Multiplicity matrices for the affine graded Hecke algebra

Abstract

In this paper we are looking at the problem of determining the composition factors for the affine graded Hecke algebra via the computation of Kazhdan–Lusztig type polynomials. We review the algorithms of [G. Lusztig, Study of perverse sheaves arising from graded Lie algebras, Adv. Math. 112 (1995) 147–217; G. Lusztig, Graded Lie algebras and intersection cohomology, preprint], and use them in particular to compute, at every real central character which admits tempered modules, the geometric parameterization, the Kazhdan–Lusztig polynomials, the composition series, and the Iwahori–Matsumoto involution for the representations with Iwahori fixed vectors of the split p-adic groups of type G 2 and F 4 (and by the nature of the algorithms, for their Levi subgroups).