Crossed S-matrices and character sheaves on unipotent groups

Let k be the algebraic closure of a finite field Fq of characteristic p. Let G be a connected unipotent group over k equipped with an Fq-structure given by a Frobenius map F:G⟶G. We will denote the corresponding algebraic group defined over Fq by G0. Character sheaves on G are certain special objects in the triangulated braided monoidal category DG(G) of bounded conjugation equivariant Q‾l-complexes (where l≠p is a prime number) on G. Boyarchenko has proved that the “trace of Frobenius” functions associated with F-stable character sheaves on G form an orthonormal basis of the space of class functions on G0(Fq) and that the matrix relating this basis to the basis formed by the irreducible characters of G0(Fq) is block diagonal with “small” blocks. In particular, there is a partition of the set of character sheaves as well as a partition of the set of irreducible characters of G0(Fq) into “small” families known as L-packets. In this paper we describe these block matrices relating character sheaves and irreducible characters corresponding to each L-packet. We prove that these matrices can be described as certain “crossed S-matrices” associated with each L-packet. We will also derive a formula for the dimensions of the irreducible representations of G0(Fq) in terms of certain modular categorical data associated with the corresponding L-packet. In fact we will formulate and prove more general results which hold for possibly disconnected groups G such that G∘ is unipotent. To prove our results, we will establish a formula (which holds for any algebraic group G) which expresses the inner product of the “trace of Frobenius” function of any F-stable object of DG(G) with any character of G0(Fq) (or of any of its pure inner forms) in terms of certain categorical operations.