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

In this paper we study the mixed Poincaré polynomial of generic ${\mathrm{P}\mathrm{G}\mathrm{L}}_n(\mathbb{C})$-character stacks with coefficients in some local systems arising from the conjugacy classes of ${\mathrm{P}\mathrm{G}\mathrm{L}}_n(\mathbb{C})$ 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$(\mathcal{C}({\mathrm{P}\mathrm{G}\mathrm{L}}_n({\mathbb{F}}_q)),Loc({\mathrm{P}\mathrm{G}\mathrm{L}}_n),\ast )$ and $(\mathcal{C}({\mathrm{S}\mathrm{L}}_n({\mathbb{F}}_q)),CS({\mathrm{S}\mathrm{L}}_n),\cdot )$ where for a group $H$, $\mathcal{C}(H)$ denotes the space of complex valued class functions on $H$, $Loc({\mathrm{P}\mathrm{G}\mathrm{L}}_n)$ denotes the basis of characteristic functions of intermediate extensions of equivariant local systems on conjugacy classes of ${\mathrm{P}\mathrm{G}\mathrm{L}}_n$ and $CS({\mathrm{S}\mathrm{L}}_n)$ the basis of characteristic functions of Lusztig’s character-sheaves on ${\mathrm{S}\mathrm{L}}_n$. Our result reminds us of a non-abelian Fourier transform.