# A q-Cauchy identity for Schur functions and imprimitive complex reflection groups (Dedicated to Professor Shunichi Tanaka on his sixtieth birthday)

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Kawanaka, N. Osaka J. Math. 38 (2001), 775–810 A q-CAUCHY IDENTITY FOR SCHUR FUNCTIONS AND IMPRIMITIVE COMPLEX REFLECTION GROUPS Dedicated to Professor Shunichi Tanaka on his sixtieth birthday NORIAKI KAWANAKA (Received January 12, 2000) 1. Introduction Let ρ : −→ (C) be a finite group acting on C as a complex reflection group. For an irreducible char- acter χ of , we define a rational function (1.1) (χ; ) = | |−1 ∑ ∈ χ( 2) det(1 + ρ( ))det(1− ρ( )) 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 in- dependently), 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( ). Theorem 1.1. Let = ( 1 2 3 . . .) and = ( 1 2 3 . . .) be two infinite se- quences of independent variables. For a partition λ let λ( ) = λ( 1 2 3 . . .) and λ( ) = λ( 1 2 3 . . .) be the corresponding Schur functions in and respectively. 776 N. KAWANAKA Then we have the following identities: (1.2)∏ ∞∏ =0 1 + 2 +1 1− 2 +1 1 + 2 +1 1− 2 +1 ∏ 1 1− = ∑ λ µ |λ−µ|+|µ−λ| λµ( 2) λ( ) µ( ) and (1.3) ∏ ∞∏ =0 1 + 2 +1 1− 2 +1 1 + 2 +1 1− 2 +1 ∏ (1 + ) = ∑ λ µ |λ−µ′|−|µ′−λ| λµ′( 2) λ( ) µ( ) where µ′ is the dual partition of µ, λµ( ) is a rational function defined in Section 2.1, and the sums are taken o

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