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The center of an extended affine Hecke algebra is known to be isomorphic tothe 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 centerand Lusztig showed in [Lus15] that these characters act as translations on theKazhdan-Lusztig basis element $C\_{w\_0}$ where $w\_0$ is the longest elementof $W\_0$, that is we have $C\_{w\_0}{\sf s}\_\la =C\_{w\_0t\_\la}$. As aconsequence, the coefficients that appear whendecomposing~$C\_{w\_0t\_{\la}}{\sf s}\_\tau$ in the Kazhdan-Lusztig basis aretensor multiplicities of the Lie algebra with Weyl group $W\_0$. The aim ofthis paper is to explain how admissible subsets and Littelmann paths, which aremodels to compute such multiplicities, naturally appear when working out thisdecomposition.

Admissible subsets and Littelmann paths in affine Kazhdan-Lusztig theory