Bitraces for GL n( [formula omitted]q) and the Iwahori-Hecke algebra of type An − 1

Abstract

Let H n ( q) be the Iwahori-Hecke algebra of the symmetric group S n . Let F q be a finite field with q elements and let B be the Borel subgroup of upper triangular matrices in the general linear group G = GL n( F q). Let 1 B g denote the trivial representation of B induced to G. Then H n ( q) has a natural action on 1 B G that commutes with the G-action, and we define the bitrace btr( g, a) to be the trace of g ϵ G and a ϵ H n ( q) acting simultaneously in 1 B G . For partitions, μ, ν of n, let T μ be a standard basis element of H n ( q) corresponding to the S n -conjugacy class μ, and let u ν be a unipotent element of G with Jordan block structure ν. We give a combinatorial formula for btr( u ν , T μ ) as a weighted sum of column strict tableaux of shape ν and content μ. This bitrace also essentially counts F q-rational points in the intersection of a conjugacy class with a Schubert cell, provides a new proof of the Frobenius formula for characters of H n ( q), and gives a natural pairing between the conjugacy classes of S n and the unipotent classes in G.