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.

Generating series for the E‐polynomials of GL(n,C)$GL(n,{\mathbb {C}})$‐character varieties

“…For example, by [7] the irreducible character variety $$\mathcal {X}_{\Gamma }^{irr}GL_{n}$$ coincides precisely with the smooth locus of the full character variety $$\mathcal {X}_{\Gamma }GL_{n}$$, in the case of the free group $$\Gamma =F_{r}$$. Recently, this theme has been greatly expanded in [12].…”

“…For example, by [7] the irreducible character variety  𝑖𝑟𝑟 Γ 𝐺𝐿 𝑛 coincides precisely with the smooth locus of the full character variety  Γ 𝐺𝐿 𝑛 , in the case of the free group Γ = 𝐹 𝑟 . Recently, this theme has been greatly expanded in [12].…”