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The center of Hecke algebras of types

Reda Boumasmoud, Radhika Ganapathy

Abstract

We describe the center of the Hecke algebra of a type attached to a Bernstein block under some hypothesis. When $\bf G$ is a connected reductive group over non-archimedean local field $F$ that splits over a tamely ramified extension of $F$ and the residue characteristic of $F$ does not divide the order of the absolute Weyl group of $\bf G$, the works of Kim-Yu and Fintzen associate a type to each Bernstein block and our hypothesis is satisfied for such types. We use our results to give a description of the Bernstein center of the Hecke algebra $\mathcal{H}({\bf G } (F),K)$ when $K$ belongs to a nice family of compact open subgroups of ${\bf G}(F)$ (which includes all the Moy-Prasad filtrations of an Iwahori subgroup) via the theory of types.

The center of Hecke algebras of types

Abstract

We describe the center of the Hecke algebra of a type attached to a Bernstein block under some hypothesis. When is a connected reductive group over non-archimedean local field that splits over a tamely ramified extension of and the residue characteristic of does not divide the order of the absolute Weyl group of , the works of Kim-Yu and Fintzen associate a type to each Bernstein block and our hypothesis is satisfied for such types. We use our results to give a description of the Bernstein center of the Hecke algebra when belongs to a nice family of compact open subgroups of (which includes all the Moy-Prasad filtrations of an Iwahori subgroup) via the theory of types.
Paper Structure (23 sections, 27 theorems, 83 equations)

This paper contains 23 sections, 27 theorems, 83 equations.

Table of Contents

  1. The Bernstein center
  2. The Bernstein decomposition
  3. Type associated to a Bernstein block
  4. Some Clifford theory
  5. $\flat$-world
  6. Some results on Hecke algebras of types
  7. Isomorphism of Hecke algebras
  8. Multiplicites for types
  9. Center of Hecke algebras
  10. Criterion for the Hecke algebra of a supercuspidal type to be commutative
  11. Weyl action on (center of) Hecke algebras and a Satake isomorphism
  12. $G$-equivalence of types
  13. Normalizer of a type
  14. Weyl action on the center
  15. Some nice families of compact open subgroups
  16. ...and 8 more sections

Key Result

Lemma 2.1

Let $\tilde{H}$ be any open subgroup of $G$ and set $H=\tilde{H} \cap {}^\flat G$. Let $\tilde{\sigma}$ be a semi-simple of finite length representation of $\tilde{H}$.

Theorems & Definitions (63)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • ...and 53 more