E-polynomials of $PGL(2,\mathbb{C})$-character varieties of surface groups

TL;DR

This paper calculates the E-polynomials of $PGL(2, abla{C})$-character varieties for surface groups with one puncture, comparing them to the $SL(2, abla{C})$ case, using stratification and fibration analysis.

Contribution

It provides explicit E-polynomial computations for $PGL(2, abla{C})$-character varieties and explores their relation to the Langlands dual case, expanding understanding of these moduli spaces.

Findings

Computed E-polynomials for $PGL(2, abla{C})$-character varieties.

Compared E-polynomials with the $SL(2, abla{C})$ case.

Analyzed the behavior of E-polynomials under fibrations.

Abstract

In this paper, we compute the E-polynomials of the $PGL(2,\mathbb{C})$-character varieties associated to surfaces of genus $g$ with one puncture, for any holonomy around it, and compare it with its Langlands dual case, $SL(2,\mathbb{C})$. The study is based on the stratification of the space of representations and on the analysis of the behaviour of the E-polynomial under fibrations.