Bitraces for Gl,(f,) and the Iwahori-hecke Algebra of Type A

Let H,,(q) be the Iwahori-Hecke algebra of the symmetric group S,,. Let F, be a finite field with y elements and let B be the Bore1 subgroup of upper triangular matrices in the general linear group G = GL,(F,). Let 1: denote the trivial representation of B induced to G. Then H,,(q) has a natural action on 1; that commutes with the G-action, and we define the bitrace btr(g, a) to be the trace of g E G and a E H,(q) acting simultaneously in lg. For partitions, p, v of n, let Tw be a standard basis element of H,,(q) corresponding to the S,-conjugacy class p, and let u, be a unipotent element of G with Jordan block structure V. We give a combinatorial formula for btr(u,, Tp) as a weighted sum of column strict tableaux of shape v and content p. This bitrace also essentially counts Ifs-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,(q), and gives a natural pairing between the conjugacy classes of S, and the unipotent classes in G. 0. INTRODUCTION Motivation for this paper In the original work of Frobenius, where he determined the irreducible characters of the symmetric groups, one of the key features was a formula ‘Research supported in part by National Science Foundation grant DMS-9622985, a Postdoctoral Fellowship at Mathematical Sciences Research Institute and an Australian Research Council Fellowship.