Multiplicative Higgs bundles and involutions (Q6592204)

The purpose of this paper is two-fold. On the one hand, the authors introduce a generalization of multiplicative Higgs bundles from groups to pairs \((G,\theta)\) where \(G\) is a reductive group and \(\theta\in\mbox{Aut}_2(G)\) is an involution of \(G\). On the other hand, for any such pair \((G,\theta)\), they consider involutions of the moduli space of simple multiplicative \(G\)-Higgs bundles and study their fixed points. \N\NThe aim of this paper is to generalize the theory of multiplicative \(G\)-Higgs bundles over a curve to pairs \((G,\theta)\). 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. The authors study how these objects appear as fixed points of involutions of the moduli space of multiplicative \(G\)-Higgs bundles, induced by the involution \(\theta\). \N\NThis paper is organized as follows: Section 1 is an introduction to the subject. Section 2 is devoted to involutions, symmetric varieties and symmetric embeddings and covers all the preliminary notions that needed about reductive groups with involutions, symmetric varieties and their embeddings, and their invariant theory. The authors begin by reviewing some general facts about reductive groups with involutions. They introduce \(\theta\)-split tori and \(\theta\)-split parabolic subgroups, and recall the notion of a quasisplit involution and the Iwasawa decomposition. They recall the construction of the restricted root system and its associated root and weight lattices and 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. They also recall the construction of the dual group of a symmetric variety, as introduced by \textit{Y. Sakellaridis} and \textit{A. Venkatesh} [Periods and harmonic analysis on spherical varieties. Paris: Société Mathématique de France (SMF) (2017; Zbl 1479.22016)] and state its most relevant properties. \N\NThe authors then review \textit{N. Guay}'s classification of a certain class of symmetric embeddings, which they call very flat and his construction of the enveloping embedding [Transform. Groups 6, No. 4, 333--352 (2001; Zbl 1006.14020)]. Guay's enveloping embedding is an affine variety associated to a semisimple simplyconnected group with involution that generalizes Vinberg's enveloping monoid. They also introduce the compactification of a symmetric variety, and explain that the Guay embedding can be obtained as the spectrum of its Cox ring. \N\NAfter this, the authors study the invariant theory of symmetric embeddings: first they recall \textit{R. W. Richardson}'s results on the invariant theory of a symmetric variety [Invent. Math. 66, 287--312 (1982; Zbl 0508.20021)], and extend them to very flat symmetric embeddings. They finish the section by describing the loop parametrization of a symmetric variety, and extend it to the enveloping embedding, thus giving a generalization of \textit{J. Chi} [J. Inst. Math. Jussieu 21, No. 1, 1--65 (2022; Zbl 1537.22022)]. \N\NSection 3 deals with multiplicative Higgs bundles for symmetric varieties and introduces the main objects from several perspectives that the authors compare. In the first part of the section they emulate the construction of the multiplicative Hitchin fibration associated to a reductive monoid 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 the authors introduce the notion of a multiplicative Higgs bundle associated to a reductive group with involution. They finish the section by relating the objects defined in the second part with semisimple group with the objects defined in the first part. \N\NSection 4 deals with multiplicative Higgs bundles, involutions and fixed points. The authors study a certain type of involutions in the moduli space of simple multiplicative Higgs bundles, how the objects introduced in the previous section appear as their fixed points, and the relation between these involutions and the Hurtubise-Markman symplectic form. The paper is supported by an appendix to cover the basic facts the authors used about root systems and to establish some notations.