Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parameters of Hecke algebras for Bernstein components of p-adic groups

Maarten Solleveld

TL;DR

The paper develops a unified framework to compute the $q$-parameters of affine Hecke algebras $\mathcal{H}(\mathcal{O},G)$ attached to Bernstein blocks $\mathrm{Rep}(G)^{\mathfrak s}$ for reductive $p$-adic groups, by relating $\mathrm{Rep}(G)^{\mathfrak s}$ to endomorphism algebras of progenerators. It reduces the problem to absolutely simple simply connected groups and leverages characteristic-zero techniques via close local fields, enabling a broad computation of $q_\alpha$ and $q_{\alpha*}$ for principal series and many other blocks across classical and exceptional types. The results confirm Lusztig's conjecture that these parameters are powers of the residue characteristic base $q_F$ (with specified exceptions) and relate the label data $\lambda,\lambda^*$ to unipotent and enhanced Langlands parameters. The work provides substantial new computations for split, quasi-split, inner forms of type $A$, and many classical groups, and establishes comprehensive partial results for $G_2,F_4,E_6,E_7,E_8$, strengthening the link between Hecke-algebra parameters, Bernstein components, and the local Langlands program.

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.

Parameters of Hecke algebras for Bernstein components of p-adic groups

TL;DR

The paper develops a unified framework to compute the -parameters of affine Hecke algebras attached to Bernstein blocks for reductive -adic groups, by relating to endomorphism algebras of progenerators. It reduces the problem to absolutely simple simply connected groups and leverages characteristic-zero techniques via close local fields, enabling a broad computation of and for principal series and many other blocks across classical and exceptional types. The results confirm Lusztig's conjecture that these parameters are powers of the residue characteristic base (with specified exceptions) and relate the label data to unipotent and enhanced Langlands parameters. The work provides substantial new computations for split, quasi-split, inner forms of type , and many classical groups, and establishes comprehensive partial results for , strengthening the link between Hecke-algebra parameters, Bernstein components, and the local Langlands program.

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.

Paper Structure

This paper contains 11 sections, 33 theorems, 176 equations, 2 tables.

Table of Contents

  1. Progenerators and endomorphism algebras for Bernstein blocks
  2. Reduction to simply connected groups
  3. Reduction to characteristic zero
  4. Hecke algebra parameters for simple groups
  5. Principal series of split groups
  6. Principal series of quasi-split groups
  7. Inner forms of Lie type $A_n$
  8. Classical groups
  9. Groups of Lie type $G_2$
  10. Groups of Lie type $F_4$
  11. Groups of Lie type $E_6, E_7, E_8$

Key Result

Theorem B

(see Theorem thm:3.9 and Corollary cor:3.12) Conjecture conj:1 holds for all Bernstein blocks in the principal series of a quasi-split connected reductive group over $F$. For $X_\alpha \in \Sigma_{\mathcal{O}}^\vee$ (with one exception in type ${}^2 A_{2n}$ that we analyse as well) $q_{\alpha *} = 1

Theorems & Definitions (53)

  • Conjecture A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Proposition 1.1
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • ...and 43 more