Crossed $S$-matrices and Character Sheaves on Unipotent Groups

Tanmay Deshpande

Published 2014-10-13Version 1

Let $\mathtt{k}$ be an algebraic closure of a finite field $\mathbb{F}_{q}$ of characteristic $p$. Let $G$ be an algebraic group over $\mathtt{k}$ equipped with an $\mathbb{F}_q$-structure given by a Frobenius map $F:G\to G$. We will denote the corresponding algebraic group defined over $\mathbb{F}_q$ by $G_0$. Character sheaves on $G$ are supposed to be certain objects in the triangulated braided monoidal category $\mathscr{D}_G(G)$ of bounded conjugation equivariant $\bar{\mathbb{Q}}_l$-complexes (where $l\neq p$ is a prime number) on $G$. If $C\in \mathscr{D}_G(G)$ is any object equipped with an isomorphism $\psi:F^*(C){\cong} C$ and $g\in G$ then using Grothendieck's sheaf-function correspondence we can define the "trace of Frobenius" class function $t^g_{C,\psi}:G^g_0(\mathbb{F}_q)\to \bar{\mathbb{Q}}_l$ on each pure inner form $G^g_0$ of $G_0$ corresponding to the modified Frobenius, $ad(g)\circ F:G\to G$. Boyarchenko has proved that if the neutral connected component $G^\circ$ is unipotent, then the functions associated with $F$-stable character sheaves on $G$ form an orthonormal basis of the space of class functions on all pure inner forms $G^g_0(\mathbb{F}_q)$ and that the matrix relating this basis to the basis formed by the irreducible characters of the pure inner forms $G^g_0(\mathbb{F}_q)$ is block diagonal with "small" blocks. In this paper we describe these block matrices and interpret them as certain "crossed $S$-matrices".