Symmetric spaces for groups over involutive algebras and applications to Higgs bundles

Abstract

We study symplectic groups and indefinite orthogonal groups over involutive, possibly noncommutative, algebras $(A, σ)$. In the case when the algebra $(A, σ)$ is Hermitian, or the complexification $(A_{\mathbb C}, σ_{\mathbb C})$ of a Hermitian involutive algebra, one can identify maximal compact subgroups of such groups, and consider their associated Riemannian symmetric spaces. This new perspective allows for the realization of various geometric models for the symmetric space. We describe explicitly the complexified tangent space for each of the models, as well as the diffeomorphisms between them and their differentials. In the second part of the article, we give a number of applications of this theory. The geometric realizations of the Riemannian symmetric spaces described in the first part provide new geometric interpretations of Higgs bundle data that can be used for the study of fundamental group representations into symplectic or into indefinite orthogonal groups over Hermitian involutive algebras. We give an exact component count for the moduli spaces of $\rm{Sp}_2(A_{\mathbb C}, σ_{\mathbb C})$-Higgs bundles and of $\rm O(A_{\mathbb C}, σ_{\mathbb C})$-Higgs bundles, using the topology of the corresponding maximal compact subgroups rather than Morse-Bott theory techniques. Furthermore, we use the noncommutative symmetric-space models to construct a factorization of the Hitchin morphism for $\rm{Sp}_2(A_{\mathbb C},σ_{\mathbb C})$-Higgs bundles, together with analogous factorizations for the real groups $\rm{Sp}_2(A,σ)$ and $\rm O_{(1,1)}(A,σ)$. These factorizations are induced by quadratic norm maps from the corresponding tangent models to Jordan-algebraic targets and pass through intermediate affine GIT quotients. As a consequence, they reduce the algebraic complexity required in order to characterize the Hitchin base explicitly.