Multiplicity matrices for the affine graded Hecke algebra

TL;DR

This paper advances the understanding of the affine graded Hecke algebra by computing Kazhdan-Lusztig polynomials and composition factors for specific p-adic groups, enhancing representation classification methods.

Contribution

It introduces algorithms for computing geometric parameters and Kazhdan-Lusztig polynomials for affine graded Hecke algebras associated with split p-adic groups of types G2 and F4.

Findings

Computed Kazhdan-Lusztig polynomials for real central characters with tempered modules.

Determined composition series and Iwahori-Matsumoto involution for specific groups.

Provided a geometric parameterization of representations with Iwahori fixed vectors.

Abstract

In this paper, we look 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 \cite{L1,L2}, 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 $F_4$.