Bad Representations and Homotopy of Character Varieties

Clément Guérin; Sean Lawton; Daniel Ramras

doi:10.5802/ahl.119

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arXiv · 1908.02915

Bad Representations and Homotopy of Character Varieties

Clément Guérin·Sean Lawton·Daniel Ramras

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Abstract

Let G be a connected reductive complex affine algebraic group, and let X denote the moduli space of G-valued representations of a rank r free group. We first characterize the singularities in X, 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, 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.

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BibTeX RIS

Clément Guérin, Sean Lawton, Daniel Ramras. 2019-08-08. Bad Representations and Homotopy of Character Varieties. https://doi.org/10.5802/ahl.119

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