Hecke group algebras as degenerate affine Hecke algebras
Identifieur interne : 000A82 ( France/Analysis ); précédent : 000A81; suivant : 000A83Hecke group algebras as degenerate affine Hecke algebras
Auteurs : Florent Hivert [France] ; Anne Schilling [États-Unis] ; Nicolas M. Thiéry [France]Source :
English descriptors
Abstract
The Hecke group algebra H W of a finite Coxeter group W , as introduced by the first and last author, is obtained from 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 W is the classical Weyl group associated to an affine Weyl group W. Namely, we prove that, for q not a root of unity, H W is the natural quotient of the affine Hecke algebra 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 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. \par
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Affiliations:
- France, États-Unis
- Haute-Normandie, Région Normandie
- Le Havre, Rouen
- Université de Rouen, Université du Havre
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In this paper, we give an equivalent alternative construction in the case when W is the classical Weyl group associated to an affine Weyl group W. Namely, we prove that, for q not a root of unity, H W is the natural quotient of the affine Hecke algebra 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 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. \par</div></front></TEI><affiliations><list><country><li>France</li><li>États-Unis</li></country><region><li>Haute-Normandie</li><li>Région Normandie</li></region><settlement><li>Le Havre</li><li>Rouen</li></settlement><orgName><li>Université de Rouen</li><li>Université du Havre</li></orgName></list><tree><country name="France"><region name="Région Normandie"><name sortKey="Hivert, Florent" sort="Hivert, Florent" uniqKey="Hivert F" first="Florent" last="Hivert">Florent Hivert</name></region><name sortKey="Thiery, Nicolas M" sort="Thiery, Nicolas M" uniqKey="Thiery N" first="Nicolas M." last="Thiéry">Nicolas M. Thiéry</name></country><country name="États-Unis"><noRegion><name sortKey="Schilling, Anne" sort="Schilling, Anne" uniqKey="Schilling A" first="Anne" last="Schilling">Anne Schilling</name></noRegion></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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