Kawanaka Invariants for Representations of Weyl Groups

Let W be a Weyl group and let V be the natural W-module, i.e., the reflection representation. For a complex irreducible character χ of W, we consider the invariant introduced by N. Kawanaka. We determine I(χ; q) explicitly. Looking over these results, we observe a relation between Kawanakaʹs 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.