Parameters of Hecke algebras for Bernstein components of p-adic groups
arXiv:2103.13113
Abstract
Let G be a reductive group over a non-archimedean local field F. Consider an arbitrary Bernstein block Rep(G)^s in the category of complex smooth G-representations. In earlier work the author showed that there exists an affine Hecke algebra H(O,G) whose category of right modules is closely related to Rep(G)^s. In many cases this is in fact an equivalence of categories, like for Iwahori-spherical representations. In this paper we study the q-parameters of the affine Hecke algebras H(O,G). We compute them in many cases, in particular for principal series representations of quasi-split groups and for classical groups. Lusztig conjectured that the q-parameters are always integral powers of q_F and that they coincide with the q-parameters coming from some Bernstein block of unipotent representations. We reduce this conjecture to the case of simple p-adic groups, and we prove it for most of those.
Various minor improvements in Section 2. The proof of the previous Theorem 3.3 was flawed (in part c). Now that theorem is stretched over Lemma 3.3--Theorem 3.5. Proposition 4.10 was incorrect and has been repaired. At the same time, Theorem 4.9 has been simplified a little. In version 3, the paragraph 4.6 on F4 has been rewritten, now with more complete results