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

AbstractThe Hecke group algebra HW˚ 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, HW˚ 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˚)(0) and that of the affine Hecke algebra H(W)(q) at this specialization