Cohomology of non-generic character stacks

We study (compactly supported) cohomology of character stacks of punctured Riemann surface with prescribed semisimple local monodromies at punctures. In the case of generic local monodromies, the cohomology of these character stacks has been studied in [23, 39]. In this paper we extend the results and conjectures of [23] to the non-generic case. In particular we compute the E-series and give a conjectural formula for the mixed Poincaré series. We prove our conjecture in the case of the projective line with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>4</mml:mn> </mml:math> punctures.