Kawanaka invariants for representations of Weyl groups
Let \(W\) be a Weyl group and let \(V\) be the complex space affording the reflection representation \(\rho\) of \(W\). For a complex irreducible character \(\chi\) of \(W\), the rational function in an indeterminate \(q\) given by \[ I_W(\chi,q)=|W|^{-1}\sum_{w\in W}\chi(\omega^2)\frac{\text{det}(1+q\rho(w))}{\text{det}(1-q\rho(w))} \] is called by the authors the Kawanaka invariant of \(\chi\). In the case where \(W\) is of type \(A\), \textit{N.~Kawanaka} expressed \(I_W(\chi,q)\) explicitly in the form \[ \prod^l_{i=1}\frac{1+q^{h_i}}{1-q^{h_i}} \] for some integers \(h_1,\dots,h_l\) [see Osaka J. Math. 36, No. 1, 157-176 (1999; Zbl 0915.05008)]. In the paper under review, the authors generalize Kawanaka's result by giving an explicit description of all the \(I_W(\chi,q)\)'s for all the other irreducible Weyl groups \(W\). Moreover, they study some properties of Kawanaka invariants and the relations between Kawanaka invariants and the two-sided cells of characters of \(W\).
scientific article; zbMATH DE number 1429472