A q-Cauchy identity for Schur functions and imprimitive complex reflection groups

Abstract

in an indeterminate . Note that, at = 0, this reduces to the Frobenius-Schur index of χ. When is the symmetric group on letters, we have [6] an explicit formula for (1.1). In a recent work [4], [5] (this and the present work were done largely independently), A. Gyoja, K. Nishiyama and K. Taniguchi explicitly calculated (1.1) in the cases of real reflection groups of type 4 2( ) and ; in the case of type , their proof depends upon one of the main result (Theorem 1.1 below) of the present paper. The authors of [4], [5] also observed a mysterious connection between (χ; ), Lusztig’s cells and modular representations of Iwahori Hecke algebras. The main purpose of this paper is to calculate (χ; ) explicitly when is an imprimitive complex reflection group ( ) (in the notation of G.C. Shephard and J.A. Todd [12]). This includes, as special cases, the cases of real reflection groups of type and 2( ).