Admissible subsets and Littelmann paths in affine Kazhdan-Lusztig theory

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.