Parameters of Hecke algebras for Bernstein components of p-adic groups
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.
Keywords
Cite
@article{arxiv.2103.13113,
title = {Parameters of Hecke algebras for Bernstein components of p-adic groups},
author = {Maarten Solleveld},
journal= {arXiv preprint arXiv:2103.13113},
year = {2025}
}
Comments
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