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 W0. The set of Weyl characters sλ forms a basis of the center and Lusztig showed in [11] that these characters act as translations on the Kazhdan–Lusztig basis element $$ {C}_{w_0} $$ where w0 is the longest element of W0, that is, we have $$ {C}_{w_0}{\mathrm{s}}_{\uplambda}={C}_{w_0{t}_{\uplambda}} $$ . As a consequence, the coefficients that appear when decomposing $$ {C}_{w_0{t}_{\uplambda}}{\mathrm{s}}_{\tau } $$ in the Kazhdan–Lusztig basis are tensor multiplicities of the Lie algebra with Weyl group W0. 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.