The Iwahori--Matsumoto dual for tempered representations of Lusztig's geometric Hecke algebras

The Iwahori--Matsumoto involution IM is an algebra involution on an affine Hecke algebra. To a connected complex reductive group G, Lusztig associated various geometric graded Hecke algebras. These graded Hecke algebras are also associated to certain affine Hecke algebras. Let H be such a graded Hecke algebra associated to an affine Hecke algebra H. We obtain an involution IM on H induced by IM on H via Lusztig's reduction theorems. We also denote by IM the involution on the Grothendieck group of complex finite-dimensional representations of H induced by IM. The irreducible representations of H are parametrised by the set M of G-conjugacy classes of quadruples (e,s,r₀,ψ) where r₀ ∈ C, e ∈ Lie(G) is nilpotent, s ∈ Lie(G) is semisimple, and ψ is an irreducible representation of the group of components of the simultaneous centraliser of $(e,s)$ in G. Let Y be an irreducible tempered representation of H with real infinitesimal character. In this paper, we give an explicit algorithm that computes the G-orbit of the nilpotent element in the quadruple in M that parametrises the irreducible representation IM( Y) for G = Sp(2n,C) and G = SO(N,C).