Cohomology of non-generic character stacks

TL;DR

This paper extends the understanding of the cohomology of character stacks of punctured Riemann surfaces to non-generic local monodromies, providing new formulas and conjectures verified in specific cases.

Contribution

It generalizes previous results on generic monodromies to non-generic cases, including computing E-series and proposing a conjectural formula for the mixed Poincaré series.

Findings

Computed E-series for non-generic character stacks

Proposed a conjectural formula for the mixed Poincaré series

Verified the conjecture for a specific case of four punctures on the projective line

Abstract

We study (compactly supported) cohomology of character stacks of punctured Riemann surfaces with prescribed semisimple local monodromies at punctures. In the case of generic local monodromies, the cohomology of these character stacks has already been studied by Hausel, Letellier and Rodriguez-Villegas and by Mellit. In this paper, we extend the results of Hausel, Letellier and Rodriguez-Villegas to the non-generic case. In particular, we compute the E-series and we give a conjectural formula for the mixed Poincar\'e series. Moreover, we verify our conjecture in the case of the projective line with 4 punctures and a certain choice of a non-generic quadruple.