Character stacks are PORC count

TL;DR

This paper proves that the number of points over finite fields for character stacks related to surface groups and reductive groups is a PORC function, using advanced representation theory and algebraic geometry tools.

Contribution

It establishes that the point count of character stacks is a PORC function and derives an explicit expression for their E-polynomial.

Findings

Number of points over finite fields is a PORC function.

Derived an explicit formula for the E-polynomial of the character stack.

Connected the point count with Lusztig's Jordan decomposition and genus numbers.

Abstract

We compute the number of points over finite fields of the character stack associated to a compact surface group and a reductive group with connected centre. We find that the answer is a Polynomial On Residue Classes (PORC). The key ingredients in the proof are Lusztig's Jordan decomposition of complex characters and Deriziotis's results on genus numbers of finite reductive groups. As a consequence of our main theorem, we obtain an expression for the $E$-polynomial of the character stack.