Character sheaves and characters of unipotent groups over finite fields

TL;DR

This paper establishes a correspondence between Frobenius-invariant character sheaves on unipotent groups over finite fields and irreducible characters of their finite group points, forming a basis for conjugation-invariant functions.

Contribution

It proves that Frobenius-invariant character sheaves on unipotent groups correspond to pure perverse sheaves on the base field, and these form an orthonormal basis related to irreducible characters, extending to disconnected groups.

Findings

Frobenius-invariant character sheaves correspond to pure perverse sheaves of weight 0.

The functions from these sheaves form an orthonormal basis of conjugation-invariant functions.

The basis relates to irreducible characters via a block-diagonal matrix aligned with L-packets.

Abstract

Let G_0 be a connected unipotent algebraic group over a finite field F_q, and let G be the unipotent group over an algebraic closure F of F_q obtained from G_0 by extension of scalars. If M is a Frobenius-invariant character sheaf on G, we show that M comes from an irreducible perverse sheaf M_0 on G_0, which is pure of weight 0. As M ranges over all Frobenius-invariant character sheaves on G, the functions defined by the corresponding perverse sheaves M_0 form a basis of the space of conjugation-invariant functions on the finite group G_0(F_q), which is orthonormal with respect to the standard unnormalized Hermitian inner product. The matrix relating this basis to the basis formed by irreducible characters of G_0(F_q) is block-diagonal, with blocks corresponding to the L-packets (of characters, or, equivalently, of character sheaves). We also formulate and prove a suitable generalization of this result to the case where G_0 is a possibly disconnected unipotent group over F_q. (In general, Frobenius-invariant character sheaves on G are related to the irreducible characters of the groups of F_q-points of all pure inner forms of G_0.)