Algebraic geometry/Differential geometry
Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars
Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 666-673.

For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups, are the only previously know cases where natural invariants do not fully distinguish connected components. In this note we announce the existence of new such exotic components in the moduli spaces for the groups SO(p,q) with 2<p<q. These groups lie outside formerly know classes of groups associated with exotic components.

Pour les groupes de Lie semisimples, les espaces de modules de fibrés de Higgs sur une surface de Riemann sont en correspondance avec les variétés de représentations du groupe fondamental de la surface. Pour de nombreux groupes, les invariants topologiques naturels distinguent les composantes connexes de l'espace des modules. Les représentations de Hitchin dans un groupe réel déployé et des représentations maximales dans un groupe hermitien fournissaient les seuls exemples connus jusqu'ici dans lesquels les invariants primitifs étaient insuffisants. Cette note a pour objet d'annoncer l'existence de nouvelles composantes exotiques pour les espaces de modules pour les groupes SO(p,q), pour 2<p<q.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.04.024
Aparicio-Arroyo, Marta 1; Bradlow, Steven 2; Collier, Brian 3; García-Prada, Oscar 4; Gothen, Peter B. 5; Oliveira, André 5

1 Raet — HR software and services, Avenida de Bruselas 7, 28108 Alcobendas, Madrid, Spain
2 Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
3 Department of Mathematics, University of Maryland, College Park, MD 20742, USA
4 Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Nicolás Cabrera, 13–15, 28049 Madrid, Spain
5 Centro de Matemática da Universidade do Porto, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n, 4169-007 Porto, Portugal
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Aparicio-Arroyo, Marta; Bradlow, Steven; Collier, Brian; García-Prada, Oscar; Gothen, Peter B.; Oliveira, André. Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars. Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 666-673. doi : 10.1016/j.crma.2018.04.024. http://www.numdam.org/articles/10.1016/j.crma.2018.04.024/

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