Parameters of Hecke algebras for Bernstein components of \(p\)-adic groups (Q6663162)

The affine Hecke algebras play an important role in the representation of reductive groups \(G\) over non-Archimedean local fields \(F\). A Bernstein block \(\mathrm{Rep}(G)^{\mathfrak{s}}\) in the category of smooth complex \(G\)-representations is equivalent, in many cases, to the module category of an affine Hecke algebra eventually extended with some finite group. The author, in [J. Algebra 606, 371--470 (2022; Zbl 1517.22010)], proved that there exists an affine Hecke algebra \(\mathcal{H}(\mathcal{O},G)\) whose category of right modules is closely related to \(\mathrm{Rep}(G)^{\mathfrak{s}}\).\N\NIn this paper the author studies the \(q\)-parameters of the affine Hecke algebras \(\mathcal{H}(\mathcal{O},G)\). He compute them in many important cases, in particular for principal series representations of quasi-split groups and for classical groups. \textit{G. Lusztig}, in [``Open problems on Iwahori-Hecke algebras'', Preprint, \url{arXiv:2006.08535}], conjectured that the \(q\)-parameters are always integral powers of the cardinality of the residue field of \(F\), and that they coincide with the \(q\)-parameters coming from some Bernstein block of unipotent representations. The author reduces this conjecture to the case of absolutely simple \(p\)-adic groups and he proves it for most of those.