Representations of Weyl groups and their Hecke algebras on virtual character modules of a semisimple Lie group (Q1118046)

Let G be a connected semisimple Lie group with finite center and \({\mathfrak g}\) its Lie algebra. In a preceeding paper [J. Math. Soc. Japan 37, 719- 740 (1985; Zbl 0589.22012)], we defined a Weyl group action on virtual character modules with regular infinitesimal characters (recall that a virtual character is by definition a linear combination of irreducible characters on G). There, the representations of Weyl groups were completely decomposed by means of induced representations. However, in the case of singular infinitesimal character, representations of Weyl groups cannot be canonically realized on virtual character modules. In this paper, we will define representations of Hecke algebras on virtual character modules with singular infinitesimal characters. These representations are natural ones and can be considered as the ``limits'' of the representations of Weyl groups.