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

The Hecke group algebra H (W) over dot of a finite Coxeter group k as introduced by the first and last authors, is obtained from (W) over dot 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) over dot is the finite Weyl group associated to an affine Weyl group W. Namely, we prove that, for q not a root of unity of small order, H (W) over dot 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 H (W) (0) 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 0-Hecke algebra H ((W) over dot) (0) and that of the affine Hecke algebra H (W) (q) at this specialization. (C) 2008 Elsevier Inc. All rights reserved.