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arXiv · 1606.05542

Admissible subsets and Littelmann paths in affine Kazhdan-Lusztig theory

Jeremie Guilhot
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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.

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Jeremie Guilhot. 2016-06-17. Admissible subsets and Littelmann paths in affine Kazhdan-Lusztig theory. https://arxiv.org/abs/1606.05542

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