$\mathrm{PGL}_n(\mathbb{C})$-character stacks and Langlands duality over finite fields

TL;DR

This paper investigates the structure of $ ext{PGL}_n( ext{C})$-character stacks, proposing a conjectural formula for their mixed Poincaré polynomial, and connects it to Langlands duality over finite fields through interpolation of structure coefficients.

Contribution

It introduces a conjectural formula for the mixed Poincaré polynomial of $ ext{PGL}_n( ext{C})$-character stacks and proves its validity under Euler specialization, linking it to Langlands duality.

Findings

Conjectural formula for the mixed Poincaré polynomial proven under Euler specialization.

Interpolates structure coefficients of two based rings related to finite groups.

Reminds of a non-abelian Fourier transform.

Abstract

In this paper we study the mixed Poincar\'e polynomial of generic $\mathrm{PGL}_n(\mathbb{C})$-character stacks with coefficients in some local systems arising from the conjugacy classes of $\mathrm{PGL}_n(\mathbb{C})$ which have non-connected stabiliser. We give a conjectural formula that we prove to be true under the Euler specialisation. We then prove that this conjectured formula interpolates the structure coefficients of the two based rings$ \left(\mathcal{C}(\mathrm{PGL}_n(\mathbb{F}_q)),Loc(\mathrm{PGL}_n),*\right)$ and $\left(\mathcal{C}(\mathrm{SL}_n(\mathbb{F}_q)), CS(\mathrm{SL}_n),\cdot\right) $ where for a group $H$, $\mathcal{C}(H)$ denotes the space of complex valued class functions on $H$, $Loc(\mathrm{PGL}_n)$ denotes the basis of characteristic functions of intermediate extensions of equivariant local systems on conjugacy classes of $\mathrm{PGL}_n$ and $CS(\mathrm{SL}_n)$ the basis of characteristic functions of Lusztig's character-sheaves on $\mathrm{SL}_n$. Our result reminds us of a non-abelian Fourier transform.