Representations of Weyl group and its subgroups on the virtual character modules (Q761546)

Let G be a connected, acceptable, reductive Lie group. Take a regular infinitesimal character \(\lambda\). For a Cartan subgroup H of G, there corresponds naturally a space \(V_ H(\lambda)\) of virtual characters of G with the infinitesimal character \(\lambda\). The author defines, for each H and a regular \(\lambda\), a subgroup \(W_ H(\lambda)\) of the complex Weyl group W called the integral Weyl group for \(\lambda\) and H. He then constructs a representation of \(W_ H(\lambda)\) on the space \(V_ H(\lambda)\) and decomposes it thoroughly in terms of induced representations. If \(\lambda\) is integral, then the group \(W_ H(\lambda)\) coincides with W itself and the representation is equivalent to the one which Zuckerman has defined [\textit{A. W. Knapp} and \textit{G. J. Zuckerman}, Ann. Math., II. Ser. 116, 389-455 (1982; Zbl 0516.22011)]. Recently, even for a singular \(\lambda\), the author has succeeded in constructing a representation of the Hecke algebra \({\mathcal H}(W_ H(\lambda),W_{\lambda})\) on the space \(V_ H(\lambda)\) in a natural way. It turns out that this representation is closely related to Zuckerman's translation functors.