Localization of the Parabolic Hecke Algebra at a Strictly Positive Element

TL;DR

This paper establishes a localization relationship between the Hecke algebras of a parabolic subgroup and its Levi component over a non-archimedean local field, providing a new perspective on module correspondence.

Contribution

It proves that the Hecke algebra of the Levi subgroup is a left ring of fractions of the Hecke algebra of the parabolic subgroup, offering a novel algebraic characterization.

Findings

Hecke algebra of the Levi is a left ring of fractions of the parabolic Hecke algebra.

Characterization of modules over the parabolic Hecke algebra that originate from Levi modules.

Provides a new algebraic framework for understanding module relationships in p-adic groups.

Abstract

Let $\mathbf{P}$ be a parabolic subgroup with Levi $\mathbf{M}$ of a connected reductive group defined over a locally compact non-archimedean field $F$. Given a certain compact open subgroup $\Gamma$ of $\mathbf{P}(F)$, this note proves that the Hecke algebra $\mathcal{H}(\mathbf{M}(F))$ of $\mathbf{M}(F)$ with respect to $\Gamma\cap \mathbf{M}(F)$ is a left ring of fractions of the Hecke algebra $\mathcal{H}(\mathbf{P}(F))$ of $\mathbf{P}(F)$ with respect to $\Gamma$. This leads to a characterization of $\mathcal{H}(\mathbf{P}(F))$-modules that come from $\mathcal{H}(\mathbf{M}(F))$-modules.