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

The Hecke group algebra H W ˚ of a finite Coxeter group W ˚ , as introduced by the first and last authors, 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 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 ˚ is the natural quotient of the affine Hecke algebra H ( W ) ( q ) through its level 0 representation. oof 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 ˚ ) ( 0 ) and that of the affine Hecke algebra H ( W ) ( q ) at this specialization.