Bad Representations and Homotopy of Character Varieties

Abstract

Let G be a connected reductive complex affine algebraic group, and let 𝔛 r denote the moduli space of G-valued representations of a rank r free group. We first characterize the singularities in 𝔛 r , extending a theorem of Richardson and proving a Mumford-type result about topological singularities; this resolves conjectures of Florentino–Lawton. In particular, we compute the codimension of the orbifold singular locus using facts about Borel–de Siebenthal subgroups. We then use the codimension bound to calculate higher homotopy groups of the smooth locus of 𝔛 r , proving conjectures of Florentino–Lawton–Ramras. Lastly, using the earlier analysis of Borel–de Siebenthal subgroups, we prove a conjecture of Sikora about centralizers of irreducible representations in Lie groups.