The most singular loci of twisted character varieties

Abstract

For any simply connected almost simple group $G$ and any compact Riemann surface of genus $g>1$, we study the singular locus of the character variety which parametrises fundamental group representations with a twist by a central element $z$ of $G$. While for $z=1$ the most singular representations have values in a maximal torus (and are invariant for the Weyl group action) and for generic $z$ every $z$-twisted representation is irreducible, for a general $z$ we find that the most singular representations have values in a particular Levi subgroup $L_z$ up to conjugation. From the explicit Lie-theoretic description of $L_z$, we determine the structure of the maximal stabiliser groups, and prove that the most singular loci is a disjoint union of generic type $A$ character varieties.