1}$
to obtain a monopole. These equations can be solved provided the stability
condition on the holomorphic bundle on the $4$-manifold, which translates into
an ad-hoc stability condition for the multiplicative Higgs bundle. The
Charbonneau–Hurtubise–Smith correspondence is widely extended in Mochizuki’s
book [mochizuki]. The moduli space of singular monopoles on $\mathbb{C}\times
S^{1}$ is known to be a hyperkähler manifold [cherkis-kapustin]. Using
Mochizuki’s correspondence, Elliot and Pestun [elliot-pestun] show that the
holomorphic symplectic structure defined by that hyperkähler structure
coincides with the Hurtubise–Markman algebraic symplectic structure on the
moduli space of simple multiplicative Higgs bundles on $\mathbb{A}^{1}$.
Furthermore, they study the twistor space for this hyperkähler manifold by
introducing the notion of a _$q$ -difference connection_.

###  1.2. Multiplicative Higgs bundles and involutions

The purpose of this paper is two-fold.

On the one hand, we introduce a generalization of multiplicative Higgs bundles
from groups to pairs $(G,\theta)$ where $G$ is a reductive group and
$\theta\in\operatorname{Aut}_{2}(G)$ is an involution of $G$. A
_multiplicative $(G,\theta)$-Higgs bundle_ on $X$ is formed by a principal
$G^{\theta}$-bundle $E\rightarrow X$ and a section $\varphi$ of the associated
bundle $E(G/G^{\theta})$ over the complement of a finite subset of $X$. Here,
$G^{\theta}\subset G$ is the subgroup of fixed points of $\theta$ and
$G/G^{\theta}$ is the corresponding symmetric variety. The group $G^{\theta}$
acts on $G/G^{\theta}$ by left multiplication and thus
$E(G/G^{\theta})\rightarrow X$ denotes the bundle of symmetric varieties
associated to this action.

Moreover, we generalize the description of the multiplicative Hitchin map in
terms of the theory of reductive monoids, as developed in [bouthier_Springer,
bouthier_fibration, bouthier-chi, chi, frenkel-ngo, wang], to this involutive
situation. We do this by studying the theory of embeddings of symmetric
varieties and using Guay’s enveloping embedding [guay], and by extending the
results of Richardson [richardson].

On the other hand, for any such pair $(G,\theta)$, we consider the involutions
of the moduli space of (simple) multiplicative $G$-Higgs bundles given by

$\iota_{\pm}:(E,\varphi)\mapsto(\theta(E),\theta(\varphi)^{\pm 1})$

and study their fixed points. We show that the fixed points of $\iota_{+}$
define an algebraic symplectic submanifold of the moduli space of
multiplicative Higgs bundles, while the fixed points of $\iota_{-}$ define a
Lagrangian submanifold. We are specially interested in the involution
$\iota_{-}$ and we show that multiplicative $(G,\theta)$-Higgs bundles appear
as their fixed points, although also other components show up.

One of the reasons why multiplicative $(G,\theta)$-Higgs bundles are important
is that when $k=\mathbb{C}$ they are the multiplicative object analogous to
Higgs bundles for _real forms_ of $G$. We recall that if $G_{\mathbb{R}}$ is a
real form of $G$, a ($L$-twisted) $G_{\mathbb{R}}$-Higgs bundle on $X$ is a
pair $(E,\varphi)$, where $E$ is an $H$-bundle on $X$ and $\varphi$ is a
section of $E(\mathfrak{m})\otimes L$. Here, $H$ is the complexification of a
maximal compact subgroup $H_{\mathbb{R}}\subset G_{\mathbb{R}}$ and
$\mathfrak{m}$ is the complexification of the space
$\mathfrak{m}_{\mathbb{R}}$ appearing in the Cartan decomposition
$\mathfrak{g}_{\mathbb{R}}=\mathfrak{h}_{\mathbb{R}}\oplus\mathfrak{m}_{\mathbb{R}}$.

There is a well known correspondence between real forms an involutions of $G$.
If one regards a real form as the fixed points of an anti-holomorphic
involution $\sigma:G\rightarrow G$, then one can find a commuting compact real
form $\tau:G\rightarrow G$ and consider the involution
$\theta=\sigma\circ\tau$. In this language, if $G_{\mathbb{R}}=G^{\sigma}$,
then $H=G^{\theta}$ and $\mathfrak{m}$ is the $(-1)$-eigenspace in the
decomposition $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$. Note that
$\mathfrak{m}$ is the tangent space of $G/G^{\theta}$ at $1\in G$, so a
multiplicative $(G,\theta)$-Higgs bundle is indeed a multiplicative analogue
of a $G_{\mathbb{R}}$-Higgs bundle. In [oscar-ramanan], $G_{\mathbb{R}}$-Higgs
bundles receive the name of $(G^{\theta},-)$-Higgs bundles and are shown to
appear as fixed points of the involution
$(E,\varphi)\mapsto(\theta(E),-\theta(\varphi))$.

In any case, the definition of $G_{\mathbb{R}}$-Higgs bundles originated from
the nonabelian Hodge correspondence, through which they yield representations
of the fundamental group of $X^{\operatorname{an}}$ in $G_{\mathbb{R}}$. A
similar correspondence is not known in the multiplicative situation. We expect
to obtain some clarification of this by looking at multiplicative
$(G,\theta)$-Higgs bundles or studying the fixed points of $\iota_{-}$ from
the other side of the Charbonneau–Hurtubise–Smith correspondence. This will be
the topic of a forthcoming paper.

Another reason to study the objects and involutions related to multiplicative
Higgs bundles and pairs $(G,\theta)$ is to study _branes_. In mirror symmetry,
symplectic subvarieties of holomorphic symplectic manifolds are the support of
_$B$ -branes_, while Lagrangian subvarieties are the support of of _$A$
-branes_. In the language of hyperkähler geometry, holomorphic $B$-branes are
$(B,B,B)$-branes, while holomorphic $A$-branes are $(B,A,A)$-branes.

For the classical Hitchin fibration, _$S$ -duality_ is expected to give a
correspondence between $B$-branes in the moduli space of $G$-Higgs bundles and
$A$-branes in the moduli space of $\check{G}$-Higgs bundles, for $\check{G}$
the Langlands dual group of $G$. More precisely, since the Hitchin bases for
$G$ and $\check{G}$ are canonically identified, the two moduli spaces fibre
over the same space and Fourier–Mukai transforms on the fibres are expected to
give the construction of $A$-branes on one of them from $B$-branes on the
other. In particular, it is suggested that Higgs bundles for real forms
$G_{\mathbb{R}}$ of $G$, that give $A$-branes in the moduli space of $G$-Higgs
bundles, should correspond to $\check{G}_{G_{\mathbb{R}}}$-Higgs bundles, for
$\check{G}_{G_{\mathbb{R}}}\subset\check{G}$ the Nadler dual group of the real
form $G_{\mathbb{R}}$ [nadler]. This conjectures arise from the work of
Kapustin and Witten [kapustin-witten] relating the geometric Langlands program
with gauge theory, from the results on the duality of Hitchin systems given by
Donagi and Pantev [donagi-pantev], and from a gauge-theoretical description of
the construction of the Nadler group, given by Gaiotto and Witten [gaiotto-
witten], and has been further studied in several works [baraglia-schaposnik,
oscar-biswas, oscar-biswas-hurtubise, branco, hitchin_hirzebruch]. The Nadler
dual group coincides with $\check{G}_{G/G^{\theta}}$, the _dual group_ of the
symmetric variety $G/G^{\theta}$, for $\theta$ the involution of $G$
corresponding to the real form $G_{\mathbb{R}}$, introduced in the context of
spherical varieties [sake-venka, knop-schalke]. The dual group of a spherical
variety plays an important role in the recent extension of the Langlands
program (both in its classical and geometric flavours) to spherical varieties;
this is the _relative_ Langlands program of Ben-Zvi, Sakellaridis and
Venkatesh [BZSV]. See Section 2.6 for more about dual groups.

In [elliot-pestun]*Pseudo-Conjecture 3.9, Elliot and Pestun suggest the
existence of a similar correspondence between $A$-branes and $B$-branes in the
moduli space of multiplicative $G$-Higgs bundles, when $X$ is Calabi–Yau and
$G$ a Langlands self-dual group. If $G$ is a Langlands self-dual group, we can
identify a maximal torus $T\subset G$ with its dual
$\check{T}=\operatorname{Spec}k[e^{\mathrm{X}_{*}(T)}]$, aswell as the
lattices of characters and cocharacters
$\mathrm{X}^{*}(T)\leftrightarrow\mathrm{X}_{*}(T)$ and the roots and coroots
$\Phi_{T}\leftrightarrow\Phi_{T}^{\vee}$. Moreover, in that case, the dual
group $\check{G}_{G/G^{\theta}}$ of $G/G^{\theta}$ is naturally a subgroup of
$G$. We thus suggest the following.

######  Conjecture 1.1.

If $X$ is Calabi–Yau and $G$ is a Langlands self-dual group, for any
involution $\theta\in\operatorname{Aut}_{2}(G)$, multiplicative
$(G,\theta)$-Higgs bundles inside the moduli space of multiplicative $G$-Higgs
bundles define the support of an $A$-brane dual to the $B$-brane given by
multiplicative $\check{G}_{G/G^{\theta}}$-Higgs bundles.

###  1.3. Outline of the paper

This paper consists of three different sections (apart from this introduction)
and an appendix.

Section 2 covers all the preliminary notions that we need about reductive
groups with involutions, symmetric varieties and their embeddings, and their
invariant theory. We begin by reviewing some general facts about reductive
groups with involutions. We then introduce $\theta$-split tori and
$\theta$-split parabolic subgroups, and recall the notion of a _quasisplit_
involution and the Iwasawa decomposition. We then recall the construction of
the restricted root system and its associated root and weight lattices. We
continue by describing the weight lattices and the weight semigroups of
symmetric varieties and their embeddings, in the context of the theory of
spherical varieties. We also recall the construction of the dual group of a
symmetric variety, as introduced by Sakellaridis and Venkatesh [sake-venka]
and state its most relevant properties for us. We then review Guay’s
classification of a certain class of symmetric embeddings, which we call “very
flat” (in analogy with very flat monoids), and his construction of the
“enveloping” embedding [guay]. Guay’s enveloping embedding is an affine
variety associated to a semisimple simply-connected group with involution that
generalizes Vinberg’s enveloping monoid. After this, we study the invariant
theory of symmetric embeddings: first we recall Richardson’s results on the
invariant theory of a symmetric variety [richardson], and in Proposition 2.29
we extend them to very flat symmetric embeddings. We finish the section by
describing the loop parametrization of a symmetric variety, and extend it to
the enveloping embedding in Proposition 2.34, thus giving a generalization of
[chi]*2.5.

Section 3 introduces the main objects we are interested in, from several
perspectives that we compare. In the first part of the section we emulate the
construction of the multiplicative Hitchin fibration associated to a reductive
monoid, as explained in [wang], in order to construct a multiplicative Hitchin
map associated to a very flat symmetric embedding acted by a semisimple
simply-connected group. In the second part of the section we introduce the
notion of a multiplicative Higgs bundle associated to a reductive group with
involution. We finish the section by relating the objects defined in the
second part (with semisimple group) with the objects defined in the first
part.

Finally, in Section 4 we study a certain type of involutions in the moduli
space of simple multiplicative Higgs bundles, how the objects introduced in
Section 3 appear as their fixed points, and the relation between these
involutions and the Hurtubise–Markman symplectic form. More precisely, the
section is dedicated to the study of the two involutions
$\iota_{\pm}:(E,\varphi)\mapsto(\theta(E),\theta(\varphi)^{\pm 1})$, of the
moduli space of simple multiplicative $G$-Higgs bundles, associated to an
involution $\theta\in\operatorname{Aut}_{2}(G)$. We begin by recalling
Hurtubise and Markman’s [hurtubise-markman] description of this moduli space
of simple multiplicative $G$-Higgs bundles. We then show how multiplicative
$(G,\theta)$-Higgs bundles are naturally mapped inside the fixed points of
$\iota_{-}$ in Proposition 4.5 and give a detailed description of the fixed
points in Theorem 4.6 and Corollary 4.11. We finish the section by reviewing
the definition of the Hurtubise–Markman algebraic symplectic form in the
moduli space of simple multiplicative $G$-Higgs bundles when the curve $X$ is
assumed to be Calabi–Yau, and show in Theorem 4.12 how this symplectic form
behaves under the above involutions. More precisely we show that the fixed
points of $\iota_{+}$ define a symplectic submanifold, while the fixed points
of $\iota_{-}$ give a Lagrangian submanifold. This is then a first result that
motivates Conjecture 1.1 stated above.

At the end of the paper, we have included an Appendix (A) to cover the basic
facts that we use about root systems, and to establish our notation.

###  1.4. Some notations

We establish now some notations regarding characters and cocharacters of
algebraic tori. Given an algebraic torus $T$ we denote by
$\mathrm{X}^{*}(T)=\operatorname{Hom}(T,\mathbb{G}_{m})$ its lattice of
characters and by $\mathrm{X}_{*}(T)=\operatorname{Hom}(\mathbb{G}_{m},T)$ the
lattice of cocharacters. Throughout the text we will use additive notation for
characters and cocharacters. Thus, given an element $t\in T$ and a character
$\chi\in\mathrm{X}^{*}(T)$, we denote by $t^{\chi}\in\mathbb{G}_{m}$ the image
of the corresponding morphism $T\rightarrow\mathbb{G}_{m}$. On the other hand,
given an element $z\in\mathbb{G}_{m}$ and a cocharacter
$\lambda\in\mathrm{X}_{*}(T)$, we denote by $z^{\lambda}\in T$ the image of
the corresponding $1$-parameter subgroup $\mathbb{G}_{m}\rightarrow T$. We
consider the vector space
$\mathcal{E}_{T}=\mathrm{X}^{*}(T)\otimes_{\mathbb{Z}}\mathbb{R}$, endowed
with the scalar product $(\cdot,\cdot)$ induced by the choice of an invariant
bilinear form on $T$ (which we also choose to be invariant under $\theta$ and
under the Weyl group $W_{T}$ of $T$), and identify $\mathrm{X}^{*}(T)$ and
$\mathrm{X}_{*}(T)$ as lattices inside $\mathcal{E}_{T}$ and
$\mathcal{E}_{T}^{*}$, respectively. We can also consider the root system of
$T$ inside $(\mathcal{E}_{T},(\cdot,\cdot))$; this is the set
$\Phi_{T}\subset\mathrm{X}^{*}(T)$ of the weights of the adjoint
representation of $T$ in its Lie algebra . See the Appendix A for more
notations regarding root systems.

#### Acknowledgements

We would like to thank Guillermo Barajas, Thomas Hameister, Jacques Hurtubise,
Benedict Morrisey, Ngô Bao Châu and Griffin Wang for fruitful discussions,
suggestions and references.

##  2\. Involutions, symmetric varieties and symmetric embeddings

###  2.1. Some generalities on involutions

Let $G$ be a reductive algebraic group over $k$. By an _involution_ of $G$ we
mean an order $2$ automorphism $\theta\in\operatorname{Aut}_{2}(G)$. To any
such involution $\theta$ we can associate the subgroups

$\displaystyle G^{\theta}$ $\displaystyle=\left\\{g\in
G:\theta(g)=g\right\\},$ $\displaystyle G_{\theta}$
$\displaystyle=\left\\{g\in G:g\theta(g)^{-1}\in Z_{G}\right\\},$

where $Z_{G}\subset G$ is the centre of $G$. The group $G^{\theta}$ is the
group of fixed points of $\theta$ while $G_{\theta}$ can be shown to be its
normalizer [deconcini-procesi]*1.7. We can also consider
$G_{0}^{\theta}=(G^{\theta})^{0}$ the neutral connected component of
$G^{\theta}$, and one can show that if $G$ is of simply-connected type then
$G^{\theta}$ is connected, so $G_{0}^{\theta}=G^{\theta}$
[steinberg_endomorphisms]*Theorem 8.1.

An involution $\theta$ of $G$ defines the _$\theta$ -twisted conjugation
action_ of $G$ on itself

$\displaystyle G\times G$ $\displaystyle\longrightarrow G$
$\displaystyle(g,s)$ $\displaystyle\longmapsto g*_{\theta}s=gs\theta(g)^{-1}.$

For each $s\in G$ we denote by $\tau_{s}:G\rightarrow G$ the map
$\tau^{\theta}_{s}(g)=g*_{\theta}s$ sending $G$ to the $\theta$-twisted
$G$-orbit $\tau^{\theta}_{s}(G)=G*_{\theta}s$. We also denote these
$\theta$-twisted orbits by $M^{\theta}_{s}=\tau^{\theta}_{s}(G)$. In
particular, we denote $\tau^{\theta}=\tau^{\theta}_{1}$ and
$M^{\theta}=\tau^{\theta}(G)$. One can easily check that the isotropy subgroup
of $s\in G$ is the fixed point subgroup $G^{\theta_{s}}$ of the automorphism

$\theta_{s}=\operatorname{Int}_{s}\circ\theta.$

Here, $\operatorname{Int}_{s}$ stands for the automorphism of $G$ given by
conjugation by $s$; the elements of this form are the so called _inner
automorphisms_ , that form a subgroup
$\operatorname{Int}(G)\subset\operatorname{Aut}(G)$.

From the above, we conclude that the $\theta$-twisted orbits $M^{\theta}_{s}$
are homogeneous spaces of the form $G/G^{\theta_{s}}$, but not necessarily
symmetric varieties. In particular, note that the orbit $M^{\theta}$ is
naturally identified with the symmetric variety $G/G^{\theta}$. Moreover, this
identification is a $G$-equivariant isomorphism, where $G$ acts on
$M^{\theta}$ by $\theta$-twisted conjugation and on $G/G^{\theta}$ by left
multiplication. We can ask ourselves for which $s\in G$ the automorphism
$\theta_{s}$ is an involution. A simple computation shows that
$\theta_{s}^{2}=\operatorname{id}_{G}$ if and only if $s\theta(s)$ is an
element of the centre $Z_{G}$. The set of such $s$ is denoted by

$S_{\theta}=\left\\{s\in G:s\theta(s)\in Z_{G}\right\\}.$

For $s\in S_{\theta}$, the $\theta$-twisted orbit $M^{\theta}_{s}$ gets
explicitly identified with $M^{\theta_{s}}$ as

$M^{\theta}_{s}=M^{\theta_{s}}s.$

In particular, note that if $s\in M^{\theta}$, then
$M^{\theta_{s}}s=M^{\theta}_{s}=M^{\theta}$, so for $s\in M^{\theta}$ the
symmetric varieties $G/G^{\theta}$ and $G/G^{\theta_{s}}$ can be identified.
Moreover, if $s=g\theta(g)^{-1}$, we can identify
$\theta_{s}=\operatorname{Int}_{g}\circ\theta\circ\operatorname{Int}_{g}^{-1}$
and $G^{\theta_{s}}=gG^{\theta}g^{-1}$.

It makes sense then to define an equivalence relation $\sim$ on
$\operatorname{Aut}_{2}(G)$ :

$\theta\sim\theta^{\prime}$ if and only if there exists
$\alpha\in\operatorname{Int}(G)$ such that
$\theta^{\prime}=\alpha\circ\theta\circ\alpha^{-1}$,

and describe the quotient set $\operatorname{Aut}_{2}(G)/\sim$. We recall the
description given in [oscar-ramanan] in terms of what is called the “clique
map”.

The natural projection
$\pi:\operatorname{Aut}_{2}(G)\rightarrow\operatorname{Out}_{2}(G):=\operatorname{Aut}_{2}(G)/\operatorname{Int}(G)$
factors through $\operatorname{Aut}_{2}(G)/\sim$. Indeed, this follows from
the fact that

$\operatorname{Int}_{g}\circ\theta\circ\operatorname{Int}_{g}^{-1}=\operatorname{Int}_{g\theta(g)^{-1}}\circ\theta.$

We obtain a surjective map

$\displaystyle\operatorname{cl}:\operatorname{Aut}_{2}(G)/\sim$
$\displaystyle\longrightarrow\operatorname{Out}_{2}(G),$

called the _clique map_. The inverse image $\operatorname{cl}^{-1}(a)$ of a
class $a\in\operatorname{Out}_{2}(G)$ is called the _clique_ of $a$. It is
easy to check that for any $\theta\in\pi^{-1}(a)$ the clique
$\operatorname{cl}^{-1}(a)$ is in bijection with the orbit set
$S_{\theta}/(G\times Z_{G})$, where $(G\times Z_{G})$ acts on $S_{\theta}$
through the natural extension of the $\theta$-twisted conjugation action

$\displaystyle(G\times Z_{G})\times G$ $\displaystyle\longrightarrow G$
$\displaystyle((g,z),s)$ $\displaystyle\longmapsto zgs\theta(g)^{-1}.$

The $\theta$-twisted conjugation action clearly preserves $S_{\theta}$ and
also, for each $z\in Z_{G}$, the subset

$S^{\theta}_{z}=\left\\{s\in G:s\theta(s)=z\right\\}\subset S_{\theta}.$

In particular, it preserves the subset of “anti-fixed” points of $\theta$,

$S^{\theta}=S^{\theta}_{1}=\left\\{s\in G:s=\theta(s)^{-1}\right\\}.$

As explained in [oscar-ramanan], the quotient sets $S^{\theta}/G$ and
$S_{\theta}/(G\times Z_{G})$ admit an interpretation in terms of nonabelian
group cohomology:

$S^{\theta}/G=H^{1}_{\theta}(\mathbb{Z}/2,G)\text{ and }S_{\theta}/(G\times
Z_{G})=H^{1}_{\theta}(\mathbb{Z}/2,G^{\mathrm{ad}}),$

where $G^{\mathrm{ad}}=G/Z_{G}$ and the subscript $\theta$ indicates that
$\mathbb{Z}/2$ acts on $G$ and $G^{\mathrm{ad}}$ through $\theta$. Moreover,
Richardson [richardson2] shows that $H^{1}_{\theta}(\mathbb{Z}/2,G)$ (and thus
$H^{1}_{\theta}(\mathbb{Z}/2,G)$) is finite.

######  Remark 2.1.

Consider the case where $k=\mathbb{C}$. For any involution
$\theta\in\operatorname{Aut}_{2}(G)$ we can find a maximal compact real form,
defined by some conjugation $\sigma_{c}$ of $G$ commuting with $\theta$ and
consider the composition $\sigma=\sigma_{c}\circ\theta$. This $\sigma$ defines
a real form $G^{\sigma}$ of $G$. In [adams-taibi], Adams and Taïbi show that
the nonabelian cohomology set $H^{1}_{\theta}(\mathbb{Z}/2,G)$ is isomorphic
to the Galois cohomology
$H^{1}_{\sigma}(\mathrm{Gal}(\mathbb{C}/\mathbb{R}),G)$. In particular, for
$G^{\mathrm{ad}}$ one recovers Cartan’s classification of real forms of $G$.
Tables with $H^{1}_{\theta}(\mathbb{Z}/2,G)$ computed for semisimple simply-
connected groups can be found in [adams-taibi]*Section 10. We also refer the
reader to [oscar-ramanan]*Section 2.3 for more details about the
correspondence between involutions and real forms.

######  Example 2.2 (The diagonal case).

Groups with involution are in a certain sense a generalization of groups.
Indeed, if $G$ is any linear algebraic group, we can consider the pair
$(G\times G,\Theta)$, where $\Theta\in\operatorname{Aut}_{2}(G\times G)$ is
the involution $\Theta(g_{1},g_{2})=(g_{2},g_{1})$. Let us denote by
$\Delta:G\rightarrow G\times G$ the _diagonal_ map $\Delta(g)=(g,g)$ and by
$\tilde{\Delta}$ the _antidiagonal_ $\tilde{\Delta}(g)=(g,g^{-1})$. The fixed
point subgroup $(G\times G)^{\Theta}$ is the diagonal $\Delta(G)$, which can
be naturally identified with $G$, whereas the orbit $M^{\Theta}$ is the
antidiagonal $\tilde{\Delta}(G)$, which is again identified with $G$. The set
of anti-fixed points $S^{\Theta}$ is easily shown to be equal to $M^{\Theta}$.
We also have

$\displaystyle(G\times G)_{\Theta}$ $\displaystyle=\left\\{(zg,g):g\in
G\right\\},\text{ and }S_{\Theta}=\left\\{(zg,g^{-1}):g\in G\right\\}.$

Therefore, the orbit sets $S^{\Theta}/(G\times G)$ and $S_{\Theta}/(Z\times
G\times G)$ are just singletons. The real form of $G\times G$ corresponding to
$\Theta$ is simply the group $G$ regarded as a real group.

######  Example 2.3 ($G=\operatorname{SL}_{n},n>2$).

In this case we have $\operatorname{Out}(G)\cong\mathbb{Z}/2$, so we
distinguish two cases for $a\in\operatorname{Out}_{2}(G)$, $a=1$ and $a=-1$.
For $a=1$ we have

$\operatorname{cl}^{-1}(1)=\left\\{\theta_{p,q}:0\leq p\leq q\leq
n,p+q=n\right\\},$

with $\theta_{p,q}(g)=I_{p,q}gI_{p,q}$, for

$I_{p,q}=\begin{pmatrix}I_{p}&0\\\ 0&-I_{q}\end{pmatrix},$

where $I_{p}$ denotes the identity matrix of rank $p$. The corresponding
groups of fixed points are

$G^{\theta_{p,q}}=\mathrm{S}(\operatorname{GL}_{p}\times\operatorname{GL}_{q}).$

Here, the $\mathrm{S}$ stands for taking the subset of matrices of determinant
equal to $1$. Consider now the case $a=-1$. The clique
$\operatorname{cl}^{-1}(-1)$ consists of a sigle element if $n$ is odd and of
two elements if $n=2m$ is even. In both cases we have the involution
$\theta_{0}(g)=(g^{T})^{-1}$. The fixed points subgroup is

$G^{\theta_{0}}=\mathrm{SO}_{n}.$

When $n=2m$ is even, we also have the involution
$\theta_{1}(g)=J_{m}\theta_{0}(g)J_{m}^{-1}$, where $J_{m}$ is the symplectic
matrix

$J_{m}=\begin{pmatrix}0&I_{m}\\\ -I_{m}&0\end{pmatrix}.$

The fixed points subgroup is

$G^{\theta_{1}}=\mathrm{Sp}_{2m}.$

When $k=\mathbb{C}$, we can also consider the real forms corresponding to
these involutions. The compact real form of
$\operatorname{SL}_{n}(\mathbb{C})$ is $\mathrm{SU}(n)$, which sits inside
$\operatorname{SL}_{n}(\mathbb{C})$ as the fixed points of the conjugation
$\sigma_{K}(g)=(g^{\dagger})^{-1}$, where † stands for taking transpose and
complex-conjugation. Therefore, the real forms corresponding to the involution
$\theta_{p,q}$ are $\sigma_{p,q}(g)=I_{p,q}(g^{\dagger})^{-1}I_{p,q}$, and
$G^{\sigma_{p,q}}=\mathrm{SU}(p,q)$. Similarly, we have
$\sigma_{0}(g)=\bar{g}$, so
$G^{\sigma_{0}}=\operatorname{SL}_{n}(\mathbb{R})$, and, if $n=2m$,
$\sigma_{1}(g)=J_{m}\bar{g}J_{m}^{-1}$, so
$G^{\sigma_{1}}=\mathrm{SU}^{*}(2m)$.

We refer to Table 26.3 in [timashev] for a complete classification of the
involutions of the simple groups.

###  2.2. Split tori and split parabolics

Let $G$ be a reductive group over $k$ and $\theta\in\operatorname{Aut}_{2}(G)$
an involution. We say that a torus $A\subset G$ is _$\theta$ -split_ if
$\theta(a)=a^{-1}$ for every $a\in A$, and we say that it is _maximal
$\theta$-split_ if it is maximal among $\theta$-split tori.

######  Remark 2.4.

The name “$\theta$-split” comes from the correspondence with real groups when
$k=\mathbb{C}$. Indeed, a $\theta$-split torus is just the complexification of
an $\mathbb{R}$-split torus $A_{\mathbb{R}}\subset G^{\sigma}$, for $\sigma$
the real form corresponding to $\theta$.

We now state some of the results of Vust [vust_cones] on $\theta$-split tori.

######  Proposition 2.5 (Vust).

  1. (1)

Non-trivial $\theta$-split tori exist.

  2. (2)

Any maximal torus $T$ containing a maximal $\theta$-split torus $A$ is
_$\theta$ -stable_, meaning that $\theta(T)\subset T$.

  3. (3)

All maximal $\theta$-split tori are pairwise conjugated by elements of
$G^{\theta}$.

  4. (4)

For any maximal $\theta$-split torus $A\subset G$, the group $G^{\theta}$
decomposes uniquely as $G^{\theta}=F^{\theta}G_{0}^{\theta}$, for

$F^{\theta}=A\cap G^{\theta}=\left\\{a\in A:a^{2}=1\right\\}.$

Related to point $(4)$ in the proposition above, we also have the following
result of Richardson.

######  Lemma 2.6 ([richardson]*Lemma 8.1.(a)).

For any maximal $\theta$-split torus $A\subset G$, the group $G_{\theta}$
decomposes uniquely as $G_{\theta}=F_{\theta}G_{0}^{\theta}$, for

$F_{\theta}=A\cap G_{\theta}=\left\\{a\in A:a^{2}\in Z_{G}\right\\}.$

A parabolic subgroup $P\subset G$ is _$\theta$ -split_ if $P$ and $\theta(P)$
are opposite, that is, if $P\cap\theta(P)$ is a Levi subgroup of both $P$ and
$\theta(P)$. If $P\subset G$ is a minimal $\theta$-split parabolic subgroup
one can show [vust_cones] that there exists a maximal $\theta$-split torus $A$
and a dominant cocharacter $\lambda\in\mathrm{X}_{*}(A)_{+}$ such that $P$ is
of the form

$P=P(\lambda)=\left\\{p\in G:\lim_{t\rightarrow 0}t^{\lambda}pt^{-\lambda}\in
G\right\\}.$

Equivalently, if one takes $T\supset A$ a maximal torus, and considers
$\Phi_{T}$ the associated root system and
$\mathfrak{g}=\mathfrak{t}\oplus\bigoplus_{\alpha\in\Phi_{\mathfrak{g}}}\mathfrak{g}^{\alpha}$
the corresponding root space decomposition, then $P(\lambda)$ is the parabolic
subgroup with Lie algebra

$\mathfrak{p}(\lambda)=\mathfrak{t}\oplus\bigoplus_{\langle\alpha,\lambda\rangle\geq
0}\mathfrak{g}^{\alpha}.$

This $P(\lambda)$ admits the Levi decomposition

$\mathfrak{l}=\mathfrak{t}\oplus\bigoplus_{\langle\alpha,\lambda\rangle=0}\mathfrak{g}^{\alpha}=\mathrm{Lie}(Z_{G}(\lambda))\
\ \text{ and }\ \
\mathfrak{p}(\lambda)^{u}=\bigoplus_{\langle\alpha,\lambda\rangle>0}\mathfrak{g}^{\alpha},$

for $Z_{G}(\lambda)=Z_{G}(\left\\{z^{\lambda}:z\in\mathbb{C}^{*}\right\\})$
the centralizer of the uniparametric subgroup induced by $\lambda$. As a
consequence, the Levi subgroup $P\cap\theta(P)$ is equal to

$P\cap\theta(P)=Z_{G}(\lambda)=Z_{G}(A).$

######  Remark 2.7.

Again, the name “$\theta$-split” comes from the correspondence with real
groups when $k=\mathbb{C}$. Indeed, a parabolic subgroup is $\theta$-split if
and only if its corresponding real form is split. We can also consider
parabolic subgroups such that their corresponding real form is quasisplit,
which motivates the following.

######  Definition 2.8.

An involution $\theta\in\operatorname{Aut}_{2}(G)$ is _quasisplit_ if there
exists a $\theta$-split Borel subgroup $B\subset G$. Equivalently, $\theta$ is
quasisplit if there exists a maximal $\theta$-split torus $A$ such that
$T=Z_{G}(A)$ is a maximal torus of $G$.

We refer the reader to [ABV] for the proof of the following fact.

######  Lemma 2.9.

Any class $a\in\operatorname{Out}_{2}(G)$ can be represented by a quasisplit
involution.

A consequence of the existence of minimal $\theta$-split parabolics is the
_Iwasawa decomposition_ (see [timashev] for a proof).

######  Lemma 2.10 (Iwasawa decomposition).

Let $P\subset G$ be a minimal $\theta$-split parabolic subgroup and $P^{u}$
its unipotent radical. Then the product $G^{\theta}_{0}AP^{u}$ is an open
subset of $G$.

###  2.3. The restricted root system

Let $A\subset G$ be a maximal $\theta$-split torus and $T\subset G$ a maximal
$\theta$-stable torus containing it. Let $r$ denote the rank of $T$ and $l$
the rank of $A$. The number $l$ is called the _rank of the symmetric variety_
$G/G^{\theta}$. Let us consider $\Phi_{T}$ the root system of $T$.

The involution $\theta$ naturally induces an involution
$\chi\mapsto\chi^{\theta}$ on the characters $\mathrm{X}^{*}(T)$. For each
$\chi\in\mathrm{X}^{*}(T)$ we define $\bar{\chi}=(\chi-\chi^{\theta})/2$,
which is a well defined element of $\mathcal{E}_{T}$. Now, the lattice formed
by the elements $\bar{\chi}$ of this form is naturally identified with the
group of characters $\mathrm{X}^{*}(A)$.

It is easy to check that the involution on characters of $T$ sends roots to
roots. The elements of the set $\Phi_{T}^{\theta}$ of roots fixed under this
involution are called the _imaginary roots_. A choice of positive roots
$\Phi_{T}^{+}\subset\Phi_{T}$ can be made in such a way that if
$\alpha\in\Phi_{T}^{+}$ is not an imaginary root, then $\alpha^{\theta}$ is a
negative root. Making this choice amounts to choosing a Borel subgroup
$B\subset G$. Indeed, this is the Borel subgroup $B$ with Lie algebra

$\mathfrak{b}=\mathfrak{t}\oplus\sum_{\alpha\in\Phi_{T}^{+}}\mathfrak{g}_{\alpha}.$

The following follows easily.

######  Lemma 2.11.

If $\theta$ is quasisplit, then $\Phi_{T}^{\theta}=\varnothing$.

We consider now the vector space
$\mathcal{E}_{\theta}=\mathrm{X}^{*}(A)\otimes_{\mathbb{Z}}\mathbb{R}$, which
is naturally a subspace of $\mathcal{E}_{T}$ and thus inherits the scalar
product $(\cdot,\cdot)$.

######  Definition 2.12.

The set of _restricted roots_ of $\theta$ is

$\Phi_{\theta}=\left\\{\bar{\alpha}=\frac{\alpha-\alpha^{\theta}}{2}\in\mathcal{E}_{\theta}:\alpha\in\Phi_{T}\setminus\Phi_{T}^{\theta}\right\\}.$

The following is well known (see, for example [richardson]).

######  Lemma 2.13.

The set $\Phi_{\theta}$ is a (possibly non-reduced) root system in
$\mathcal{E}_{\theta}$ with Weyl group the _little Weyl group_
$W_{\theta}=N_{G}(A)/Z_{G}(A)=N_{G_{0}^{\theta}}(A)/Z_{G_{0}^{\theta}}(A)$.

Simple roots for $\Phi_{\theta}$ can be constructed from the simple roots
$\Delta_{T}$ of $\Phi_{T}$. Indeed, these roots can be split into two groups

$\Delta_{T}\cap\Phi_{T}\setminus\Phi_{T}^{\theta}=\left\\{\alpha_{1},\dots,\alpha_{m}\right\\},\
\
\Delta_{T}\cap\Phi_{T}^{\theta}=\left\\{\beta_{1},\dots,\beta_{r-m}\right\\}.$

Now, one can order (see [deconcini-procesi] for more details) the simple roots
$\left\\{\alpha_{1},\dots,\alpha_{m}\right\\}$ in such a way that the
$\bar{\alpha}_{i}$ are mutually distinct for all $i\leq l$ and for each $j>l$
there exists some $i\leq l$ with $\bar{\alpha}_{j}=\bar{\alpha}_{i}$. Thus, we
obtain the set of _restricted simple roots_

$\Delta_{\theta}=\left\\{\bar{\alpha}_{1},\dots,\bar{\alpha}_{l}\right\\}.$

There is a natural inclusion of the root lattice
$\mathrm{R}_{\theta}:=\mathrm{R}(\Phi_{\theta})=\mathbb{Z}\langle\Phi_{\theta}\rangle$
in the group of characters $\mathrm{X}^{*}(A)$, and the cokernel of this
inclusion is the group of characters of the subgroup

$\left\\{a\in A:\bar{\alpha}(a)=1\text{ for all
}\bar{\alpha}\in\Phi_{\theta}\right\\}.$

This group can be easily shown to be equal to $F_{\theta}=A\cap G_{\theta}$,
and thus, if we denote $A_{G_{\theta}}=A/F_{\theta}$, we get

$\mathrm{R}_{\theta}=\mathrm{X}^{*}(A_{G_{\theta}}).$

Dually, we get
$\mathrm{P}^{\vee}_{\theta}=\mathrm{P}^{\vee}(\Phi_{\theta})=\mathrm{X}_{*}(A_{G_{\theta}})$.

We describe now the weight lattice
$\mathrm{P}_{\theta}=\mathrm{P}(\Phi_{\theta})$ of $\Phi_{\theta}$. Let us
denote by $\varpi_{1},\dots,\varpi_{l}$ the corresponding fundamental dominant
weights (that is, we want these weights to satisfy
$\langle\bar{\alpha}_{i}^{\vee},\varpi_{j}\rangle=\delta_{ij}$ for
$i=1,\dots,l$). When $G$ is semisimple simply-connected, these $\varpi_{i}$
can be determined in terms of the fundamental dominant weights of $\Phi_{T}$.
We can partition these into two sets

$\left\\{\omega_{1},\dots,\omega_{m}\right\\},\ \
\left\\{\zeta_{1},\dots,\zeta_{r-m}\right\\}$

with $\langle\alpha_{i}^{\vee},\omega_{j}\rangle=\delta_{ij}$,
$\langle\beta_{i}^{\vee},\zeta_{j}\rangle=\delta_{ij}$ and all the
$\langle\alpha_{i}^{\vee},\zeta_{j}\rangle=\langle\beta_{i}^{\vee},\omega_{j}\rangle=0$.

Since a root $\alpha$ and its image ${\alpha}^{\theta}$ must have the same
length, we can have three posible cases: (1) $\alpha^{\theta}=-\alpha$, (2)
$\langle\alpha^{\vee},\alpha^{\theta}\rangle=0$, and (3)
$\langle\alpha^{\vee},\alpha^{\theta}\rangle=1$. Note that if there is a root
$\alpha$ of type (3) then $s_{\alpha^{\theta}}(\alpha)=\alpha-\alpha^{\theta}$
must be a root in $\Phi_{T}$, and this root restricts to itself, which is
equal to $2\bar{\alpha}$. This implies that $\Phi_{\theta}$ is nonreduced and
that we can take the restricted simple roots $\Delta_{\theta}$ coming just
from simple roots of types (1) and (2).

Now, for any $\chi\in\mathrm{X}^{*}(T)$ we have

$\langle\bar{\alpha}^{\vee},\chi-\chi^{\theta}\rangle=\begin{cases}\langle\alpha^{\vee},\chi\rangle&\text{
in case (1)}\\\ \langle\alpha^{\vee},\chi-\chi^{\theta}\rangle&\text{ in case
(2)}\\\ 2\langle\alpha^{\vee},\chi-\chi^{\theta}\rangle&\text{ in case
(3)}.\end{cases}$

As in [deconcini-procesi], it is easy to see that there exists some
permutation $\sigma\in\mathfrak{S}_{m}$ of order $2$ such that for each
$i=1,\dots,l$, we have that
$\alpha_{i}^{\theta}+\alpha_{\sigma(i)}=\alpha_{\sigma(i)}^{\theta}+\alpha_{i}$
is a positive imaginary root. Moreover, we have
$\omega_{i}^{\theta}=-\omega_{\sigma(i)}$. Then it is straightforward from the
above that we get

$\varpi_{i}=\begin{cases}2\omega_{i}&\text{ if $\alpha_{i}$ is of type (1)}\\\
\omega_{i}+\omega_{\sigma(i)}&\text{ if $\alpha_{i}$ is of type (2) and
$i\neq\sigma(i)$}\\\ \omega_{i}&\text{ if $\alpha_{i}$ is of type (2) and
$i=\sigma(i)$}.\end{cases}$

Note that if $\Phi_{T}^{\theta}=\varnothing$, then $i=\sigma(i)$ if and only
if $\alpha_{i}^{\theta}=-\alpha_{i}$, and thus we do not have the third
situation in the formula above.

######  Lemma 2.14.

When $G$ is semisimple simply connected, the weight lattice
$\mathrm{P}_{\theta}=\mathbb{Z}\langle\varpi_{1},\dots,\varpi_{l}\rangle$ is
equal to the group of characters $\mathrm{X}^{*}(A_{G^{\theta}})$ of the torus
$A_{G^{\theta}}=A/F^{\theta}$. Dually, we get
$\mathrm{R}^{\vee}_{\theta}=\mathrm{R}^{\vee}(\Phi_{\theta})=\mathrm{X}_{*}(A_{G^{\theta}})$.

###### Proof.

The lattice $\mathrm{X}^{*}(A_{G^{\theta}})$ can be identified with
$2\mathrm{X}^{*}(A)=\left\\{\chi-\chi^{\theta}:\chi\in\mathrm{X}^{*}(T)\right\\}$.
It is clear from the above that for every $\chi\in\mathrm{X}^{*}(T)$ and
$\bar{\alpha}\in\Phi_{\theta}$, the number
$\langle\bar{\alpha}^{\vee},\chi-\chi^{\theta}\rangle$ is an integer. On the
other hand, $2\omega_{i}$ and $\omega_{i}+\omega_{\sigma(i)}$ are clearly of
the form $\chi-\chi^{\theta}$, while clearly $\omega_{i}$ restricts to $1$ on
$F^{\theta}=A\cap G^{\theta}$ if $\omega_{i}^{\theta}=-\omega_{i}$. ∎

To finish the section, note that

$\mathrm{P}_{\theta,+}:=\mathrm{P}_{+}(\Delta_{\theta})=\mathrm{P}_{\theta}\cap
C_{\Delta_{\theta}}^{+}=\mathrm{P}_{\theta}\cap
C_{\Delta}^{+}=\mathrm{X}^{*}(A_{G^{\theta}})\cap\mathrm{X}^{*}(T)_{+}=:\mathrm{X}^{*}(A_{G^{\theta}})_{+}.$

By an analogous argument, we get
$\mathrm{P}_{\theta,+}^{\vee}=\mathrm{X}_{*}(A_{G_{\theta}})\cap\mathrm{X}_{*}(T)_{+}=\mathrm{X}_{*}(A_{G_{\theta}})_{+}$.

######  Example 2.15 (The diagonal case).

In the diagonal case from Example 2.2, a maximal $\Theta$-split torus is given
by $A=\left\\{(t^{-1},t):t\in T\right\\}\subset T\times T$, with character
lattice
$\mathrm{X}^{*}(A)=\left\\{(-\chi,\chi):\chi\in\mathrm{X}^{*}(T)\right\\}$.
The restricted roots are then identified with $\Phi_{T}$.

######  Example 2.16 ($\operatorname{SL}_{n},n>2$).

Recall the classification of the involutions of $\operatorname{SL}_{n}$ given
in Example 2.3. We determine now the restricted root systems associated to
each of these involutions. We begin by considering the inner involutions
$\theta_{p,q}(g)=I_{p,q}gI_{p,q}$. Under a change of basis, we can write these
involutions as $\theta_{p,q}(g)=I^{\prime}_{p,q}gI^{\prime}_{p,q}$, for

$I^{\prime}_{p,q}=\begin{pmatrix}0&0&\tilde{I}_{p}\\\ 0&I_{q-p}&0\\\
\tilde{I}_{p}&0&0\end{pmatrix},$

where $\tilde{I}_{p}$ is the $p\times p$ matrix

$\tilde{I}_{p}=\begin{pmatrix}0&\dots&1\\\ 0&\reflectbox{$\ddots$}&0\\\
1&\dots&0\end{pmatrix}.$

Now it is clear that the elements of the form

$a=\mathrm{diag}(a_{1},\dots,a_{p},1,\dots,1,a_{p}^{-1},\dots,a_{1}^{-1})$

form a maximal $\theta_{p,q}$-split torus $A$. This torus sits inside the
standard maximal torus $T$ of $\operatorname{SL}_{n}$, which is
$\theta$-stable. The corresponding simple roots are
$\Delta_{\mathfrak{sl}_{n}}=\left\\{\alpha_{1},\dots,\alpha_{n-1}\right\\}$,
with

$\alpha_{i}=\varepsilon_{i}-\varepsilon_{i+1},$

for $\varepsilon_{i}(\mathrm{diag}(a_{1},\dots,a_{n}))=a_{i}$. The imaginary
roots are those of the form $\varepsilon_{i}-\varepsilon_{j}$, for $p<i\neq
j\leq q$; thus, $\theta_{p,q}$ is quasisplit if and only if $q=p$ or $q=p+1$.
The restricted root system is

$\Phi_{\theta}=\left\\{\pm\bar{\varepsilon}_{i}\pm\bar{\varepsilon}_{j},\pm
2\bar{\varepsilon}_{i},\pm\bar{\varepsilon}_{i}:1\leq i\neq j\leq p\right\\},$

when $p\neq n/2$, and

$\Phi_{\theta}=\left\\{\pm\bar{\varepsilon}_{i}\pm\bar{\varepsilon}_{j},\pm
2\bar{\varepsilon}_{i}:1\leq i\neq j\leq p\right\\},$

when $p=n/2$. Therefore, the restricted root system is nonreduced, of type BCp
if $p\neq n/2$, and reduced of type Cp if $p=n/2$.

For the outer involution $\theta_{0}$, the standard torus is maximal
$\theta_{0}$-split and $\Phi_{\theta}=\Phi_{T}$. On the other hand, if $n=2m$
is even, we can choose a basis such that
$\theta_{1}(g)=I_{m,m}(g^{\tilde{T}})^{-1}I_{m,m}$, where $\tilde{T}$ denotes
transposition with respect to the secondary diagonal. It is now easy to see
that the elements of the form

$a=\mathrm{diag}(a_{1},a_{2},\dots,a_{2},a_{1}),$

with $a_{1}a_{2}\cdots a_{m}=1$ form a maximal $\theta$-split torus $A$. The
restricted root system now is

$\Phi_{\theta}=\left\\{\bar{\varepsilon}_{i}-\bar{\varepsilon}_{j}:1\leq i\neq
j\leq m\right\\},$

so it has type Am-1. Table 1 summarizes this example and Example 2.3.

Table 1. Involutions of $\mathrm{SL}_{n}$, for $n>2$. Notation for real forms
following Helgason [helgason].

Involution $G^{\theta}$ Real form (when $k=\mathbb{C}$) Split or quasisplit?
$\Phi_{\theta}$ Inner $\theta_{p,q}(g)=I_{p,q}gI_{p,q}$
$\mathrm{S}(\mathrm{GL}_{p}\times\mathrm{GL}_{q})$ $\mathrm{SU}(p,q)$
Quasisplit iff $q=p$ or $q=p+1$ BCp if $p\neq q$ Cp if $p=q$ Outer
$\theta_{0}(g)=(g^{T})^{-1}$ $\mathrm{SO}_{n}$ $\mathrm{SL}_{n}(\mathbb{R})$
Split An-1 $\theta_{1}(g)=J_{m}(g^{T})^{-1}J_{m}^{-1}$ (only if $n=2m$)
$\mathrm{Sp}_{2m}$ $\mathrm{SU}^{*}(n)$ No Am-1

###  2.4. Symmetric pairs and symmetric varieties

######  Definition 2.17.

Any closed subgroup $H\subset G$ of a reductive group $G$ with

$G_{0}^{\theta}\subset H\subset G_{\theta}$

for some involution $\theta\in\operatorname{Aut}_{2}(G)$ is called a
_symmetric subgroup_ (associated to $\theta$). A pair $(G,H)$ where $G$ is a
reductive group and $H\subset G$ is a symmetric subgroup is called a
_symmetric pair_ , while the corresponding algebraic homogenoeus space $G/H$
is called a _symmetric variety_.

An _embedding of a symmetric variety_ or _symmetric embedding_ is a normal
$G$-variety $\Sigma$ with a $G$-equivariant open embedding
$O_{\Sigma}\hookrightarrow\Sigma$, where $O_{\Sigma}=G/H$ is a symmetric
variety.

######  Remark 2.18.

Note that the Lie algebra of any symmetric subgroup associated to $\theta$ is
equal to $\mathfrak{g}^{\theta}$, the fixed points in $\mathfrak{g}$ of the
involution induced by $\theta$.

If $G$ is a reductive group, we can decompose it as $G=G^{\prime}Z_{G}^{0}$,
for $G^{\prime}$ the derived subgroup and $Z_{G}^{0}$ the connected centre of
$G$, which is a torus. Now, any involution
$\theta\in\operatorname{Aut}_{2}(G)$ preserves the centre and its connected
components, so $\theta$ acts on $G^{\prime}$ as the involution
$\theta^{\prime}=\theta|_{G^{\prime}}\in\operatorname{Aut}_{2}(G^{\prime})$.
Moreover, there exists a torus $Z$ such that any symmetric variety $G/H$
associated to $\theta$ is isomorphic to one associated to the involution
$(g,z)\mapsto(\theta(g),z^{-1})$ on $G^{\prime}\times Z$.

Thus, from now on we will assume that any symmetric variety is of the form
$G_{Z}/H_{Z}$ where $G_{Z}$ is the reductive group $G_{Z}=G\times Z$, with $G$
a semisimple group and $Z$ a torus, and that $G_{Z}/H_{Z}$ is associated to
the involution $(g,z)\mapsto(\theta(g),z^{-1})$ for
$\theta\in\operatorname{Aut}_{2}(G)$. The group
$H=H_{Z}\cap(G\times\left\\{1\right\\})$ is a symmetric subgroup of $G$
associated to $\theta$. We denote $(G_{Z}/H_{Z})^{\prime}=G/H$ and call it the
_semisimple part_ of $G_{Z}/H_{Z}$.

###  2.5. Symmetric varieties as spherical varieties

Let $\Sigma$ be a normal $G$-variety. We denote by $k[\Sigma]$ its ring of
regular functions and by $k(\Sigma)$ its function field. For any subgroup
$H\subset G$ we denote by $k[\Sigma]^{H}$ (respectively $k(\Sigma)^{H}$) the
corresponding ring (resp. field) of $H$-invariant functions; that is, of
functions $f$ with

$f(h\cdot x)=f(x)\text{ for any }h\in H\text{ and }x\in\Sigma.$

We denote by $k[\Sigma]^{(H)}$ (respectively $k(\Sigma)^{(H)}$) the
corresponding ring (resp. field) of $H$-_semiinvariants_ of $\Sigma$; these
are the functions $f$ such that there exists a character
$\chi_{f}\in\mathrm{X}^{*}(H)$ with

$f(h\cdot x)=h^{\chi_{f}}f(x)\text{ for any }h\in H.$

Let $B\subset G$ be a Borel subgroup. The characters that arise from
$B$-semiinvariants of $\Sigma$ are called the _weights_ of $\Sigma$. More
precisely, we define the _weight lattice_ and the _weight semigroup_ of
$\Sigma$ to be, respectively

$\displaystyle\mathrm{P}(\Sigma)$
$\displaystyle=\left\\{\chi\in\mathrm{X}^{*}(B):\exists f\in
k(\Sigma)^{(B)},\chi_{f}=\chi\right\\},$ $\displaystyle\mathrm{P}_{+}(\Sigma)$
$\displaystyle=\left\\{\chi\in\mathrm{X}^{*}(B):\exists f\in
k[\Sigma]^{(B)},\chi_{f}=\chi\right\\}.$

The rank of the weight lattice $\mathrm{P}(\Sigma)$ is called the _rank_ of
$\Sigma$. When $\Sigma$ is affine, $\mathrm{P}(\Sigma)$ can be determined in
terms of $\mathrm{P}_{+}(\Sigma)$ as
$\mathrm{P}(\Sigma)=\mathbb{Z}\mathrm{P}_{+}(\Sigma)$.

A normal $G$-variety $\Sigma$ is _spherical_ if there exists a Borel subgroup
$B\subset G$ and an open $B$-orbit in $\Sigma$. Such a $B$-orbit is of the
form $Bx_{0}$, for some $x_{0}\in\Sigma$, and it is birational to $\Sigma$, so
$k(\Sigma)^{B}=k(Bx_{0})^{B}=k^{\times}$. Moreover, if $H\subset G$ is the
stabilizer of $x_{0}$, we have $Bx_{0}\cong B/H$ and thus
$k(\Sigma)^{(B)}\cong k(Bx_{0})^{(B)}\cong k(B/H)^{(B)}$, so

$\mathrm{P}(\Sigma)=\mathrm{X}^{*}(B/(B\cap H)).$

Restricting characters to a maximal torus $T\subset B$, we conclude that
$\mathrm{P}(\Sigma)=\mathrm{X}^{*}(T_{\Sigma})$, where $T_{\Sigma}$ is the
torus $T_{\Sigma}=T/(T\cap H)$. If $\Sigma$ is affine, its coordinate ring is
naturally a $G$-module, and as such it decomposes as a sum of irreducible
representations of $G$

$k[\Sigma]=\bigoplus_{\chi\in\mathrm{X}^{*}(T)_{+}}m_{\chi}V_{\chi},$

for certain multiplicities $m_{\chi}$. This allows us to identify

$\mathrm{P}_{+}(\Sigma)=\left\\{\chi\in\mathrm{X}^{*}(T)_{+}:m_{\chi}\geq
1\right\\}.$

Moreover, if $\Sigma$ is spherical, then one can show that all the
multiplicities $m_{\chi}$ are $\leq 1$, so $k[\Sigma]$ decomposes as

$k[\Sigma]=\bigoplus_{\chi\in\mathrm{P}_{+}(\Sigma)}V_{\chi}.$

Finally, one can also prove that the weight semigroup of an affine spherical
variety $\Sigma$ can be recovered as the lattice points on the cone generated
by it; that is,

$\mathrm{P}_{+}(\Sigma)=\mathbb{Q}_{+}\mathrm{P}_{+}(\Sigma)\cap\mathbb{Z}\mathrm{P}_{+}(\Sigma).$

It follows from the Iwasawa decomposition that for any involution
$\theta\in\operatorname{Aut}_{2}(G)$ and any Borel subgroup $B$ of a minimal
$\theta$-split parabolic subgroup $P\subset G$, there is an open
$G$-equivariant embedding $B\hookrightarrow G/G^{\theta}_{0}$. This implies
that all symmetric varieties and their embeddings are spherical. A first
consequence of this is that, if $\Sigma$ is a symmetric embedding with
$O_{\Sigma}=G/H$ a symmetric variety associated to an involution
$\theta\in\operatorname{Aut}_{2}(G)$, then

$\mathrm{P}(\Sigma)=\mathrm{X}^{*}(T_{H}),$

where $T_{H}=T/(T\cap H)$ for any $\theta$-stable maximal torus $T\subset G$.
Note that if $A\subset G$ is a maximal $\theta$-split torus contained in $T$,
then we also have

$T_{H}=A_{H}=A/(A\cap H).$

In particular, we have

$\mathrm{P}(G/G_{\theta})=\mathrm{X}^{*}(A_{G_{\theta}})=\mathrm{R}_{\theta}.$

Moreover, we have

$\mathrm{P}(G/G^{\theta})=\mathrm{X}^{*}(A_{G^{\theta}})$

which, if $G$ is semisimple simply-connected, is equal to the weight lattice
$\mathrm{P}_{\theta}$.

In general, one can recover the weight semigroup of a symmetric variety $G/H$
as the dominant weights of the corresponding torus $A_{H}$. More precisely, we
have

$\mathbb{Q}_{+}\mathrm{P}_{+}(G/H)=C^{+}_{\Delta_{\theta}}.$

Therefore,

$\mathrm{P}_{+}(G/H)=\mathrm{X}^{*}(A_{H})\cap
C^{+}_{\Delta_{\theta}}=\mathrm{X}^{*}(A_{H})\cap
C^{+}_{\Delta}=:\mathrm{X}^{*}(A_{H})_{+}.$

###  2.6. The dual group

Table 2. Simple groups with their involutions, real forms and associated dual
groups. Notation for types and real forms following Helgason [helgason].

Type $\mathfrak{g}$ $\check{\mathfrak{g}}$ $\mathfrak{g}_{\mathbb{R}}$ (when
$k=\mathbb{C}$) $\mathfrak{g}^{\theta}$ $\Delta_{\theta}$
$\Delta_{\theta}^{\mathrm{red}}$ $\Delta_{G/G^{\theta}}$
$\check{\mathfrak{g}}_{G/G^{\theta}}$ Notes AI $\mathfrak{sl}_{n}$
$\mathfrak{sl}_{n}$ $\mathfrak{sl}_{n}(\mathbb{R})$ $\mathfrak{so}_{n}$ An-1
An-1 An-1 $\mathfrak{sl}_{n}$ split AII $\mathfrak{sl}_{n}$
$\mathfrak{sl}_{n}$ $\mathfrak{so}^{*}(2n)$ $\mathfrak{sp}_{2n}$ An-1 An-1
An-1 $\mathfrak{sl}_{n}$ AIII $\mathfrak{sl}_{n}$ $\mathfrak{sl}_{n}$
$\mathfrak{so}(p,q)$ $\mathfrak{s}(\mathfrak{gl}_{p}\oplus\mathfrak{gl}_{q})$
BCp Bp Bp $\mathfrak{sp}_{2p}$ $p+q=n$, $p<q$ QS if $q=p+1$ AIV
$\mathfrak{sl}_{2n}$ $\mathfrak{sl}_{2n}$ $\mathfrak{su}(n,n)$
$\mathfrak{s}(\mathfrak{gl}_{n}\oplus\mathfrak{gl}_{n})$ Cn Cn Bn
$\mathfrak{sp}_{2n}$ quasisplit
$\Delta_{\theta}^{\mathrm{red}}\neq\Delta_{G/G^{\theta}}$ BI
$\mathfrak{so}_{2n+1}$ $\mathfrak{sp}_{2n}$ $\mathfrak{so}(p,q+1)$
$\mathfrak{so}_{p}\oplus\mathfrak{so}_{q}$ Bp Bp Bp $\mathfrak{sp}_{2p}$
$p+q=2n$, $p<q$ BII $\mathfrak{so}_{2n+1}$ $\mathfrak{sp}_{2n}$
$\mathfrak{so}(n,n+1)$ $\mathfrak{so}_{n}\oplus\mathfrak{so}_{n+1}$ Bn Bn Bn
$\mathfrak{sp}_{2n}$ split CI $\mathfrak{sp}_{2n}$ $\mathfrak{so}_{2n+1}$
$\mathfrak{sp}_{2n}(\mathbb{R})$ $\mathfrak{gl}_{n}$ Cn Cn Cn
$\mathfrak{so}_{2n+1}$ split CII $\mathfrak{sp}_{2n}$ $\mathfrak{so}_{2n+1}$
$\mathfrak{sp}(2p,2q)$ $\mathfrak{sp}_{2p}\oplus\mathfrak{sp}_{2q}$ BCp Bp Bp
$\mathfrak{sp}_{2p}$ $p+q=n$, $p<q$ $\mathfrak{sp}_{4n}$
$\mathfrak{so}_{4n+1}$ $\mathfrak{sp}(2n,2n)$
$\mathfrak{sp}_{2n}\oplus\mathfrak{sp}_{2n}$ Cn Cn Bn $\mathfrak{sp}_{2n}$
$\Delta_{\theta}^{\mathrm{red}}\neq\Delta_{G/G^{\theta}}$ DI
$\mathfrak{so}_{2n}$ $\mathfrak{so}_{2n}$ $\mathfrak{so}(n,n)$
$\mathfrak{so}_{n}\oplus\mathfrak{so}_{n}$ Dn Dn Dn $\mathfrak{so}_{2n}$ split
DII $\mathfrak{so}_{2n}$ $\mathfrak{so}_{2n}$ $\mathfrak{so}(p,q)$
$\mathfrak{so}_{p}\oplus\mathfrak{so}_{q}$ Bp Bp Cp $\mathfrak{so}_{2p+1}$
$p+q=2n$, $p<q$ QS if $q=p+2$
$\Delta_{\theta}^{\mathrm{red}}\neq\Delta_{G/G^{\theta}}$ DIII
$\mathfrak{so}_{2n}$ $\mathfrak{so}_{2n}$ $\mathfrak{so}^{*}(2n)$
$\mathfrak{gl}_{n}$ BC(n-1)/2 B(n-1)/2 B(n-1)/2 $\mathfrak{sp}_{n-1}$ $n$ odd
$\mathfrak{so}_{2n}$ $\mathfrak{so}_{2n}$ $\mathfrak{so}^{*}(2n)$
$\mathfrak{gl}_{n}$ Cn/2 Cn/2 Bn/2 $\mathfrak{sp}_{n}$ $n$ even
$\Delta_{\theta}^{\mathrm{red}}\neq\Delta_{G/G^{\theta}}$ EI
$\mathfrak{e}_{6}$ $\mathfrak{e}_{6}$ $\mathfrak{e}_{6(6)}$
$\mathfrak{sp}_{8}$ E6 E6 E6 $\mathfrak{e}_{6}$ split EII $\mathfrak{e}_{6}$
$\mathfrak{e}_{6}$ $\mathfrak{e}_{6(2)}$
$\mathfrak{sl}_{6}\oplus\mathfrak{sl}_{2}$ F4 F4 F4 $\mathfrak{f}_{4}$
quasisplit EIII $\mathfrak{e}_{6}$ $\mathfrak{e}_{6}$ $\mathfrak{e}_{6(-14)}$
$\mathfrak{so}_{11}\oplus\mathbb{C}$ BC2 B2 C2 $\mathfrak{so}_{5}$
$\Delta_{\theta}^{\mathrm{red}}\neq\Delta_{G/G^{\theta}}$ EIV
$\mathfrak{e}_{6}$ $\mathfrak{e}_{6}$ $\mathfrak{e}_{6(-26)}$
$\mathfrak{f}_{4}$ A2 A2 A2 $\mathfrak{sl}_{3}$ EV $\mathfrak{e}_{7}$
$\mathfrak{e}_{7}$ $\mathfrak{e}_{7(7)}$ $\mathfrak{sl}_{8}$ E7 E7 E7
$\mathfrak{e}_{7}$ split EVI $\mathfrak{e}_{7}$ $\mathfrak{e}_{7}$
$\mathfrak{e}_{7(-5)}$ $\mathfrak{so}_{6}\oplus\mathfrak{sl}_{2}$ F4 F4 F4
$\mathfrak{f}_{4}$ EVII $\mathfrak{e}_{7}$ $\mathfrak{e}_{7}$
$\mathfrak{e}_{7(-25)}$ $\mathfrak{e}_{6}\oplus\mathbb{C}$ C3 C3 B3
$\mathfrak{sp}_{6}$ $\Delta_{\theta}^{\mathrm{red}}\neq\Delta_{G/G^{\theta}}$
EVIII $\mathfrak{e}_{8}$ $\mathfrak{e}_{8}$ $\mathfrak{e}_{8(8)}$
$\mathfrak{so}_{16}$ E8 E8 E8 $\mathfrak{e}_{8}$ split EIX $\mathfrak{e}_{8}$
$\mathfrak{e}_{8}$ $\mathfrak{e}_{8(-24)}$
$\mathfrak{e}_{7}\oplus\mathfrak{sl}_{2}$ F4 F4 F4 $\mathfrak{f}_{4}$ FI
$\mathfrak{f}_{4}$ $\mathfrak{f}_{4}$ $\mathfrak{f}_{4(4)}$
$\mathfrak{sp}_{6}\oplus\mathfrak{sl}_{2}$ F4 F4 F4 $\mathfrak{f}_{4}$ split
FII $\mathfrak{f}_{4}$ $\mathfrak{f}_{4}$ $\mathfrak{f}_{4(-20)}$
$\mathfrak{so}_{9}$ BC1 A1 A1 $\mathfrak{sl}_{2}$ G $\mathfrak{g}_{2}$
$\mathfrak{g}_{2}$ $\mathfrak{g}_{2(2)}$
$\mathfrak{sl}_{2}\oplus\mathfrak{sl}_{2}$ G2 G2 G2 $\mathfrak{g}_{2}$ split

Sakellaridis and Venkatesh [sake-venka] describe a canonical way of
associating a _dual group_ $\check{G}_{\Sigma}$ to any spherical $G$-variety
$\Sigma$, conjecturally with a natural homomorphism
$\check{G}_{\Sigma}\rightarrow\check{G}$ to the Langlands dual group
$\check{G}$ of $G$. Knop and Schalke [knop-schalke] later proved that, indeed,
$\check{G}_{\Sigma}$ is contained in $\check{G}$ up to isogeny. Their
construction consists on taking the weight lattice $\mathrm{P}(\Sigma)$ on
$\Sigma$ and a natural set of simple roots $\Delta_{\Sigma}$ associated to
$\Sigma$, in such a way that
$(\mathrm{P}(\Sigma),\Delta_{\Sigma},\mathrm{P}(\Sigma)^{\vee},\Delta_{\Sigma}^{\vee})$
is a based root datum. The dual group $\check{G}_{\Sigma}$ is defined as the
reductive group with based root datum equal to
$(\mathrm{P}(\Sigma)^{\vee},\Delta_{\Sigma}^{\vee},\mathrm{P}(\Sigma),\Delta_{\Sigma})$.
The simple roots $\Delta_{\Sigma}$ are obtained by taking the generators of
intersections of the weight lattice with the negative of the dual of the
_central valuation cone_ , and then applying some process of normalization,
consisting in further intersecting the real span of each generator with the
root lattice of $G$, as explained in [sake-venka, knop-schalke].

If $\theta\in\operatorname{Aut}_{2}(G)$ is an involution, there is a natural
reduced root system associated to $\theta$, namely

$\Phi_{\theta}^{\mathrm{red}}=\left\\{\alpha\in\Phi_{\theta}:\alpha/2\not\in\Phi_{\theta}\right\\},$

the reduced version of the restricted root system obtained by taking the
shortest roots. If $A$ is the corresponding maximal $\theta$-split torus, the
resulting based root datum is
$(\mathrm{X}^{*}(A),\Delta_{\theta}^{\mathrm{red}},\mathrm{X}_{*}(A),(\Delta_{\theta}^{\mathrm{red}})^{\vee})$.
When $k=\mathbb{C}$, the group $\tilde{G}_{\theta}$ corresponding to this
based root datum is the complexification of a _maximal $\mathbb{R}$-split
subalgebra_
$\tilde{\mathfrak{g}}_{\mathbb{R}}\subset\mathfrak{g}_{\mathbb{R}}$ inside the
Lie algebra $\mathfrak{g}_{\mathbb{R}}$ of the real form of $G$ corresponding
to $\theta$. See [ana-oscar-ramanan]*Section 2.2 for more details.

In most cases $\Delta_{G/G^{\theta}}=\Delta_{\theta}^{\mathrm{red}}$, but in
some cases there are normalization factors involved, that change some roots by
multiples of them by factors of $1/2$ or $2$. In specific cases, we can find
that $\Delta_{\theta}^{\mathrm{red}}$ is of type $B_{m}$ or $C_{m}$, whereas
$\Delta_{G/G^{\theta}}$ is of type $C_{m}$ or $B_{m}$, respectively. In Table
2, we recollect the different involutions of the simple Lie algebras with
their fixed points, their associated real forms and root systems, and the Lie
algebra of the corresponding dual groups. The reader can compare Table 2 with
Table 1 in [ana-oscar-ramanan] and with Table 1 in [nadler]. For us, the most
relevant properties of the dual group $\check{G}_{G/G^{\theta}}$ are the
following:

  1. (1)

$\check{G}_{G/G^{\theta}}$ contains
$\check{A}_{G^{\theta}}=\operatorname{Spec}(k[e^{\mathrm{X}_{*}(A_{G^{\theta}})}])$
as a maximal torus,

  2. (2)

the Weyl group of $\check{G}_{G/G^{\theta}}$ is the little Weyl group
$W_{\theta}$,

  3. (3)

the dominant Weyl chamber of $\check{G}_{G/G^{\theta}}$ coincides with
$\mathrm{X}_{*}(A_{G^{\theta}})_{+}$.

The dual group $\check{G}_{G/G^{\theta}}$ of the symmetric variety
$G/G^{\theta}$ was previously discovered by Nadler [nadler] as a group
$\check{G}_{G_{\mathbb{R}}}$ naturally associated to the corresponding real
form $G_{\mathbb{R}}$, obtained as the Tannaka group of a certain neutral
Tannakian category of perverse sheaves in the _real loop Grassmannian_ of
$G_{\mathbb{R}}$, thus giving a generalization of the geometric Satake
correspondence. More generaly, by a similar procedure, Gaitsgory and Nadler
[gaitsgory-nadler] associated a reductive group
$\check{G}_{\Sigma,\mathrm{GN}}$ to any spherical $G$-variety $\Sigma$, with a
natural inclusion $\check{G}_{\Sigma,\mathrm{GN}}\subset\check{G}$. The
equality of $\check{G}_{\Sigma,\mathrm{GN}}$ and $\check{G}_{\Sigma}$ remains
conjectural in the general case of a spherical variety [BZSV]*Page 75.

###  2.7. Very flat symmetric embeddings and the Guay embedding

A $G$-variety $\Sigma$ is _simple_ if it has only one closed $G$-orbit. Let
$\Sigma$ be a simple affine symmetric embedding. Recall from our previous
discussions that the corresponding symmetric variety $O_{\Sigma}$ can be
written in the form $O_{\Sigma}=G_{Z}/H_{Z}$, for $G_{Z}$ a reductive group
$G_{Z}=G\times Z$, with $G$ semisimple and $Z$ a torus and $G_{Z}/H_{Z}$
associated to the involution $\vartheta:(g,z)\mapsto(\theta(g),z^{-1})$ for
$\theta\in\operatorname{Aut}_{2}(G)$.

Consider the natural projections $\mathrm{pr}_{1}:G_{Z}\rightarrow G$ and
$\mathrm{pr}_{2}:G_{Z}\rightarrow Z$. We can now define the tori

$A_{\Sigma}=O_{\Sigma}/G=Z/\mathrm{pr}_{2}(H_{Z}),$

and

$Z_{\Sigma}=Z/(\mathrm{pr}_{2}(H_{Z})\cap Z_{2}),\text{ for
}Z_{2}=\left\\{z\in Z:z^{2}=1\right\\}.$

Naturally, we have a projection $Z_{\Sigma}\rightarrow A_{\Sigma}$ and an
inclusion $\mathrm{X}^{*}(A_{\Sigma})\rightarrow\mathrm{X}^{*}(Z_{\Sigma})$.

Let $A\subset G$ be a maximal $\theta$-split torus and $T\subset G$ a maximal
torus containing it. Then, $A_{Z}=A\times Z$ is a maximal $\vartheta$-split
torus of $G_{Z}$ and $T_{Z}$ a maximal $\vartheta$-stable torus containing it.
Recall that we have another torus associated to $\Sigma$

$T_{\Sigma}=T_{Z}/(T_{Z}\cap H_{Z})=A_{Z}/(A_{Z}\cap H_{Z})$

such that $\mathrm{P}(\Sigma)=\mathrm{X}^{*}(T_{\Sigma})$. We denoted
$H=H_{Z}\cap(G\times\left\\{1\right\\})$. Let us put now
$\tilde{H}=\mathrm{pr}_{1}(H_{Z})$. Then,

$T_{Z}\cap H_{Z}\subset(T\cap\tilde{H})\times\mathrm{pr}_{2}(H_{Z}),$

so we have natural projections

$\mathrm{pr}_{1}:T_{\Sigma}\rightarrow A/(A\cap\tilde{H})=:A_{\tilde{H}}\text{
and }\mathrm{pr}_{2}:T_{\Sigma}\rightarrow
Z/\mathrm{pr}_{2}(H_{Z})=A_{\Sigma}.$

and inclusions
$i_{1}:\mathrm{X}^{*}(A_{\tilde{H}})\hookrightarrow\mathrm{X}^{*}(T_{\Sigma})$
and
$i_{2}:\mathrm{X}^{*}(A_{\Sigma})\hookrightarrow\mathrm{X}^{*}(T_{\Sigma})$.
Moreover, the kernel of the projection $\mathrm{pr}_{2}:T_{\Sigma}\rightarrow
A_{\Sigma}$ is the torus

$T_{\Sigma}\cap(G\times\left\\{1\right\\})=T/(T\cap H)=A/(A\cap H)=A_{H}.$

Therefore, we obtain a natural projection
$p:\mathrm{X}^{*}(T_{\Sigma})\rightarrow\mathrm{X}^{*}(A_{H})$. Note that
since $\Sigma$ is affine we also have a natural inclusion
$\mathrm{P}_{+}(\Sigma)\subset\mathrm{P}_{+}(O_{\Sigma})=\mathrm{X}^{*}(T_{\Sigma})_{+}.$

######  Definition 2.19.

The GIT quotient $\mathbf{A}_{\Sigma}:=\Sigma\mathbin{/\mkern-6.0mu/}G$ is an
$A_{\Sigma}$-toric variety, which we call the _abelianization_ of $\Sigma$.
The natural projection $\alpha_{\Sigma}:\Sigma\rightarrow\mathbf{A}_{\Sigma}$
is called the _abelianization map_ of $\Sigma$.

######  Remark 2.20.

The abelianization $\mathbf{A}_{\Sigma}$ is simply the toric variety

$\mathbf{A}_{\Sigma}=\operatorname{Spec}\left(\bigoplus_{\chi\in\mathrm{P}_{+}(\mathbf{A}_{\Sigma})}k[e^{\chi}]\right),$

where $\mathrm{P}_{+}(\mathbf{A}_{\Sigma})$ is the intersection

$\mathrm{P}_{+}(\mathbf{A}_{\Sigma})=\mathrm{P}_{+}(\Sigma)\cap
i_{2}(\mathrm{X}^{*}(A_{\Sigma})).$

######  Definition 2.21.

A _very flat symmetric embedding_ is a simple affine symmetric embedding
$\Sigma$ such that the abelianization map
$\alpha_{\Sigma}:\Sigma\rightarrow\mathbf{A}_{\Sigma}$ is dominant, flat and
with integral fibres.

We assume for the rest of the section that $G$ is simply-connected and fix
$\theta\in\operatorname{Aut}_{2}(G)$. As in the previous section, we fix a
maximal $\theta$-split torus $A\subset G$. We are interested in classifying
very flat symmetric embeddings $\Sigma$ such that the semisimple part of
$O_{\Sigma}$ is equal to $G/G^{\theta}$. These have been determined by Guay
[guay]*Proposition 7.

######  Theorem 2.22 (Guay).

A simple affine symmetric embedding $\Sigma$ with
$O_{\Sigma}^{\prime}=G/G^{\theta}$ is very flat if and only if there exists a
homomorphism

$\displaystyle\psi:\mathrm{X}^{*}(A_{G^{\theta}})$
$\displaystyle\longrightarrow\mathrm{X}^{*}(Z_{\Sigma}),$

such that, for any $(a,z)\in H_{Z}\cap A_{Z}\subset(\tilde{H}\cap
A)\times\mathrm{pr}_{2}(H_{Z})$ and for any
$\chi\in\mathrm{X}^{*}(A_{G^{\theta}})$, we have

$a^{\chi}=z^{-\psi(\chi)};$

and such that the weight semigroup $\mathrm{P}_{+}(\Sigma)$ is of the form

$\mathrm{P}_{+}(\Sigma)=\left\\{(\chi,\psi(\chi)+\eta):\chi\in\mathrm{X}^{*}(A_{G^{\theta}})_{+},\eta\in\mathrm{P}_{+}(\mathbf{A}_{\Sigma})\right\\}.$

######  Remark 2.23.

When $\Sigma$ is very flat any function $f\in k[G/G^{\theta}]$ with weight
$\chi$ can be extended to a function $f_{+}\in k[\Sigma]$ with weight
$(\chi,\psi(\chi))$. Indeed, just define

$f_{+}(g,z)=z^{\psi(\chi)}f(g).$

It is clear that $f_{+}$ has the desired weight. Now, since $\tilde{H}$ is a
symmetric subgroup for $\theta$, we have that any element $h\in\tilde{H}$ is
of the form $h=ah_{0}$ for $h\in G^{\theta}$ and $a\in A\cap\tilde{H}$. Thus,
if $(h,s)\in H_{Z}$, we have, for any $(g,z)\in G_{Z}$,

$f_{+}(hg,zs)=s^{\psi(\chi)}z^{\psi(\chi)}f(hg)=s^{\psi(\chi)}a^{\chi}f_{+}(g,z)=f_{+}(g,z).$

This allows us to describe $\Sigma$ more explicitly. For each fundamental
dominant weight $\varpi_{i}\in\mathrm{X}^{*}(A_{G^{\theta}})$, consider the
$G$-submodule $k[G/G^{\theta}]_{i}$ of $k[G/G^{\theta}]$ formed by functions
with weight $\varpi_{i}$, and take $f_{i}^{1},\dots,f_{i}^{m_{i}}$ a basis of
$k[G/G^{\theta}]_{i}$ as a $k$-vector space. On the other hand, let
$\gamma_{1},\dots,\gamma_{s}$ be generators of
$\mathrm{P}_{+}(\mathbf{A}_{\Sigma})\subset\mathrm{X}^{*}(Z)$. We can now
define the map

$\displaystyle(f_{+},\alpha_{+}):O_{\Sigma}$
$\displaystyle\longrightarrow\left(\bigoplus_{i=1}^{l}\mathbb{A}^{m_{i}}\right)\oplus\mathbb{A}^{s}$
$\displaystyle(g,z)H_{Z}$
$\displaystyle\longmapsto((f_{i+}^{1}(g,z),\dots,f_{i+}^{m_{i}}(g,z)),(z^{\gamma_{1}},\dots,z^{\gamma_{s}})).$

The symmetric embedding $\Sigma$ is then identified with the closure of the
image of this map, $\Sigma=\overline{(f_{+},\alpha_{+})(O_{\Sigma})}$.

Amongst all simple affine symmetric embeddings, those having a fixed point
will be of particular interest. By a fixed point of an $\Sigma$ of a symmetric
variety $G/H$ we mean a point $0\in\Sigma$ such that $g\cdot 0=0$ for any
$g\in G$. Since by assumption $\Sigma$ is simple, the set
$\left\\{0\right\\}\subset\Sigma$ is its unique closed orbit and thus a fixed
point, if it exists, is unique. From Guay’s classification we can obtain
necessary and sufficient conditions for the existence of a fixed point.

######  Proposition 2.24.

A very flat symmetric embedding $\Sigma$ with
$O_{\Sigma}^{\prime}=G/G^{\theta}$ has a fixed point if and only if
$\mathrm{P}_{+}(\mathbf{A}_{\Sigma})$ is pointed (meaning that if $\chi$ and
$-\chi$ are in $\mathrm{P}_{+}(\mathbf{A}_{\Sigma})$, then $\chi=0$) and the
only element in $\mathrm{P}_{+}(\Sigma)$ of the form $(\chi,\psi(\chi))$ for
$\chi\in\mathrm{X}^{*}(A_{G^{\theta}})_{+}$ is $(0,0)$.

###### Proof.

A fixed point $\Sigma$ corresponds to a maximal ideal $I\subset k[\Sigma]$
which is fixed under the action of $G_{Z}$. If such an ideal exists, it must
be of the form

$I=\bigoplus_{(\chi,\psi(\chi)+\eta)\in\mathrm{P}_{+}(\Sigma)\setminus\left\\{(0,0)\right\\}}k[O_{\Sigma}]_{(\chi,\psi(\chi)+\eta)},$

and this is only an ideal if for every
$(\chi,\psi(\chi)+\eta)\in\mathrm{P}_{+}(\Sigma)\setminus\left\\{(0,0)\right\\}$,
$(0,0)$ is not a highest weight of $k[O_{\Sigma}]_{(\chi,\psi(\chi)+\eta)}$.
That is the case and $(0,0)$ is a highest weight of
$k[O_{\Sigma}]_{(\chi,\psi(\chi)+\eta)}$ if and only if both $\eta$ and
$-\eta$ are contained in $\mathrm{P}_{+}(\mathbf{A}_{\Sigma})$. ∎

By the universal property of categorical quotients, a morphism
$\Sigma_{1}\rightarrow\Sigma_{2}$ of simple affine symmetric embeddings
induces a commutative square

${\Sigma_{1}}$${\Sigma_{2}}$${\mathbf{A}_{1}}$${\mathbf{A}_{2}.}$

######  Definition 2.25.

A morphism $\Sigma_{1}\rightarrow\Sigma_{2}$ of simple affine symmetric
embeddings is _excellent_ if the induced square is Cartesian.

We denote by $\mathcal{VF}(G/G^{\theta})$ the category with very flat
symmetric embeddings with semisimple part $G/G^{\theta}$ as objects and
excellent morphisms as arrows. We can also consider the subcategory
$\mathcal{VF}_{0}(G/G^{\theta})$ formed by very flat symmetric embeddings with
a fixed point.

Suppose that $\theta$ does not have imaginary roots (that is
$\Phi_{T}^{\theta}=\varnothing$, for $T\subset G$ a $\theta$-stable maximal
torus). For example, we can assume that $\theta$ is quasisplit. In that case,
a versal object of $\mathcal{VF}(G/G^{\theta})$ was constructed by Guay
[guay]. We recall here his construction.

We begin by taking $A\subset T\subset G$ a maximal $\theta$-split torus of $G$
and putting $G_{A}=G\times A$ and

$H_{A}=\left\\{(nh,an^{-1}):h\in G^{\theta},a\in F^{\theta},n\in
F_{\theta}\right\\},$

where we recall that we denoted $F^{\theta}=G^{\theta}\cap A$ and
$F_{\theta}=G_{\theta}\cap A$. The space $(G/G^{\theta})_{+}:=G_{A}/H_{A}$ is
a symmetric variety associated to the involution
$\vartheta:(g,a)\mapsto(\theta(g),a^{-1})$ with semisimple part isomorphic to
$G/G^{\theta}$. Indeed, we have

$H=H_{A}\cap(G\times\left\\{1\right\\})=G^{\theta},\ \ \text{ and }\ \
\tilde{H}=\mathrm{pr}_{1}(H_{A})=G_{\theta}.$

A maximal $\vartheta$-split torus of $G_{A}$ is given by $A\times A$.

######  Definition 2.26.

We define the (Guay) _enveloping embedding_ of $G/G^{\theta}$ as the symmetric
embedding $\operatorname{Env}(G/G^{\theta})$ with
$O_{\operatorname{Env}(G/G^{\theta})}=(G/G^{\theta})_{+}$ and determined by
the weight semigroup

$\mathrm{P}_{+}(\operatorname{Env}(G/G^{\theta}))=\left\\{(\chi,w_{0}\chi+\eta):\chi\in\mathrm{X}^{*}(A_{G^{\theta}})_{+},\eta\in-\mathbb{Z}_{+}\langle\Delta_{\theta}\rangle\right\\}\cup\left\\{(0,0)\right\\},$

for $\Delta_{\theta}=\left\\{\bar{\alpha}_{1},\dots,\bar{\alpha}_{l}\right\\}$
the simple restricted roots associated to $\theta$.

Note that $Z_{\operatorname{Env}(G/G^{\theta})}=A/F^{\theta}=A_{G^{\theta}}$
and $A_{\operatorname{Env}(G/G^{\theta})}=A/F_{\theta}=A_{G_{\theta}}$. Now,
we can define

$\displaystyle\psi:\mathrm{X}^{*}(A_{G^{\theta}})$
$\displaystyle\longrightarrow\mathrm{X}^{*}(A_{G^{\theta}})$
$\displaystyle\chi$ $\displaystyle\longmapsto w_{0}\chi.$

An element of $H_{A}\cap(A\times A)$ is of the form $(na_{1},a_{2}n^{-1})$ for
$a_{1},a_{2}\in F^{\theta}$ and $n\in F_{\theta}$. Therefore,

$(na_{1})^{-w_{0}\chi}=n^{-w_{0}\chi}=(n^{-1})^{w_{0}\chi}=(a_{2}n^{-1})^{\chi}.$

We conclude that $\operatorname{Env}(G/G^{\theta})$ is a very flat symmetric
embedding. Moreover, one checks easily that it has a fixed point
$0\in\operatorname{Env}(G/G^{\theta})$. Now, its complement
$\operatorname{Env}^{0}(G/G^{\theta})=\operatorname{Env}(G/G^{\theta})\setminus\left\\{0\right\\}$
is a smooth dense subvariety and it follows from [guay]*Theorem 4 that the GIT
quotient
$\operatorname{Env}^{0}(G/G^{\theta})\mathbin{/\mkern-6.0mu/}A_{G^{\theta}}$
can be shown to be isomorphic to the wonderful compactification
$\overline{G/G_{\theta}}$.

The main result of Guay’s paper is the following [guay]*Theorem 3.

######  Theorem 2.27 (Guay).

When $\Phi_{T}^{\theta}=\varnothing$, the envelopping embedding
$\operatorname{Env}(G/G^{\theta})$ is a versal object of the category
$\mathcal{VF}(G/G^{\theta})$ and a universal object of
$\mathcal{VF}_{0}(G/G^{\theta})$. That is, for any very flat symmetric
embedding $\Sigma$ with semisimple part $G/G^{\theta}$ there exists an
excellent morphism $\Sigma\rightarrow\operatorname{Env}(G/G^{\theta})$, which
is unique if $\Sigma$ has a fixed point.

###  2.8. The invariant theory of a symmetric embedding

Let $G$ be a reductive group and $\theta\in\operatorname{Aut}_{2}(G)$ an
involution. The action of $G$ on $G/G^{\theta}$ by left multiplication
restricts to an action of $G^{\theta}$. This action was widely studied by
Richardson [richardson], and we state here his main result.

######  Theorem 2.28 ([richardson]*Corollary 11.5).

Let $A\subset G$ be a maximal $\theta$-split torus and $W_{\theta}$ the little
Weyl group. The restriction homomorphism $k[G/G^{\theta}]\rightarrow
k[A_{G^{\theta}}]$ induces an isomorphism

$k[G/G^{\theta}]^{G^{\theta}}\overset{\sim}{\longrightarrow}k[A_{G^{\theta}}]^{W_{\theta}}.$

We denote

$\mathbf{C}_{G/G^{\theta}}=(G/G^{\theta})\mathbin{/\mkern-6.0mu/}G^{\theta}\cong
A_{G^{\theta}}/W_{\theta}.$

Richardson’s result can be extended easily to a symmetric embedding. Indeed,
suppose that $G$ is semisimple, $\theta\in\operatorname{Aut}_{2}(G)$ and let
$\Sigma$ be a simple affine symmetric embedding with $O_{\Sigma}^{\prime}$
isomorphic to $G/G^{\theta}$. The left action of $G^{\theta}$ extends to an
action on $\Sigma$. Let us denote
$\mathbf{C}_{\Sigma}=\Sigma\mathbin{/\mkern-6.0mu/}G^{\theta}$.

######  Proposition 2.29.

If $\Sigma$ is very flat, there is an isomorphism

$\mathbf{C}_{\Sigma}\cong\mathbf{C}_{G/G^{\theta}}\times\mathbf{A}_{\Sigma}.$

###### Proof.

Since every element of $\mathrm{P}_{+}(\Sigma)$ is of the form
$(\chi,\psi(\chi)+\eta)$ for $\chi\in\mathrm{X}^{*}(A_{G^{\theta}})_{+}$ and
$\eta\in\mathrm{P}_{+}(\mathbf{A}_{\Sigma})$,

$k[\Sigma]^{G^{\theta}}=\bigoplus_{(\chi,\psi(\chi)+\eta)\in\mathrm{P}_{+}(\Sigma)}k[O_{\Sigma}]_{(\chi,\psi(\chi)+\eta)}^{G^{\theta}}=k[G/G^{\theta}]^{G^{\theta}}\otimes
k[\mathbf{A}_{\Sigma}],$

as we wanted to show. ∎

It follows from the discussion in [richardson]*Sections 13 and 14 that when
$G$ is semisimple simply-connected, the ring $k[A_{G^{\theta}}]^{W_{\theta}}$
is a polynomial algebra and thus
$A_{G^{\theta}}\mathbin{/\mkern-6.0mu/}W_{\theta}$ is an affine space.
Moreover, since $k[A_{G^{\theta}}]=k[e^{\mathrm{X}^{*}(A_{G^{\theta}})}]$ and
the lattice $\mathrm{X}^{*}(A_{G^{\theta}})$ is generated by
$\varpi_{1},\dots,\varpi_{l}$, for the $\varpi_{i}$ defined in Section 2.3, we
get an isomorphism

$k[G/G^{\theta}]^{G^{\theta}}\cong k[b_{1},\dots,b_{l}],$

where each $b_{i}$ is a function in $k[G/G^{\theta}]$ with weight
$\varpi_{i}$. In general, for $G$ semisimple not simply-connected, Richardson
[richardson]*Section 15 characterizes which involutions satisfy that
$k[A_{G^{\theta}}]^{W_{\theta}}$ is a polynomial algebra.

###  2.9. Loop parametrization

Let $\mathcal{O}=k[[z]]$ denote the ring of formal power series in a formal
variable $z$ and $F=\mathrm{qf}(\mathcal{O})=k((z))$ the field of formal
Laurent series. For any variety $\Sigma$, we let
$\Sigma(F)=\mathrm{Maps}_{k}(\operatorname{Spec}F,\Sigma)$ be the set of
_formal loops_ and
$\Sigma(\mathcal{O})=\mathrm{Maps}_{k}(\operatorname{Spec}\mathcal{O},\Sigma)$
the set of _positive formal loops_. Both these sets can be regarded as the
spaces of $k$-points of the functors $\Sigma_{F}$ and $\Sigma_{\mathcal{O}}$
sending any $k$-algebra $R$ to
$\mathrm{Maps}_{k}(\operatorname{Spec}(R\otimes_{k}F),\Sigma)$ and
$\mathrm{Maps}_{k}(\operatorname{Spec}(R\otimes_{k}\mathcal{O}),\Sigma)$,
respectively. These functors can be endowed with the structure of an _ind-
scheme_. If $G$ is an algebraic group, the space $G(F)$ is called the _formal
loop group_ and $G(\mathcal{O})$ is the _formal positive loop group_. The
homogeneous space $\mathrm{Gr}_{G}=G(F)/G(\mathcal{O})$ is known as the
_affine Grassmannian of_ $G$.

If $G$ is a reductive group and $T\subset G$ is a maximal torus, for any
cocharacter $\lambda\in\mathrm{X}_{*}(T)$ we can obtain an element
$z^{\lambda}\in G(F)$ as the image of the formal variable $z$ under the
induced morphism

$\lambda(F):F^{\times}=\mathbb{G}_{m}(F)\rightarrow T(F)\subset G(F).$

The Cartan decomposition of the loop group states that

$G(F)=\bigsqcup_{\lambda\in\mathrm{X}_{*}(T)_{+}}G(\mathcal{O})z^{\lambda}G(\mathcal{O}).$

Moreover, the closure of any of these orbits
$G(\mathcal{O})z^{\lambda}G(\mathcal{O})$ is equal to

$\overline{G(\mathcal{O})z^{\lambda}G(\mathcal{O})}=\bigsqcup_{\mu\in\mathrm{X}_{*}(T)_{+},\mu\leq\lambda}G(\mathcal{O})z^{\mu}G(\mathcal{O}).$

See [zhu] for references on these facts.

The results stated above for reductive groups can be easily generalized to
symmetric varieties. We thus begin by taking an involution
$\theta\in\operatorname{Aut}_{2}(G)$, a maximal $\theta$-split torus $A\subset
G$, a maximal torus $T\subset G$ containing $A$, and a Borel subgroup
$B\subset G$ contained in a minimal $\theta$-split parabolic subgroup and
containing $T$. The following is well known.

######  Proposition 2.30.

Given any symmetric subgroup $H\subset G$ associated to $\theta$, we can
decompose

$(G/H)(F)=\bigsqcup_{\lambda\in\mathrm{X}_{*}(A_{H})_{-}}G(\mathcal{O})z^{\lambda}.$

######  Remark 2.31.

Note that $z^{\lambda}$ is a well defined element of $(G/H)(F)$ since there is
a natural inclusion $A_{H}=T/(T\cap H)\hookrightarrow G/H$.

###### Proof.

For completion, we recall a proof given by Nadler [nadler_matsuki]. We begin
by reducing to the case $H=G_{\theta}$. Thus, suppose that there is an element
of $z^{\mathrm{X}_{*}(A_{G_{\theta}})_{-}}$ in the orbit of every element of
$(G/G_{\theta})(F)$. Since, by definition of a symmetric subgroup, we have
$H\subset G_{\theta}$, there is a natural projection $G/H\rightarrow
G/G_{\theta}$ and thus $(G/H)(F)\rightarrow(G/G_{\theta})(F)$. Moreover, since
the inverse image of $A_{G_{\theta}}$ under the projection $G/H\rightarrow
G/G_{\theta}$ is $A_{H}$, the inverse image of
$z^{\mathrm{X}_{*}(A_{G_{\theta}})_{-}}$ is $z^{\mathrm{X}_{*}(A_{H})_{-}}$.
Since the projection is $G(\mathcal{O})$-equivariant, we conclude that there
is an element of $z^{\mathrm{X}_{*}(A_{H})_{-}}$ in the orbit of every element
of $(G/H)(F)$. Uniqueness follows from the fact that anti-dominant weights of
$A_{H}$ are anti-dominant for $T$ and from the Cartan decomposition of $G(F)$.

It remains to show that for every $\phi\in(G/G_{\theta})(F)$ there exists some
$\lambda\in\mathrm{X}_{*}(A_{G_{\theta}})_{-}$ such that $z^{\lambda}$ is in
the $G(\mathcal{O})$-orbit of $\phi$. We start by considering the _wonderful
compactification_ $\overline{G/G_{\theta}}$ of $G/G_{\theta}$, introduced by
De Concini and Procesi in [deconcini-procesi]. This is a projective open
$G$-equivariant embedding of $G/G_{\theta}$ satisfying several properties, of
which we are only interested in what is called the _local structure theorem_
[deconcini-procesi]*2.3. This result implies that every element of
$\overline{G/G_{\theta}}$ is in the $G$-orbit of an element of the toric
variety $\mathbf{A}$, defined as

$\mathbf{A}=\operatorname{Spec}\left(\bigoplus_{\bar{\alpha}\in\Delta_{\theta}}k[e^{-2\bar{\alpha}}]\right),$

and which can be identified with the $l$-dimensional affine space
$\mathbb{A}^{l}$ in such a way that the embedding $A\hookrightarrow\mathbf{A}$
is given explicitly as
$a\mapsto(a^{-2\bar{\alpha}_{1}},\dots,a^{-2\bar{\alpha}_{l}})$. Since
$\overline{G/G_{\theta}}$ is projective, by the valuative criterion of
properness, every formal loop $\phi\in(G/G_{\theta})(F)$ extends to a formal
arc $\bar{\phi}\in\overline{G/G_{\theta}}(\mathcal{O})$. By the local
structure theorem, there exist some formal arcs $g\in G(\mathcal{O})$ and
$\bar{a}\in\mathbf{A}(\mathcal{O})$ such that

$\bar{\phi}=g\bar{a}.$

A cocharacter $\lambda:\mathbb{G}_{m}\rightarrow A$ extends to a morphism
$\mathbb{A}^{1}\rightarrow\mathbf{A}$ if and only if
$\langle\lambda,2\bar{\alpha}_{i}\rangle\leq 0$ for every $i=1,\dots,l$, that
is, if
$\lambda\in\mathrm{P}_{\theta,-}^{\vee}=\mathrm{X}_{*}(A_{G_{\theta}})_{-}$.
Therefore, $\bar{a}=az^{\lambda}$ for some
$\lambda\in\mathrm{X}_{*}(A_{G_{\theta}})_{-}$. ∎

######  Remark 2.32.

Let $H\subset G$ be a symmetric subgroup associated to $\theta$ and $f\in
k[G/H]_{\chi}$ a function with weight $\chi\in\mathrm{X}^{*}(A_{H})_{+}$.
Recall that for any dominant weight $\chi$, if $V_{\chi}$ is the irreducible
representation with highest weight $\chi$, we have an isomorphism
$V_{\chi}^{*}=V_{-w_{0}\chi}$, for $w_{0}$ the longest element of the Weyl
group, so $f(gt)=t^{-w_{0}\chi}f(g)$ for any $g\in G$ and $t\in T$. Therefore,
if $\phi=gz^{\lambda}\in G(\mathcal{O})z^{\lambda}$, we have

$f(\phi)=(z^{\lambda})^{-w_{0}\chi}f(g)=z^{-\langle\lambda,w_{0}\chi\rangle}f(g).$

Thus, $f(\phi)$ is a Laurent series with highest pole order less or equal than
$\langle\lambda,w_{0}\chi\rangle$.

Let us describe now the closures of the orbits $G(\mathcal{O})z^{\lambda}$.
Recall that we can define an order in $\mathrm{X}_{*}(A_{H})_{-}$ by putting
$\lambda\leq\mu$ if and only if
$\lambda-\mu\in-\mathbb{N}\langle\Delta_{\theta}^{\vee}\rangle$.

######  Proposition 2.33.

For any $\lambda\in\mathrm{X}_{*}(A_{H})_{-}$, the closure of
$G(\mathcal{O})z^{\lambda}$ is equal to

$\overline{G(\mathcal{O})z^{\lambda}}=\bigsqcup_{\mu\in\mathrm{X}_{*}(A_{H})_{-},\mu\leq\lambda}G(\mathcal{O})z^{\mu}.$

###### Proof.

Let $f\in k[G/H]_{\chi}$ be a function with weight
$\chi\in\mathrm{X}^{*}(A_{H})_{+}$. If $\phi\in G(\mathcal{O})z^{\mu}$ is such
that $\phi$ belongs to the closure of $G(\mathcal{O})z^{\lambda}$ then the
highest pole order of $f(\phi)$ must be less or equal than
$\langle\lambda,w_{0}\chi\rangle$, so

$\langle\mu,w_{0}\chi\rangle\leq\langle\lambda,w_{0}\chi\rangle\implies\langle\lambda-\mu,w_{0}\chi\rangle\geq
0.$

Now, since $w_{0}\mathrm{X}^{*}(A_{H})_{+}=\mathrm{X}^{*}(A_{H})_{-}$, we
conclude that $\lambda-\mu\in-\mathbb{N}\langle\Delta_{\theta}^{\vee}\rangle$.

Reciprocally, we want to show that for every $\mu\leq\lambda$,
$G(\mathcal{O})z^{\mu}\in\overline{G(\mathcal{O})z^{\lambda}}$. It suffices to
find an element $\phi\in G(\mathcal{O})z^{\mu}$ with
$\phi\in\overline{G(\mathcal{O})z^{\lambda}}$. The argument is analogous to
the one given for the affine Grassmannian (see [zhu]). We do it for
$G=\mathrm{SL}_{2}$ and the general result follows by considering the
canonical homomorphism $\mathrm{SL}_{2}\rightarrow G$ associated to the root
$\bar{\alpha}$ with $\lambda-\mu=\bar{\alpha}^{\vee}$. Given any
$m\in\mathbb{N}$ we can consider the family

$\begin{pmatrix}z^{m}&0\\\
z^{-m}+t^{-1}z^{-m+1}&z^{-m}\end{pmatrix}\in\operatorname{SL}_{2}(\mathcal{O})z^{-m},$

for $t\in k^{\times}$. One can easily check that

$\begin{pmatrix}t^{-1}&0\\\ 0&t\end{pmatrix}\begin{pmatrix}1&-tz^{2m-1}\\\
0&1\end{pmatrix}\begin{pmatrix}z^{m}&0\\\
z^{-m}+t^{-1}z^{-m+1}&z^{-m}\end{pmatrix}=\begin{pmatrix}-z^{m-1}&-z^{m-1}\\\
tz^{-m}+z^{-m+1}&tz^{-m}\end{pmatrix},$

which lies in the same orbit $\operatorname{SL}_{2}(\mathcal{O})z^{-m}$ as the
original matrix. Now, the limit as $t\rightarrow 0$ clearly has to lie in the
closure $\overline{\operatorname{SL}_{2}(\mathcal{O})z^{-m}}$, but this limit
is

$\begin{pmatrix}-z^{m-1}&-z^{m-1}\\\
z^{-m+1}&0\end{pmatrix}=\begin{pmatrix}-z^{2m}&-z^{2m}\\\
1&0\end{pmatrix}z^{-m+1}\in\operatorname{SL}_{2}(\mathcal{O})z^{-(m-1)}$

We conclude that
$\operatorname{SL}_{2}(\mathcal{O})z^{-(m-1)}\subset\overline{\operatorname{SL}_{2}(\mathcal{O})z^{-m}}$
∎

Let now $G$ be semisimple simply-connected and
$\theta\in\operatorname{Aut}_{2}(G)$ an involution. We finish the section by
describing the loop space of the enveloping embedding
$\operatorname{Env}(G/G^{\theta})$; this is a generalization of [chi]*2.5. We
begin by noticing, as in the proof of Proposition 2.30 that if $\mathbf{A}$ is
an $A$-toric variety with weight semigroup $\mathrm{P}_{+}(\mathbf{A})\subset
A$, then an element $a\in A(F)$ extends to an element
$\bar{a}\in\mathbf{A}(\mathcal{O})$ if and only if $a\in
A(\mathcal{O})z^{\lambda}$, for $\lambda$ in the dual semigroup

$\mathrm{P}_{+}(\mathbf{A})^{\vee}=\left\\{\lambda\in\mathrm{X}^{*}(A):\langle\lambda,\chi\rangle\in\mathbb{N},\forall\chi\in\mathrm{P}_{+}(\mathbf{A})\right\\}.$

In other words,

$A(F)\cap\mathbf{A}(\mathcal{O})=\bigsqcup_{\lambda\in\mathrm{P}_{+}(\mathbf{A})^{\vee}}A(\mathcal{O})z^{\lambda}.$

In particular, we get

$A_{G_{\theta}}(F)\cap\mathbf{A}_{\operatorname{Env}(G/G^{\theta})}(\mathcal{O})=\bigsqcup_{\lambda\in\mathrm{X}^{*}(A_{G_{\theta}})_{-}}A_{G_{\theta}}(\mathcal{O})z^{\lambda},$

since
$\mathrm{X}^{*}(A_{G_{\theta}})_{-}=(-\mathbb{N}\langle\Delta_{\theta}\rangle)^{\vee}$.

Consider an anti-dominant cocharacter
$\lambda\in\mathrm{X}_{*}(A_{G_{\theta}})_{-}$ and define
$\operatorname{Env}^{\lambda}(G/G^{\theta})$ as the fibered product

${\operatorname{Env}^{\lambda}(G/G^{\theta})}$${A_{G_{\theta}}\times\operatorname{Env}(G/G^{\theta})}$${\operatorname{Spec}\mathcal{O}}$${\mathbf{A}_{\operatorname{Env}(G/G^{\theta})},}$$\scriptstyle{z^{-w_{0}\lambda}}$

where the vertical map on the right is the multiplication of the
abelianization map $\alpha_{\operatorname{Env}(G/G^{\theta})}$ with the
natural embedding
$A_{G_{\theta}}\hookrightarrow\mathbf{A}_{\operatorname{Env}(G/G^{\theta})}$.
Replacing $\operatorname{Env}(G/G^{\theta})$ by the open subvariety
$\operatorname{Env}^{0}(G/G^{\theta})$ we define an open subvariety
$\operatorname{Env}^{\lambda,0}(G/G^{\theta})$. The above stratification
induces

$\operatorname{Env}(G/G^{\theta})(\mathcal{O})\cap(G/G^{\theta})_{+}(F)=\bigsqcup_{\lambda\in\mathrm{X}^{*}(A_{G_{\theta}})_{-}}\operatorname{Env}^{\lambda}(G/G^{\theta})(\mathcal{O}).$

We also note that

$\operatorname{Env}^{\lambda,0}(G/G^{\theta})(\mathcal{O})=\operatorname{Env}^{\lambda}(G/G^{\theta})(\mathcal{O})\cap\operatorname{Env}^{0}(G/G^{\theta})(\mathcal{O})$

######  Proposition 2.34.

For any $\phi\in(G/G^{\theta})_{+}(F)$, we have
$\phi\in\operatorname{Env}^{\lambda}(G/G^{\theta})(\mathcal{O})$ if and only
if the image of $\phi$ in $(G/G_{\theta})(F)$ belongs to
$\overline{G(\mathcal{O})z^{\lambda}}$. Moreover,
$\phi\in\operatorname{Env}^{\lambda,0}(G/G^{\theta})(\mathcal{O})$ if and only
if the image of $\phi$ in $(G/G_{\theta})(F)$ belongs to
$G(\mathcal{O})z^{\lambda}$.

###### Proof.

It suffices to show it for $\phi=(z^{\mu},z^{\eta})$ with
$\mu,\eta\in\mathrm{X}_{*}(A_{G_{\theta}})_{-}$. First, note that by
construction we must have $z^{\eta}=z^{-w_{0}\lambda},$ so we get
$\eta=-w_{0}\lambda$. On the other hand,
$\phi\in\operatorname{Env}(G/G^{\theta})(\mathcal{O})$ if and only if
$f_{+}(\phi)\in\mathcal{O}$ for any $f\in k[G/G^{\theta}]$. Thus, for any
$\chi\in\mathrm{X}^{*}(A_{G^{\theta}})_{+}$ and for any $f\in
k[G/G^{\theta}]_{\chi}$ we have

$\mathcal{O}\ni f_{+}(z^{\mu},z^{\eta})=z^{\langle
w_{0}\chi,\eta\rangle}z^{\langle\chi,\mu\rangle}.$

Therefore,

$0\leq\langle w_{0}\chi,\eta\rangle+\langle\chi,\mu\rangle=\langle
w_{0}\chi,-w_{0}\lambda\rangle+\langle\chi,\mu\rangle=\langle\chi,\mu-\lambda\rangle.$

We have this for any $\chi\in\mathrm{X}^{*}(A_{G^{\theta}})_{+}$, so
$\mu-\lambda\in\mathbb{N}(\Delta_{\theta}^{\vee})$ and $\lambda\geq\mu$.
Finally, $\phi\in\operatorname{Env}^{0}(G/G^{\theta})(\mathcal{O})$ if and
only if $f_{+}(\phi)\in\mathcal{O}$ does not vanish, thus, if and only if
$0\geq\langle\chi,\mu-\lambda\rangle$, so $\lambda=\mu$. ∎

##  3\. Multiplicative Higgs bundles for symmetric varieties

###  3.1. The multiplicative Hitchin map for symmetric embeddings

We begin by taking a semisimple simply-connected group $G$ over $k$ and an
involution $\theta\in\operatorname{Aut}_{2}(G)$. Let $\Sigma$ be a very flat
symmetric embedding with $O_{\Sigma}^{\prime}=G/G^{\theta}$.

The left multiplication action of $G^{\theta}$ on $G/G^{\theta}$ extends to an
action of $G^{\theta}$ on $\Sigma$. Moreover, if $O_{\Sigma}$ is a symmetric
variety the form $G_{Z}/H_{Z}$ for some torus $Z$, then the torus
$Z_{\Sigma}=Z/Z_{2}$ also acts on $\Sigma$ through the action of $Z$.

We can now consider the quotient stacks $[\Sigma/G^{\theta}]$ and
$[\Sigma/(G^{\theta}\times Z_{\Sigma})]$. The quotient map
$\chi_{\Sigma}:\Sigma\rightarrow\mathbf{C}_{\Sigma}:=\Sigma\mathbin{/\mkern-6.0mu/}G^{\theta}$
induces a natural map
$\chi_{\Sigma}:[\Sigma/G^{\theta}]\rightarrow\mathbf{C}_{\Sigma}$, that we
also denote by $\chi_{\Sigma}$. We can also descend this to a map
$[\Sigma/(G^{\theta}\times
Z_{\Sigma})]\rightarrow[\mathbf{C}_{\Sigma}/Z_{\Sigma}]$.

As we explained in the previous section, the abelianization map
$\alpha_{\Sigma}:\Sigma\rightarrow\mathbf{A}_{\Sigma}:=\Sigma\mathbin{/\mkern-6.0mu/}G$
factors through
$\Sigma\overset{\chi_{\Sigma}}{\rightarrow}\mathbf{C}_{\Sigma}\rightarrow\mathbf{A}_{\Sigma}$.
The torus $Z_{\Sigma}$ clearly acts on the abelianization
$\mathbf{A}_{\Sigma}$, so composing with the natural map
$\mathbf{A}_{\Sigma}\rightarrow\operatorname{Spec}(k)$, this sequence induces
a sequence of quotient stacks

$[\Sigma/(G^{\theta}\times
Z_{\Sigma})]$$[\mathbf{C}_{\Sigma}/Z_{\Sigma}]$$[\mathbf{A}_{\Sigma}/Z_{\Sigma}]$${\mathbb{B}Z_{\Sigma}.}$$\scriptstyle{\chi_{\Sigma}}$

Here, $\mathbb{B}Z_{\Sigma}$ denotes the quotient stack
$[\operatorname{Spec}(k)/Z_{\Sigma}]$, which is the classifying stack of
$Z_{\Sigma}$-bundles.

Let $X$ be a smooth algebraic curve over $k$. Let $\mathcal{M}_{X}(\Sigma)$,
$\mathcal{B}_{X}(\Sigma)$ and $\mathcal{A}_{X}(\Sigma)$ denote the stacks of
maps from $X$ to $[\Sigma/(G^{\theta}\times Z_{\Sigma})]$,
$[\mathbf{C}_{\Sigma}/Z_{\Sigma}]$ and $[\mathbf{A}_{\Sigma}/Z_{\Sigma}]$,
respectively. We obtain a sequence

${\mathcal{M}_{X}(\Sigma)}$${\mathcal{B}_{X}(\Sigma)}$${\mathcal{A}_{X}(\Sigma)}$${\mathrm{Bun}_{Z_{\Sigma}}(X).}$$\scriptstyle{h_{X}}$

If we let $L\rightarrow X$ be a $Z_{\Sigma}$-bundle, the natural map
$X\rightarrow\mathbb{B}Z_{\Sigma}$ associated to $L$ induces the following
diagram, where all squares are Cartesian

${L(\text{$[\Sigma/G^{\theta}]$})}$${L(\mathbf{C}_{\Sigma})}$${L(\mathbf{A}_{\Sigma})}$${X}$$[\Sigma/(G^{\theta}\times
Z_{\Sigma})]$$[\mathbf{C}_{\Sigma}/Z_{\Sigma}]$$[\mathbf{A}_{\Sigma}/Z_{\Sigma}]$${\mathbb{B}Z_{\Sigma}.}$$\scriptstyle{\chi_{L}}$$\scriptstyle{L}$$\scriptstyle{\chi}$

These $L([\Sigma/G^{\theta}])$, $L(\mathbf{C}_{\Sigma})$ and
$L(\mathbf{A}_{\Sigma})$ are naturally stacks over $X$ and we denote by
$\mathcal{M}_{L}(\Sigma)$, $\mathcal{B}_{L}(\Sigma)$ and
$\mathcal{A}_{L}(\Sigma)$ their stacks of sections. We obtain a sequence

${\mathcal{M}_{L}(\Sigma)}$${\mathcal{B}_{L}(\Sigma)}$${\mathcal{A}_{L}(\Sigma).}$$\scriptstyle{h_{L}}$

This sequence is the fibre over $L\in\mathrm{Bun}_{Z_{\Sigma}}(X)$ of the
sequence defined above. Fixing a section $s$ in $\mathcal{A}_{L}(\Sigma)$,
over it we obtain a morphism of stacks

$\mathcal{M}_{s}(\Sigma)\overset{h_{s}}{\longrightarrow}\mathcal{B}_{s}(\Sigma).$

######  Definition 3.1.

The map $h_{s}:\mathcal{M}_{s}(\Sigma)\rightarrow\mathcal{B}_{s}(\Sigma)$ is
the _multiplicative Hitchin map_ over $X$ associated to $\Sigma$ and $s$.

We can give a more explicit description of all the objects taking part in this
definition. The bundle $L(\mathbf{A}_{\Sigma})$ is just the associated bundle
over $X$ defined by the action of $Z_{\Sigma}$ on the $A_{\Sigma}$-toric
variety $\mathbf{A}_{\Sigma}$. Since $\mathbf{A}_{\Sigma}$ is an affine space,
$L(\mathbf{A}_{\Sigma})$ is a vector bundle of rank equal to the rank of
$A_{\Sigma}$. The bundle $L(\mathbf{C}_{\Sigma})$ is also an associated
bundle, this time to the action of $Z_{\Sigma}$ on $\mathbf{C}_{\Sigma}$ and,
since $\mathbf{C}_{\Sigma}$ is also an affine space, $L(\mathbf{C}_{\Sigma})$
is also a vector bundle, of rank equal to the rank of $A_{\Sigma}$ plus the
rank of $G/G^{\theta}$. The stacks
$\mathcal{A}_{L}(\Sigma)=H^{0}(X,L(\mathbf{A}_{\Sigma}))$ and
$\mathcal{B}_{L}(\Sigma)=H^{0}(X,L(\mathbf{C}_{\Sigma}))$ are just the spaces
of sections of these vector bundles.

More precisely, $\mathbf{A}_{\Sigma}$ is the $A_{\Sigma}$-toric variety with
weight semigroup
$\mathrm{P}_{+}(A_{\Sigma})\subset\mathrm{X}^{*}(A_{\Sigma})\hookrightarrow\mathrm{X}^{*}(Z_{\Sigma})$.
If we take $\gamma_{1},\dots,\gamma_{s}$ to be generators of
$\mathrm{P}_{+}(A_{\Sigma})$, then
$L(\mathbf{A}_{\Sigma})=\bigoplus_{i=1}^{s}L_{\gamma_{i}}$, for
$L_{\gamma_{i}}$ the associated line bundle to the action of $Z_{\Sigma}$ on
$\mathbb{G}_{m}$ defined by $\gamma_{i}$. On the other hand,
$k[\mathbf{C}_{\Sigma}]$ is generated by the same $e^{\gamma_{i}}$ and by some
functions $b_{i}\in k[G/G^{\theta}]$ with weight $\varpi_{i}$, so,
$L(\mathbf{C}_{\Sigma})=L(\mathbf{A}_{\Sigma})\oplus\bigoplus_{i=1}^{l}(\psi^{*}L)_{\varpi_{i}}$.
Here, $\psi^{*}L$ is the $A_{G^{\theta}}$-bundle on $X$ obtained as the image
of $L$ through the map
$\mathbb{B}Z_{\Sigma}\rightarrow\mathbb{B}A_{G^{\theta}}$ induced by
$\psi:\mathrm{X}^{*}(A_{G^{\theta}})\rightarrow\mathrm{X}^{*}(Z_{\Sigma})$.

Now, a morphism $X\rightarrow[\Sigma/(G^{\theta}\times Z_{\Sigma})]$ consists
of a pair $(E,\varphi)$, where $E\rightarrow X$ is a $G^{\theta}$-bundle and
$\varphi\in H^{0}(X,E(\Sigma))$ is a section of the associated bundle
$E(\Sigma)$ defined by the action of $G^{\theta}$ on $\Sigma$. Such a pair is
called a _mutiplicative $\Sigma$-Higgs pair_.

Now the sequence
$\mathcal{M}_{L}(\Sigma)\rightarrow\mathcal{B}_{L}(\Sigma)\rightarrow\mathcal{A}_{L}(\Sigma)$
defining the multiplicative Hitchin map can be explicitly described as

$(E,\varphi)\longmapsto(b(\varphi),\alpha_{\Sigma}(\varphi))\longmapsto\alpha_{\Sigma}(\varphi),$

for $b(\varphi)=(b_{1}(\varphi),\dots,b_{l}(\varphi))$.

###  3.2. Multiplicative $(G,\theta)$-Higgs bundles

Let $G$ be any reductive group and $\theta\in\operatorname{Aut}_{2}(G)$ an
involution. As in the previous section, we let $X$ be a smooth algebraic curve
over $k$.

For any positive integer $d$ we denote by $X_{d}=X^{d}/\mathfrak{S}_{n}$ the
$d$-th symmetric product, so that elements $D\in X_{d}$ are effective divisors
of degree $d$ on $X$. More generally, given a tuple
$\bm{d}=(d_{1},\dots,d_{n})$ of positive integers, we denote
$X_{\bm{d}}=X_{d_{1}}\times\dots\times X_{d_{n}}$ and to any element
$\bm{D}=(D_{1},\dots,D_{n})\in X_{\bm{d}}$ we can associate the divisor
$D=D_{1}+\dots+D_{n}$.

######  Definition 3.2.

Let $\bm{D}\in X_{\bm{d}}$. A _multiplicative $(G,\theta)$-Higgs bundle with
singularities in $\bm{D}$_ is a pair $(E,\varphi)$, where $E\rightarrow X$ is
a $G^{\theta}$-bundle and $\varphi$ is a section of the associated bundle of
symmetric varieties $E(G/G^{\theta})$ defined over the complement
$X\setminus\lvert D\rvert$ of the support $\lvert D\rvert$ of the divisor $D$.

We denote by $\mathcal{M}_{\bm{d}}(G,\theta)$ the moduli stack of tuples
$(\bm{D},E,\varphi)$ with $\bm{D}\in X_{\bm{d}}$ and $(E,\varphi)$ a
multiplicative $(G,\theta)$-Higgs bundle with singularities in $\bm{D}$.

To any tuple $(\bm{D},E,\varphi)$ in $\mathcal{M}_{\bm{d}}(G,\theta)$ and any
point $x\in\lvert D\rvert$ we can associate an invariant
$\operatorname{inv}_{x}(\varphi)\in\mathrm{X}_{*}(A_{G_{\theta}})_{-}$, for
$A\subset G$ a maximal $\theta$-split torus. This is constructed by
considering the completion $\mathcal{O}_{x}$ of the local ring
$\mathscr{O}_{X,x}$ of $X$ at $x$, which by picking a local parameter is
isomorphic to the ring $\mathcal{O}$ of formal power series, and its quotient
field $F_{x}=\mathrm{qf}(\mathcal{O}_{x})\cong F$. By fixing a trivialization
of $E$ around the formal disk $\operatorname{Spec}(\mathcal{O}_{x})$ and
restricting $\varphi$ to $\operatorname{Spec}(F_{x})$ we get an element of
$(G/G^{\theta})(F_{x})$ which is well defined up to the choice of the
trivialization, thus, up to the action of $G(\mathcal{O}_{x})$. That is, we
obtain a well defined element

$\operatorname{inv}_{x}(\varphi)\in(G/G^{\theta})(F_{x})/G(\mathcal{O}_{x})\cong\mathrm{X}_{*}(A_{G^{\theta}})_{-}.$

Globally, if $D=\sum_{x\in X}n_{x}x$, we get a
$\mathrm{X}_{*}(A_{G^{\theta}})_{-}$-valued divisor

$\operatorname{inv}(\varphi)=\sum_{x\in
X}n_{x}\operatorname{inv}_{x}(\varphi)x.$

Now, we can impose a constraint on the invariant $\operatorname{inv}(\varphi)$
in order to obtain a finite-type moduli stack. We define

$\mathcal{M}_{\bm{d},\bm{\lambda}}(G,\theta)=\left\\{(\bm{D},E,\varphi)\in\mathcal{M}_{\bm{d}}(G,\theta):\operatorname{inv}(\varphi)=\bm{\lambda}\cdot\bm{D}\right\\},$

where $\bm{\lambda}=(\lambda_{1},\dots,\lambda_{n})$ is a tuple of anti-
dominant cocharacters $\lambda_{i}\in\mathrm{X}_{*}(A_{G^{\theta}})_{-}$ and
$\bm{\lambda}\cdot\bm{D}=\sum_{i=1}^{n}\lambda_{i}D_{i}$. For any element
$(\bm{D},E,\varphi)$ in $\mathcal{M}_{\bm{d},\bm{\lambda}}(G,\theta)$, we say
that the pair $(E,\varphi)$ is a multiplicative $(G,\theta)$-Higgs bundle with
_singularity type_ $(\bm{D},\bm{\lambda})$. The order on
$\mathrm{X}_{*}(A_{G^{\theta}})_{-}$ induces an order on
$\mathrm{X}_{*}(A_{G^{\theta}})_{-}$-valued divisors and thus we can also
define the bigger stack

$\mathcal{M}_{\bm{d},\bar{\bm{\lambda}}}(G,\theta)=\left\\{(\bm{D},E,\varphi)\in\mathcal{M}_{\bm{d}}(G,\theta):\operatorname{inv}(\varphi)\leq\bm{\lambda}\cdot\bm{D}\right\\}.$

Suppose now that we are in the situation in which
$k[G/G^{\theta}]^{G^{\theta}}$ is a polynomial algebra and thus it is
generated by some functions $b_{1},\dots,b_{l}$ with weights
$\varpi_{1},\dots,\varpi_{l}$ respectively, for $\varpi_{1},\dots,\varpi_{l}$
the fundamental dominant weights of the restricted root system
$\Phi_{\theta}$. Consider now the bundle
$\mathcal{B}_{\bm{d},\bm{\lambda}}(G,\theta)\rightarrow X_{\bm{d}}$ whose
fibre over $\bm{D}$ is the space of sections

$\mathcal{B}_{\bm{d},\bm{\lambda}}(G,\theta)_{\bm{D}}=\bigoplus_{i=1}^{l}H^{0}(X,\mathscr{O}_{X}(\langle\varpi_{i},w_{0}\bm{\lambda}\cdot\bm{D}\rangle)).$

######  Definition 3.3.

The _multiplicative Hitchin map_ for $(G,\theta)$-Higgs bundles is the
morphism

$\displaystyle
h_{\bm{d},\bm{\lambda}}:\mathcal{M}_{\bm{d},\bm{\lambda}}(G,\theta)$
$\displaystyle\longrightarrow\mathcal{B}_{\bm{d},\bm{\lambda}}(G,\theta)$
$\displaystyle(\bm{D},E,\varphi)$
$\displaystyle\longmapsto(b_{1}(\varphi),\dots,b_{l}(\varphi)).$

We can also consider the simpler case where the $d_{i}$ are equal to $1$ and
thus we have that $\bm{D}=\bm{x}=(x_{1},\dots,x_{n})$. We define

$\mathcal{M}_{\bm{x},\bm{\lambda}}(G,\theta):=\mathcal{M}_{\vec{1},\bm{\lambda}}(G,\theta)_{\bm{x}}=\left\\{(E,\varphi)\in\mathcal{M}_{\vec{1}}(G,\theta):\operatorname{inv}(\varphi)=\bm{\lambda}\cdot\bm{x}\right\\},$

where $\vec{1}=(1,\dots,1)$, and the bigger stack

$\mathcal{M}_{\bm{x},\bar{\bm{\lambda}}}(G,\theta):=\mathcal{M}_{\vec{1},\bar{\bm{\lambda}}}(G,\theta)_{\bm{x}}=\left\\{(E,\varphi)\in\mathcal{M}_{\vec{1}}(G,\theta):\operatorname{inv}(\varphi)\leq\bm{\lambda}\cdot\bm{x}\right\\}.$

When $k[G/G^{\theta}]$ is polynomial, we can also consider the Hitchin base

$\mathcal{B}_{\bm{x},\bm{\lambda}}(G,\theta)=\mathcal{B}_{\vec{1},\bm{\lambda}}(G,\theta)_{\bm{x}}=\bigoplus_{i=1}^{l}H^{0}(X,\mathscr{O}_{X}(\langle\varpi_{i},w_{0}\bm{\lambda}\cdot\bm{x}\rangle)),$

and the Hitchin map

$\displaystyle
h_{\bm{x},\bm{\lambda}}:\mathcal{M}_{\bm{x},\bm{\lambda}}(G,\theta)$
$\displaystyle\longrightarrow\mathcal{B}_{\bm{x},\bm{\lambda}}(G,\theta)$
$\displaystyle(E,\varphi)$
$\displaystyle\longmapsto(b_{1}(\varphi),\dots,b_{l}(\varphi)).$

###  3.3. Relating the two pictures

In this section we see how, for $G$ semisimple, the multiplicative Hitchin map
for $(G,\theta)$-Higgs bundles as defined in Section 3.2 can be recovered from
the multiplicative Hitchin map as defined in Section 3.1.

#### The simply-connected case

Let us suppose that $G$ is simply-connected. Let
$\bm{\lambda}=(\lambda_{1},\dots,\lambda_{n})$ be a tuple of anti-dominant
cocharacters $\lambda_{i}\in\mathrm{X}_{*}(A_{G^{\theta}})_{-}$. The natural
projection $A_{G^{\theta}}\to A_{G_{\theta}}$ induces a projection
$\mathrm{X}_{*}(A_{G^{\theta}})_{-}\to\mathrm{X}_{*}(A_{G_{\theta}})_{-}$ and
thus $\bm{\lambda}$ defines a multiplicative map

$\displaystyle\bm{\lambda}:\mathbb{G}_{m}^{n}$ $\displaystyle\longrightarrow
A_{G_{\theta}}$ $\displaystyle(z_{1},\dots,z_{n})$ $\displaystyle\longmapsto
z_{1}^{\lambda_{1}}\dots z_{n}^{\lambda_{n}}$

which extends to a map
$\bm{\lambda}:\mathbb{A}^{n}\rightarrow\mathbf{A}_{\operatorname{Env}(G/G^{\theta})}$,
since
$\mathrm{P}_{+}(\operatorname{Env}(G/G^{\theta}))^{\vee}=\mathrm{X}_{*}(A_{G_{\theta}})_{-}$.
We can now define a very flat symmetric embedding $\Sigma^{\bm{\lambda}}$ with
semisimple part $O_{\Sigma^{\bm{\lambda}}}^{\prime}=G/G^{\theta}$ through the
Cartesian diagram

${\Sigma^{\bm{\lambda}}}$${\operatorname{Env}(G/G^{\theta})}$${\mathbb{A}^{n}}$${\mathbf{A}_{\operatorname{Env}(G/G^{\theta})}.}$$\scriptstyle{\alpha_{\operatorname{Env}(G/G^{\theta})}}$$\scriptstyle{-w_{0}\bm{\lambda}}$

Since for any symmetric embedding the tori $A_{\Sigma}$ and $Z_{\Sigma}$
differ by a finite subgroup, by construction we get
$Z_{\Sigma^{\bm{\lambda}}}=\mathbb{G}_{m}^{n}$ and
$\mathbf{A}_{\Sigma^{\bm{\lambda}}}=\mathbb{A}^{n}$, so
$\mathrm{Bun}_{Z_{\Sigma^{\bm{\lambda}}}}(X)=\mathrm{Pic}(X)^{n}$ and, for any
tuple of line bundles $\bm{L}=(L_{1},\dots,L_{n})$, the fibre of
$\mathcal{A}_{X}(\Sigma^{\bm{\lambda}})\rightarrow\mathrm{Bun}_{Z_{\Sigma^{\bm{\lambda}}}}(X)$
over $\bm{L}$ is
$\mathcal{A}_{\bm{L}}(\Sigma^{\bm{\lambda}})=\bigoplus_{i=1}^{n}H^{0}(X,L_{i})$.
Replacing $\operatorname{Env}(G/G^{\theta})$ by
$\operatorname{Env}^{0}(G/G^{\theta})$ we obtain an open dense subvariety
$\Sigma^{\bm{\lambda},0}\subset\Sigma^{\bm{\lambda}}$, and we can also
consider the open substack $\mathcal{M}_{X}(\Sigma^{\bm{\lambda},0})$ of
$\mathcal{M}_{X}(\Sigma^{\bm{\lambda}})$ of maps from $X$ to
$[M^{\bm{\lambda},0}/(G^{\theta}\times Z_{\Sigma})]$.

Consider now $\bm{d}=(d_{1},\dots,d_{n})$ a tuple of positive integers and an
element $\bm{D}=(D_{1},\dots,D_{n})\in X_{\bm{d}}$. To each $D_{i}$ we can
associate the line bundle $\mathscr{O}_{X}(D_{i})$ which, since $D_{i}$ is
effective (all its coefficients are positive), has a canonical non-vanishing
section $s_{i}\in H^{0}(X,L_{i})$. Let us denote
$\mathscr{O}_{X}(\bm{D})=\bigoplus_{i=1}^{n}\mathscr{O}_{X}(D_{i})$ and
$\bm{s}=(s_{1},\dots,s_{n})$.

######  Theorem 3.4.

The map

$\displaystyle X_{\bm{d}}$
$\displaystyle\longrightarrow\mathcal{A}_{X}(\Sigma^{\bm{\lambda}})$
$\displaystyle\bm{D}$
$\displaystyle\longmapsto(\mathscr{O}_{X}(\bm{D}),\bm{s})$

induces the following diagram, where all squares are Cartesian,

${\mathcal{M}_{\bm{d},\bar{\bm{\lambda}}}(G,\theta)}$${\mathcal{M}_{\bm{d},\bar{\bm{\lambda}}}(G,\theta)}$${\mathcal{B}_{\bm{d},\bm{\lambda}}(G,\theta)}$${X_{\bm{d}}}$${\mathcal{M}_{X}(\Sigma^{\bm{\lambda},0})}$${\mathcal{M}_{X}(\Sigma^{\bm{\lambda}})}$${\mathcal{B}_{X}(\Sigma^{\bm{\lambda}})}$${\mathcal{A}_{X}(\Sigma^{\bm{\lambda}}).}$$\scriptstyle{h_{\bm{d},\bm{\lambda}}}$$\scriptstyle{h_{X}}$

###### Proof.

From the above disussions it is clear that for any $\bm{D}\in X_{\bm{d}}$ we
can identify $\mathcal{B}_{\mathscr{O}_{X}(\bm{D})}(\Sigma)$ with the space of
sections of the bundle

$\mathscr{O}_{X}(\bm{D})\oplus\bigoplus_{i=1}^{l}(\mathscr{O}_{X}(\bm{D})\times_{w_{0}\bm{\lambda}}\mathbb{G}_{m})_{\varpi_{i}}=\mathscr{O}_{X}(\bm{D})\oplus\bigoplus_{i=1}^{l}\mathscr{O}_{X}(\langle\varpi_{i},w_{0}\bm{\lambda}\cdot\bm{D}\rangle).$

It is now clear that the rightmost square is Cartesian.

Take any point $x\in\lvert D\rvert$ and take $\lambda_{x}$ the coefficient of
the divisor $\bm{\lambda}\cdot\bm{D}$ corresponding to $x$. This $\lambda_{x}$
is an antidominant cocharacter that can be written as

$\lambda_{x}=\bm{m}_{x}\cdot\bm{\lambda},$

where $\bm{m}_{x}=(m_{1x},\dots,m_{nx})$ is a vector with components $m_{ix}$
for $D_{i}=\sum_{x}m_{ix}x$. Now, by mapping $z\mapsto
z^{\bm{m}_{x}}=(z^{m_{1x}},\dots,z^{m_{nx}})$ we obtain a morphism
$\bm{m}:\mathcal{O}\rightarrow\mathbb{A}^{n}$ and the following diagram, with
all squares Cartesian

${\operatorname{Env}^{\lambda_{x}}(G/G^{\theta})}$${\Sigma^{\bm{\lambda}}}$${\operatorname{Env}(G/G^{\theta})}$${\operatorname{Spec}\mathcal{O}}$${\mathbb{A}^{n}}$${\mathbf{A}_{\operatorname{Env}(G/G^{\theta})}.}$$\scriptstyle{\alpha_{\operatorname{Env}(G/G^{\theta})}}$$\scriptstyle{\bm{m}}$$\scriptstyle{-w_{0}\bm{\lambda}}$

Now, if we restrict an element
$(E,\varphi)\in\mathcal{M}_{X}(\Sigma^{\bm{\lambda}})$ to
$\operatorname{Spec}(F_{x})$ we obtain an element of
$(G/G^{\theta})_{+}(F_{x})$ which must lie in
$\operatorname{Env}^{\lambda_{x}}(G/G^{\theta})$. Therefore, by Proposition
2.34, the image of $\varphi|_{\operatorname{Spec}(F_{x})}$ belongs to
$\overline{G(\mathcal{O}_{x})z^{\lambda_{x}}}$ and thus
$\operatorname{inv}_{x}(\varphi)\leq\lambda_{x}$. Moreover, if we change
$\operatorname{Env}(G/G^{\theta})$ by $\operatorname{Env}^{0}(G/G^{\theta})$
in the diagram above, the element $\varphi|_{\operatorname{Spec}(F_{x})}$ lies
in $\operatorname{Env}^{\lambda_{x},0}(G/G^{\theta})$ and therefore its image
belongs to $G(\mathscr{O}_{x})z^{\lambda_{x}}$, so
$\operatorname{inv}_{x}(\varphi)=\lambda_{x}$. ∎

#### The non simply-connected case

Suppose now that $G$ is a semisimple group and consider
$\pi:\hat{G}\rightarrow G$ the simply-connected cover. This map is a central
isogeny and thus its kernel is a subgroup of the centre $\pi_{1}(G)\subset
Z_{\hat{G}}$, the fundamental group of $G$. Moreover, it is a well known
result of Steinberg [steinberg_endomorphisms]*Theorem 9.16 that any involution
$\theta\in\operatorname{Aut}_{2}(G)$ lifts to an involution
$\hat{\theta}\in\operatorname{Aut}_{2}(\hat{G})$ making the following diagram
commute

${\hat{G}}$${\hat{G}}$${G}$${G.}$$\scriptstyle{\hat{\theta}}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$$\scriptstyle{\theta}$

The following is also well-known.

######  Lemma 3.5.

$\pi(\hat{G}^{\hat{\theta}})=G_{0}^{\theta}$.

###### Proof.

The inclusion $\pi(\hat{G}^{\hat{\theta}})\subset G_{0}^{\theta}$ is clear.
Indeed, for any $g\in\hat{G}^{\hat{\theta}}$, we have

$\theta(\pi(g))=\pi(\hat{\theta}(g))=\pi(g),$

so $\pi(\hat{G}^{\hat{\theta}})\subset G^{\theta}$ and the inclusion follows
since $\pi$ is continuous and $\hat{G}^{\hat{\theta}}$ is connected.

For the other inclusion, take $k\in G_{0}^{\theta}$ and $g\in\hat{G}$ with
$\pi(g)=k$. Now,

$\pi(g\hat{\theta}(g)^{-1})=k\theta(k)^{-1}=1.$

Thus, $g\hat{\theta}(g)^{-1}\in\pi_{1}(G)\subset Z_{\hat{G}}$, so
$g\in\hat{G}_{\hat{\theta}}$. Recall now the decomposition
$G_{\hat{\theta}}=F_{\hat{\theta}}\hat{G}^{\hat{\theta}}$, which implies that
there exists some $f\in F_{\hat{\theta}}$ and some
$g_{0}\in\hat{G}^{\hat{\theta}}$ such that $g=fg_{0}$. Let
$k_{0}=\pi(g_{0})\in G_{0}^{\theta}$. Clearly, $\pi(f)\in F^{\theta}$, so we
obtain a decomposition

$k=\pi(f)k_{0},$

with $k,k_{0}\in G_{0}^{\theta}$ and $\pi(f)\in F^{\theta}$. This implies that
$\pi(f)=1$, and thus $k=k_{0}=\pi(g_{0})$, and
$k\in\pi(\hat{G}^{\hat{\theta}})$. ∎

It follows from the lemma above that we can define an action of
$G_{0}^{\theta}$ on $\hat{G}/\hat{G}^{\hat{\theta}}$ from the left
multiplication action of $\hat{G}^{\hat{\theta}}$. Indeed, for any $k\in
G_{0}^{\theta}$ we just have to take any $h\in\hat{G}^{\hat{\theta}}$ with
$\pi(h)=k$ and define $k\cdot(gG^{\theta})=hgG^{\theta}$. This is well defined
since, if $h^{\prime}$ is such that $\pi(h^{\prime})=k$, then $h^{\prime}=zh$
for some $z\in G^{\theta}\cap Z_{G}$ and thus
$h^{\prime}gG^{\theta}=hgG^{\theta}$.

For the following, we need to introduce a new object.

######  Definition 3.6.

A _multiplicative $(G,\theta)_{0}$-Higgs bundle_ is a pair $(E,\varphi)$,
where $E\rightarrow X$ is a $G_{0}^{\theta}$-bundle and $\varphi$ is a section
of the associated bundle of symmetric varieties $E(G/G^{\theta})$ defined over
$X^{\prime}$ the complement of a finite subset of $X$.

The exact sequence $1\rightarrow\pi_{1}(G)\rightarrow\hat{G}\rightarrow
G\rightarrow 1$, induces a fibration
$\Gamma\hookrightarrow\hat{G}/\hat{G}^{\hat{\theta}}\twoheadrightarrow
G/G^{\theta}$, for some finite group $\Gamma$. For any
$G_{0}^{\theta}$-bundle, we can consider the bundles $E(G/G^{\theta})$ and
$E(\hat{G}/\hat{G}^{\hat{\theta}})$, associated to the actions of
$G_{0}^{\theta}$ and we obtain a exact sequence of sets with a distinguished
element (where $H^{0}$ stands for sets of sections)

${H^{0}(X^{\prime},E(\hat{G}/\hat{G}^{\hat{\theta}}))}$${H^{0}(X^{\prime},E(G/G^{\theta}))}$${H^{1}(X^{\prime},\Gamma).}$$\scriptstyle{\delta}$

It follows that to any multiplicative $(G,\theta)_{0}$-Higgs bundle
$(E,\varphi)$ we can associate the invariant $\delta(\varphi)\in
H^{1}(X^{\prime},\Gamma)$. Now, the section $\varphi$ comes from a section of
$E(\hat{G}/\hat{G}^{\hat{\theta}})$ if and only if $\delta(\varphi)=1$.
Moreover, the map $\pi:\hat{G}^{\hat{\theta}}\rightarrow G_{0}^{\theta}$
induces a map
$\mathrm{Bun}_{\hat{G}^{\hat{\theta}}}\rightarrow\mathrm{Bun}_{G_{0}^{\theta}}$.
We conclude the following.

######  Proposition 3.7.

Any multiplicative $(G,\theta)_{0}$-Higgs bundle with $\delta(\varphi)=1$ is
induced from a multiplicative $(\hat{G},\hat{\theta})$-Higgs bundle from the
maps
$\mathrm{Bun}_{\hat{G}^{\hat{\theta}}}\rightarrow\mathrm{Bun}_{G_{0}^{\theta}}$
and $\hat{G}/\hat{G}^{\hat{\theta}}\rightarrow G/G^{\theta}$.

The next step consists on relating multiplicative $(G,\theta)_{0}$-Higgs
bundles with multiplicative $(G,\theta)$-Higgs bundles. We start by
considering a $(G,\theta)$-Higgs bundle $(E,\varphi)$. The decomposition
$G^{\theta}=F^{\theta}G_{0}^{\theta}$ allows us to define an isomorphism
$G^{\theta}\cong F^{\theta}\ltimes G_{0}^{\theta}$, and this induces a
factorization of the $G^{\theta}$-bundle $E\rightarrow X$ as

${E}$${Y=E/G_{0}^{\theta}}$${X.}$

Here, $Y\rightarrow X$ is a Galois étale cover with Galois group equal to
$F^{\theta}$, while $E\rightarrow Y$ has the structure of a principal
$G_{0}^{\theta}$-bundle. Moreover, the deck transformation $Y\rightarrow Y$
induced by an element $a\in F^{\theta}$ lifts to a map $\tilde{a}:E\rightarrow
E$ and thus defines a right $F^{\theta}$-action on the $G_{0}^{\theta}$-bundle
$E\rightarrow Y$, which clearly gives an $\operatorname{Int}$-twisted
$F^{\theta}$-equivariant structure on it. By this we mean that, for any $e\in
E$, $a\in F^{\theta}$ and $g\in G_{0}^{\theta}$, we have

$\tilde{a}(e)\cdot g=\tilde{a}(e\cdot aga^{-1}).$

A similar argument allows us to see the bundle $E(G/G^{\theta})\rightarrow X$
as a $F^{\theta}$-equivariant bundle on $Y$, and there is a bijective
correspondence between sections of $E(G/G^{\theta})\rightarrow X$ and
$F^{\theta}$-equivariant sections of $E(G/G^{\theta})\rightarrow Y$. We refer
the reader to [noconexo] for more details on these correspondences. We have
proved the following.

######  Proposition 3.8.

Any multiplicative $(G,\theta)$-Higgs bundle is induced from a multiplicative
$(G,\theta)_{0}$-Higgs bundle on a certain $F^{\theta}$-Galois cover
$Y\rightarrow X$.

##  4\. Multiplicative Higgs bundles, involutions and fixed points

###  4.1. Multiplicative Higgs bundles and moduli spaces

Let $G$ be a reductive group and $X$ a smooth algebraic curve over $k$. Let
$\bm{d}=(d_{1},\dots,d_{n})$ be a tuple of positive integers.

######  Definition 4.1.

Let $\bm{D}\in X_{\bm{d}}$. A _multiplicative $G$-Higgs bundle with
singularities in $\bm{D}$_ is a pair $(E,\varphi)$, where $E\rightarrow X$ is
a $G$-bundle and $\varphi$ is a section of the adjoint bundle of groups $E(G)$
defined over the complement $X\setminus|D|$ of the support $|D|$ of the
divisor $D$.

We denote by $\mathcal{M}_{\bm{d}}(G)$ the moduli stack of tuples
$(\bm{D},E,\varphi)$ with $\bm{D}\in X_{\bm{d}}$ and $(E,\varphi)$ a
multiplicative $G$-Higgs bundle with singularities in $\bm{D}$.

An invariant for multiplicative $G$-Higgs bundles can be defined in the same
way as in Section 3.2. We simply restrict $\varphi$ to
$\operatorname{Spec}F_{x}$ for any point $x\in\lvert D\rvert$ and obtain a
well defined orbit $G(\mathcal{O})z^{\lambda}G(\mathcal{O})$ for
$\lambda\in\mathrm{X}_{*}(T)_{+}$, and $T\subset G$ a maximal torus. We put
$\operatorname{inv}_{x}(\varphi)=\lambda$. Thus, we can also define, for any
tuple $\bm{\lambda}=(\lambda_{1},\dots,\lambda_{n})$ of dominant cocharacters
$\lambda_{i}\in\mathrm{X}^{*}(T)_{+}$, the finite-type moduli stack

$\mathcal{M}_{\bm{d},\bm{\lambda}}(G)=\left\\{(\bm{D},E,\varphi)\in\mathcal{M}_{\bm{d}}(G):\operatorname{inv}(\varphi)=\bm{\lambda}\cdot\bm{D}\right\\},$

and the bigger stack

$\mathcal{M}_{\bm{d},\bar{\bm{\lambda}}}(G)=\left\\{(\bm{D},E,\varphi)\in\mathcal{M}_{\bm{d}}(G):\operatorname{inv}(\varphi)\leq\bm{\lambda}\cdot\bm{D}\right\\}.$

It is clear that multiplicative $G$-Higgs bundles are a particular case of the
pairs introduced in Definition 3.2, for the group $G\times G$ endowed with the
involution $\Theta(g_{1},g_{2})=(g_{2},g_{1})$ (see Example 2.2 for more
details). This picture can also be related to the picture of Section 3.1 by
using _Vinberg’s enveloping monoid_ [vinberg]; this has been studied in the
works of Bouthier and J. Chi [bouthier_Springer, bouthier_fibration, bouthier-
chi, chi] and in the thesis of G. Wang [wang].

For any involution $\theta\in\operatorname{Aut}_{2}(G)$, there is a natural
map

$\displaystyle\mathcal{M}_{\bm{d}}(G,\theta)$
$\displaystyle\longrightarrow\mathcal{M}_{\bm{d}}(G)$
$\displaystyle(\bm{D},E,\varphi)$
$\displaystyle\longmapsto(\bm{D},E_{G},\varphi_{G}),$

sending a multiplicative $(G,\theta)$-Higgs bundle $(E,\varphi)$ to the
multiplicative $G$-Higgs bundle $(E_{G},\varphi_{G})$, where

$E_{G}=E\times_{G^{\theta}}G$

is the natural extension of the structure group of $E$ from $G^{\theta}$ to
$G$ and, regarding $\varphi$ as a $G^{\theta}$-equivariant map
$f:E|_{X\setminus|D|}\rightarrow M^{\theta}$, the section $\varphi_{G}$ is
determined by the map

$\displaystyle f_{G}:E_{G}|_{X\setminus|D|}$ $\displaystyle\longrightarrow G$
$\displaystyle[e,g]$ $\displaystyle\longmapsto gf(e)g^{-1}.$

Here, we are identifying the symmetric variety $G/G^{\theta}$ with the
subvariety $M^{\theta}=\tau^{\theta}(G)$ so that for any $g\in G^{\theta}$,
the image $gf(e)g^{-1}$ stays in $M^{\theta}$. Moreover, recall that for any
involution $\theta\in\operatorname{Aut}_{2}(G)$ we have a natural inclusion
$\mathrm{X}_{*}(A_{G^{\theta}})\subset\mathrm{X}_{*}(T)$, for $A\subset G$ a
maximal $\theta$-split torus and $T$ a maximal torus containing it. Now, the
map $\mathcal{M}_{\bm{d}}(G,\theta)\rightarrow\mathcal{M}_{\bm{d}}(G)$
restricts to a map

$\displaystyle\mathcal{M}_{\bm{d},\bm{\lambda}}(G,\theta)$
$\displaystyle\longrightarrow\mathcal{M}_{\bm{d},w_{0}\bm{\lambda}}(G).$

Indeed, we have the following lemma.

######  Lemma 4.2.

For any multiplicative $(G,\theta)$-Higgs bundle $(E,\varphi)$ and any
singular point $x$ of it, we have
$\operatorname{inv}_{x}(\varphi_{G})=w_{0}\operatorname{inv}_{x}(\varphi)$.

###### Proof.

The decomposition
$(G/G^{\theta})(F)=\bigsqcup_{\lambda\in\mathrm{X}_{*}(A_{G_{\theta}})_{-}}G(\mathcal{O})z^{\lambda}$
can be reformulated in terms of $M^{\theta}$ as

$M^{\theta}(F)=\bigsqcup_{\lambda\in\mathrm{X}_{*}(A_{G_{\theta}})_{-}}G(\mathcal{O})*_{\theta}z^{\lambda}.$

Now, the inclusion $M^{\theta}\subset G$ induces an inclusion
$M^{\theta}(F)\subset G(F)$ and it is clear that each orbit
$G(\mathcal{O})*_{\theta}z^{\lambda}$ is mapped inside the orbit
$G(\mathcal{O})z^{\lambda}G(\mathcal{O})=G(\mathcal{O})z^{w_{0}\lambda}G(\mathcal{O})$.
∎

Note that a section $\varphi$ of the adjoint bundle $E(G)$ is simply an
automorphism $\varphi:E\rightarrow E$. A _morphism_ of multiplicative
$G$-Higgs bundles $(E_{1},\varphi_{1})\rightarrow(E_{2},\varphi_{2})$ is then
a map $E_{1}\rightarrow E_{2}$ such that the following square commutes

${E_{1}|_{X\setminus|D|}}$${E_{2}|_{X\setminus|D|}}$${E_{1}|_{X\setminus|D|}}$${E_{2}|_{X\setminus|D|}.}$$\scriptstyle{\varphi_{1}}$$\scriptstyle{\varphi_{2}}$

We say that a multiplicative $G$-Higgs bundle $(E,\varphi)$ is _simple_ if its
only automorphisms are those given by multiplying by elements of the centre
$Z_{G}\subset G$.

A _moduli space_ for simple multiplicative $G$-Higgs bundles can be considered
after an argument by Hurtubise and Markman [hurtubise-markman] that we explain
now. Let $\rho:G\hookrightarrow\operatorname{GL}(V)$ be a faithful
representation of $G$ and, for $T\subset G$ a maximal torus, let

$\mathrm{P}(V)=\left\\{\chi\in\mathrm{X}^{*}(T):\exists v\in
V\setminus\left\\{0\right\\}\text{ such that }\rho(t)v=t^{\chi}v,\forall t\in
T\right\\}$

be its weight lattice. For any cocharacter $\lambda\in\mathrm{X}_{*}(T)$ we
define the number

$d_{\rho}(\lambda)=\min\left\\{\langle w\lambda,\chi\rangle:w\in
W_{T},\chi\in\mathrm{P}(V)\right\\},$

for $W_{T}$ the Weyl group of $T$. Now, if $\lambda\in\mathrm{X}_{*}(T)_{+}$,
for any $\phi\in G(F)$ that lies in the orbit
$G(\mathcal{O})z^{\lambda}G(\mathcal{O})$, the highest pole order in the
coefficients of the matrix $\rho(\phi)\in\operatorname{End}V\otimes F$ is
$d_{\rho}(\lambda)$. Thus, if we write $\bm{\lambda}\cdot\bm{D}=\sum_{x\in
X}\lambda_{x}x$, for $\bm{\lambda}=(\lambda_{1},\dots,\lambda_{n})$ a tuple of
dominant cocharacters and $\bm{D}\in X_{\bm{d}}$, we can consider the divisor

$(\bm{\lambda}\cdot\bm{D})_{\rho}=-\sum_{x\in X}d(\lambda_{x})x,$

and for any multiplicative $G$-Higgs bundle $(E,\varphi)$ with singularity
type $(\bm{D},\bm{\lambda})$, the representation $\rho$ induces a section

$\rho(\varphi)\in
H^{0}(X\setminus|D|,\operatorname{End}E(V)\otimes\mathscr{O}_{X}((\bm{\lambda}\cdot\bm{D})_{\rho})).$

Indeed, if $\bm{\lambda}\geq\bm{\mu}$, then $d_{\rho}(\lambda)\leq
d_{\rho}(\mu)$, so
$(\bm{\lambda}\cdot\bm{D})_{\rho}\geq(\bm{\mu}\cdot\bm{D})_{\rho}$.

A pair $(E,\varphi)$, where $E$ is a $G$-bundle on $X$ and $\varphi$ is a
section of $\operatorname{End}E(V)\otimes L$ for some representation
$\rho:G\rightarrow\operatorname{GL}(V)$ and for $L$ a line bundle, is called
an _$L$ -twisted $\rho$-Higgs bundle_. We denote the moduli stack of these
pairs by $\operatorname{Higgs}_{L,\rho}(G)$. What we have just proved is that
$\rho$ induces a well-defined inclusion

$\displaystyle\mathcal{M}_{\bm{d},\bar{\bm{\lambda}}}(G)_{\bm{D}}$
$\displaystyle\longrightarrow\operatorname{Higgs}_{\mathscr{O}_{X}((\bm{\lambda}\cdot\bm{D})_{\rho}),\rho}(G).$

The stability conditions and good moduli space theory for $L$-twisted
$\rho$-Higgs bundles are well known (the reader may refer to [oscar-gothen-
mundet] or [schmitt] for general treatments of the topic). The existence of
moduli spaces of simple $L$-twisted $\rho$-Higgs bundles guarantees the
existence of good moduli spaces of simple pairs
$\mathbf{M}_{\bm{d},\bm{\lambda}}(G)$ and
$\mathbf{M}_{\bm{d},\bar{\bm{\lambda}}}(G)$ for
$\mathcal{M}_{\bm{d},\bm{\lambda}}(G)$ and
$\mathcal{M}_{\bm{d},\bar{\bm{\lambda}}}(G)$, at least as algebraic spaces.
For more details, see Remark 2.5 in [hurtubise-markman]. Finally, for any
involution $\theta\in\operatorname{Aut}_{2}(G)$ we can consider the space
$\mathbf{M}_{\bm{d},\bm{\lambda}}(G,\theta)$ defined as the intersection of
$\mathbf{M}_{\bm{d},w_{0}\bm{\lambda}}(G)$ with the image of the map
$\mathcal{M}_{\bm{d},\bm{\lambda}}(G,\theta)\rightarrow\mathcal{M}_{\bm{d},w_{0}\bm{\lambda}}(G)$.
Analogously, we can define $\mathbf{M}_{\bm{d},\bar{\bm{\lambda}}}(G,\theta)$.

For most discussions in this section, we will be interested in fixing the
divisor $\bm{D}$, and thus we denote the fibre
$\mathbf{M}_{\bm{D},\lambda}(G)=\mathbf{M}_{\bm{d},\lambda}(G,\theta)_{\bm{D}}$,
and similarly for $\mathbf{M}_{\bm{D},\lambda}(G,\theta)$.

###  4.2. Involutions on the moduli space

Let $\theta\in\operatorname{Aut}_{2}(G)$ be an involution. To any
multiplicative $G$-Higgs bundle $(E,\varphi)$ over $X$ we can associate other
two multiplicative $G$-Higgs bundles obtained from $(E,\varphi)$ and the
action of the involution $\theta$. These are the pairs

$\iota^{\theta}_{\pm}(E,\varphi)=(\theta(E),\theta(\varphi)^{\pm 1}).$

Here, $\theta(E)$ stands for the associated principal bundle
$E\times_{\theta}G$ or, equivalently, it has the same total space as $E$ but
it is equipped with the $G$-action $e\cdot_{\theta}g=e\cdot\theta(g)$. The
section $\theta(\varphi)$ is obtained as the composition of $\varphi$ with the
natural map

$\displaystyle E(G)$ $\displaystyle\longrightarrow\theta(E)(G)$
$\displaystyle[e,g]$ $\displaystyle\longmapsto[e,\theta(g)].$

Equivalently, if we regard $\varphi$ as an automorphism
$\varphi:E|_{X^{\prime}}\rightarrow E|_{X^{\prime}}$, then $\theta(\varphi)$
is set theoretically the same map as $\varphi$, but now regarded as an
automorphism of $\theta(E)|_{X^{\prime}}$. We can give one last interpretation
of $\theta(\varphi)$ by noting that the associated $G$-equivariant maps
$f_{\varphi},f_{\theta(\varphi)}:E|_{X^{\prime}}\rightarrow G$ are related by

$f_{\theta(\varphi)}=\theta\circ f_{\varphi}.$

Let $T\subset G$ be a $\theta$-stable maximal torus and $B\subset G$ a Borel
subgroup contained in a minimal $\theta$-split parabolic and containing $T$.
Given any dominant cocharacter $\lambda\in\mathrm{X}_{*}(T)_{+}$, the
associated cocharacter $\theta(\lambda)$ is not dominant in general, but there
is a dominant cocharacter in its orbit under the action of the Weyl group
$W_{T}$. We denote this element by $\theta(\lambda)_{+}$. The map
$\lambda\mapsto\theta(\lambda)_{+}$ defines an involution on
$\mathrm{X}_{*}(T)_{+}$. One checks immediately that, if $x$ is a singular
point of $(E,\varphi)$, then

$\operatorname{inv}_{x}(\theta(E),\theta(\varphi)^{\pm})=(\pm\theta(\operatorname{inv}_{x}(E,\varphi)))_{+}.$

Therefore, if $\bm{\lambda}=(\lambda_{1},\dots,\lambda_{n})$ is a tuple of
dominant coweights $\lambda_{i}\in\mathrm{X}_{*}(T)$ with
$\theta(\lambda_{i})_{+}=\pm\lambda_{i}$, then for any $\bm{D}\in X_{\bm{d}}$,
we have the following involutions on the moduli space of simple pairs

$\displaystyle\iota^{\theta}_{\pm}:\mathbf{M}_{\bm{D},\bm{\lambda}}(G)$
$\displaystyle\longrightarrow\mathbf{M}_{\bm{D},\bm{\lambda}}(G)$
$\displaystyle[E,\varphi]$
$\displaystyle\longmapsto[\theta(E),\theta(\varphi)^{\pm 1}],$

where $\iota^{\theta}_{+}$ is well defined if
$\theta(\lambda_{i})_{+}=\lambda_{i}$ and $\iota^{\theta}_{-}$ if
$(-\theta(\lambda_{i}))_{+}=\lambda_{i}$. We will denote these two involutions
together by writing $\iota_{\varepsilon}^{\theta}$, for $\varepsilon=\pm 1$.

Moreover, note that the actions of the inner automorphisms on the moduli space
is trivial since, if $\alpha=\operatorname{Int}_{g}$ is an inner automorphism
of $G$, then the map $e\mapsto e\cdot g^{-1}$ gives an isomorphism between any
principal $G$-bundle $E$ and $E\times_{\alpha}G$, commuting with any
$\varphi$. Therefore, the above involutions are well defined at the level of
the outer class $a$ of $\theta$ in $\operatorname{Out}_{2}(G)$. That is, given
any element $a\in\operatorname{Out}_{2}(G)$, we can define

$\displaystyle\iota^{a}_{\varepsilon}:\mathbf{M}_{\bm{D},\bm{\lambda}}(G)$
$\displaystyle\longrightarrow\mathbf{M}_{\bm{D},\bm{\lambda}}(G)$
$\displaystyle[E,\varphi]$
$\displaystyle\longmapsto[\theta(E),\theta(\varphi)^{\varepsilon}],$

for any representative $\theta$ of the class $a$.

######  Remark 4.3.

Note that it follows from the Cartan decomposition of the formal loop group
$G(F)$ that if $\theta$ and $\theta^{\prime}$ are two involutions representing
the same element in $\operatorname{Out}_{2}(G)$, then
$\theta(\lambda)_{+}=\theta^{\prime}(\lambda)_{+}$.

We are interested in studying the spaces
$\mathbf{M}_{\bm{D},\bm{\lambda}}(G)^{\iota^{a}_{\varepsilon}}$ of fixed
points under these involutions. For this it will be useful for us to think
about an isomorphism between the $G$-bundles $E$ and $\theta(E)$, for
$\theta\in\operatorname{Aut}_{2}(G)$, as a _$\theta$ -twisted automorphism_.
By this we mean an automorphism $\psi:E\rightarrow E$ of the total space $E$,
such that

$\psi(e\cdot g)=\psi(e)\cdot\theta(g).$

Therefore, a multiplicative $G$-Higgs bundle $(E,\varphi)$ will be a fixed
point of the involution $\iota^{a}_{\varepsilon}$ if and only if there exists
a $\theta$-twisted automorphism $\psi:E\rightarrow E$, for any element
$\theta$ of the class $a\in\operatorname{Out}_{2}(G)$, such that the following
diagram commutes

${E|_{X^{\prime}}}$${E|_{X^{\prime}}}$${E|_{X^{\prime}}}$${E|_{X^{\prime}}.}$$\scriptstyle{\psi}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi^{\varepsilon}}$$\scriptstyle{\psi}$

Equivalently, if we define $f_{\psi}:E\rightarrow G$ as $\psi(e)=e\cdot
f_{\psi}(e)$, then
$(E,\varphi)\in\mathbf{M}_{\bm{D},\bm{\lambda}}(G)^{\iota^{a}_{\varepsilon}}$
if and only if, for any $e\in E$, we have

$f_{\psi}(e)\theta(f_{\varphi}(e))f_{\psi}(e)^{-1}=f_{\varphi}(e)^{\varepsilon}.$

###  4.3. Fixed dominant cocharacters

Before describing the fixed points of the involution $\iota^{\theta}_{-}$, it
is important that we give a good description of the involution at the level of
dominant cocharacters

$\displaystyle\mathrm{X}_{*}(T)_{+}$
$\displaystyle\longrightarrow\mathrm{X}_{*}(T)_{+}$ $\displaystyle\lambda$
$\displaystyle\longmapsto(-\theta(\lambda))_{+},$

for $T$ a $\theta$-stable maximal torus.

As we mentioned above, in general $-\theta(\lambda)$ is not a dominant
cocharacter, and that is why we need to take $(-\theta(\lambda))_{+}$ the
dominant cocharacter in its Weyl group orbit. However, there is an exception
to this, which is when there are no imaginary roots.

Indeed, the dominant Weyl chamber in
$\mathrm{X}_{*}(T)_{+}\otimes_{\mathbb{Z}}\mathbb{Q}$ is spanned by a basis of
fundamental coweights $\left\\{\mu_{1},\dots,\mu_{r}\right\\}$, which is the
dual basis of the set of simple roots
$\Delta_{T}=\left\\{\alpha_{1},\dots,\alpha_{r}\right\\}$. Since $\theta$ is
an involution, we have
$\langle\chi^{\theta},\theta(\lambda)\rangle=\langle\chi,\lambda\rangle$ for
any $\chi\in\mathrm{X}^{*}(T)$ and any $\lambda\in\mathrm{X}_{*}(T)$.
Therefore, $\left\\{\theta(\mu_{1}),\dots,\theta(\mu_{r})\right\\}$ is the
dual basis of
$\left\\{\alpha^{\theta}_{1},\dots,\alpha_{r}^{\theta}\right\\}$. Recall now
that there is an involution $\sigma$ on the set of simple roots which are not
imaginary such that $\alpha^{\theta}_{i}+\alpha_{\sigma(i)}$ is an imaginary
root. Thus, if there are no imaginary roots, we conclude that
$\alpha_{i}^{\theta}=-\alpha_{\sigma(i)}$ and
$\theta(\mu_{i})=-\mu_{\sigma(i)}$. Summing up, we have the following.

######  Lemma 4.4.

If $\Phi_{T}^{\theta}=\varnothing$, then for any dominant cocharacter
$\lambda\in\mathrm{X}_{*}(T)_{+}$, the cocharacter $-\theta(\lambda)$ is also
dominant, and thus $(-\theta(\lambda))_{+}=-\theta(\lambda)$.

Recall from Lemma 2.9 that for any involution
$\theta\in\operatorname{Aut}_{2}(G)$ there exists an inner automorphism
$u_{\theta}\in\operatorname{Int}(G)$ such that the involution
$\theta_{q}:=u_{\theta}\circ\theta$ is quasisplit. Therefore, if we let
$T_{q}$ be a maximal $\theta_{q}$-stable torus, the involution
$\lambda\mapsto(-\theta_{q}(\lambda))_{+}$ is just the involution
$\lambda\mapsto-\theta_{q}(\lambda)$. The fixed points of this involution are
simply the dominant cocharacters $\mathrm{X}_{*}(A_{q})_{+}$, where $A_{q}$ is
the maximal $\theta$-split torus formed by the anti-fixed points of
$\theta_{q}$ in $T_{q}$.

The fixed points of the involution $\lambda\mapsto-\theta(\lambda)$ in
$\mathrm{X}_{*}(T)$ are the cocharacters $\mathrm{X}_{*}(A)$ of the maximal
$\theta$-split torus $A$ of anti-fixed points of $\theta$ in $T$. Any two
maximal tori are conjugate and thus there is a canonical bijection
$\mathrm{X}_{*}(T)_{+}\rightarrow\mathrm{X}_{*}(T_{q})_{+}$. It follows from
the above that under this bijection the dominant cocharacters
$\mathrm{X}_{*}(A)_{+}$ are mapped inside $\mathrm{X}_{*}(A_{q})_{+}$, but in
general not every element of $\mathrm{X}_{*}(A_{q})_{+}$ comes from an element
of $\mathrm{X}_{*}(A)_{+}$. In fact, the image of $\mathrm{X}_{*}(A)_{+}$
under this map is the set

$\mathrm{X}_{*}(A_{q})_{+}^{u_{\theta}}=\left\\{\lambda\in\mathrm{X}_{*}(A_{q})_{+}:u_{\theta}(\lambda)=\lambda\right\\}.$

Given an element $a\in\operatorname{Out}_{2}(G)$, a quasisplit representative
$\theta_{q}$ of $a$ and a dominant cocharacter
$\lambda\in\mathrm{X}_{*}(A_{q})_{+}$, we can consider the set

$a_{\lambda}=\left\\{\theta\text{ representative of
}a:u_{\theta}(\lambda)=\lambda\right\\}.$

That is, the elements of $a_{\lambda}$ are the involutions $\theta$ for which
$\lambda$ is in the image of the cocharacters of a maximal $\theta$-split
torus. This set clearly descends to the clique $\operatorname{cl}^{-1}(a)$ and
we can define

$\operatorname{cl}^{-1}(a)_{\lambda}=\left\\{[\theta]\in\operatorname{cl}^{-1}(a):u_{\theta}(\lambda)=\lambda\right\\}.$

###  4.4. The fixed points

We move on now to describe the fixed points of the involutions