Regular elements in bases of Hecke algebras

Abstract

Let G˜=GL(n,F¯q) where q is a power of a prime. Let F be the standard Frobenius map of G˜. Let B˜ denote the Borel subgroup of G˜ of upper triangular matrices in G˜ and let T˜ denote the maximal split torus of G˜ contained in B˜. Let W denote the Weyl group of the B˜,N˜ pair where N˜=NG˜(T˜). Let w∈W and let w˙ denote an element in the pre-image of the natural surjection map from NG˜(T˜) to W. The double coset B˜w˙B˜ contains only regular elements if and only if w is a Coxeter element of minimal length. In the following we consider an element w∈W that is not of minimal length and thus B˜wB˜ contains nonregular elements. We explicitly describe certain cosets that are subsets of B˜wB˜ and contain only regular elements. These cosets are chosen for their importance in the construction of a basis of a Hecke algebra associated with G=G˜F.