Florent Hivert ; Anne Schilling ; Nicolas M. Thiéry
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Hecke group algebras as degenerate affine Hecke algebras
dmtcs:3620 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2008,
DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)
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https://doi.org/10.46298/dmtcs.3620
Hecke group algebras as degenerate affine Hecke algebrasArticle
Authors: Florent Hivert 1; Anne Schilling 2; Nicolas M. Thiéry 3
0000-0002-7531-5985##NULL##0000-0002-2735-8921
Florent Hivert;Anne Schilling;Nicolas M. Thiéry
1 Laboratoire d'Informatique, de Traitement de l'Information et des Systèmes
2 Department of Mathematics [Univ California Davis]
3 Laboratoire de Mathématiques d'Orsay
The Hecke group algebra $\operatorname{H} \mathring{W}$ of a finite Coxeter group $\mathring{W}$, as introduced by the first and last author, is obtained from $\mathring{W}$ by gluing appropriately its $0$-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when $\mathring{W}$ is the classical Weyl group associated to an affine Weyl group $W$. Namely, we prove that, for $q$ not a root of unity, $\operatorname{H} \mathring{W}$ is the natural quotient of the affine Hecke algebra $\operatorname{H}(W)(q)$ through its level $0$ representation. The proof relies on the following core combinatorial result: at level $0$ the $0$-Hecke algebra acts transitively on $\mathring{W}$. Equivalently, in type $A$, a word written on a circle can be both sorted and antisorted by elementary bubble sort operators. We further show that the level $0$ representation is a calibrated principal series representation $M(t)$ for a suitable choice of character $t$, so that the quotient factors (non trivially) through the principal central specialization. This explains in particular the similarities between the representation theory of the classical $0$-Hecke algebra and that of the affine Hecke algebra at this specialization.