Hecke group algebras as quotients of affine Hecke algebras at level 0

Abstract

The Hecke group algebra $H{W}_0$ of a finite Coxeter group $W_0$, as introduced by the first and last author, is obtained from $W_0$ by gluing appropriately its 0-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when $W_0$ is the classical Weyl group associated to an affine Weyl group $W$. Namely, we prove that, for $q$ not a root of unity, $H{W}_0$ is the natural quotient of the affine Hecke algebra through its level 0 representation. We further show that the level 0 representation is a calibrated principal series representation for a suitable choice of character, 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.

21 pages; 4 figures v2: improved presentation, 23 pages v3: final proofreading, to appear in Journal of Combinatorial Theory Series A

References in corpus (2)

• Existence of Kirillov-Reshetikhin crystals for nonexceptional types
• A combinatorial formula for Macdonald polynomials