📝 SummarXiv

Abstract

We study symplectic groups and indefinite orthogonal groups over involutive, possibly noncommutative, algebras $(A, \sigma)$. In the case when the algebra $(A, \sigma)$ is Hermitian, or the complexification $(A_{\mathbb{C}}, \sigma_{\mathbb{C}})$ of a Hermitian involutive algebra, one can identify maximal compact subgroups of such groups, and consider their associated Riemannian symmetric space. 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 an application of this theory, we introduce alternative notions of polystable Higgs bundles that can be used for the study of fundamental group representations into symplectic or into indefinite orthogonal groups over Hermitian involutive algebras. We characterize solutions to the relevant harmonicity equations with respect to these new models of the symmetric space. Reductive fundamental group representations then correspond to holomorphic pairs written with regard to these geometric incarnations of the associated symmetric space, thus allowing new insights for their study.