Kawanaka Invariants for Representations of Weyl Groups

Abstract

Let W be a Weyl group and let V be the natural CW-module, i.e., the reflection representation. For a complex irreducible character χ of W, we consider the invariantIχ;q≔|W|−1∑w∈Wχw2det1+qw|V/det1−qw|Vintroduced by N. Kawanaka. We determine I(χ;q) explicitly. Looking over these results, we observe a relation between Kawanaka's invariants I(χ;q) and the two-sided cells. For example, if a two-sided cell consists of a single element χ, then the Kawanaka invariant I(χ;q) can be expressed as ∏li=1(1+qhi)/(1−qhi) with some integers hi. This expression can be regarded as a quantization of the usual hook formula for the dimension of irreducible representations of symmetric groups.