Admissible subsets and Littelmann paths in affine Kazhdan-Lusztig theory

TL;DR

This paper explores how admissible subsets and Littelmann paths, models for computing tensor multiplicities in Lie algebra representations, naturally arise in the context of affine Kazhdan-Lusztig theory and the structure of the center of extended affine Hecke algebras.

Contribution

It demonstrates the natural appearance of Littelmann paths and admissible subsets in decomposing Kazhdan-Lusztig basis elements related to affine Hecke algebras.

Findings

Littelmann paths model tensor multiplicities.

Admissible subsets relate to basis decompositions.

Connections between representation theory and affine Hecke algebra structure.

Abstract

The center of an extended affine Hecke algebra is known to be isomorphic to the ring of symmetric functions associated to the underlying finite Weyl group $W\_0$. The set of Weyl characters ${\sf s}\_\la$ forms a basis of the center and Lusztig showed in [Lus15] that these characters act as translations on the Kazhdan-Lusztig basis element $C\_{w\_0}$ where $w\_0$ is the longest element of $W\_0$, that is we have $C\_{w\_0}{\sf s}\_\la =C\_{w\_0t\_\la}$. As a consequence, the coefficients that appear when decomposing~$C\_{w\_0t\_{\la}}{\sf s}\_\tau$ in the Kazhdan-Lusztig basis are tensor multiplicities of the Lie algebra with Weyl group $W\_0$. The aim of this paper is to explain how admissible subsets and Littelmann paths, which are models to compute such multiplicities, naturally appear when working out this decomposition.