**Prelectura de tesis **

**Automorphisms of Higgs bundle Moduli spaces for real groups.**

**Manuel Jesus Perez Garca.**

**Advisor: Oscar Garca Prada.**

**30 de Mayo, 16h., Sala 520, Módulo 17, Dpto. de Matemáticas**

**Abstract: Let G be a connected real form of a complex semisimple Lie**

**group GC with Lie algebra g. Let H be a maximal compact subgroup of**

**G and let be a Cartan involution of g such that it induces a decompo-**

**sition into 1-eigenspaces g = h m, where h is the Lie algebra of H. A**

**(G; )-Higgs bundle over a compact Riemann surface X is a pair consisting**

**on a holomorphic principal HC-bundle E and a holomorphic section ' of**

**E(mC) K where E(mC) is the bundle associated to E via the isotropy**

**representation C : HC ! GL(mC) and K is the canonical bundle over X.**

**Consider the moduli space M(G; ) of isomorphism classes of polystable**

**(G; )-Higgs bundles over X. In this talk we study the action of nite or-**

**der automorphisms of M(G) dened by combining the multiplication of**

**the Higgs eld by an nth-root of unity and the action of an element in**

**(H1(X;Z(HC) Ker(C)) o Out(g; ))n, where Out(G; ) is the group of**

**outer automorphisms of G that commute with . In addition, we describe its**

**xed points subvarieties and, through non-abelian Hodge correspondence,**

**we translate these results to the moduli**

** **