Localization of the Parabolic Hecke Algebra at a Strictly Positive Element

Claudius Heyer

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arXiv · 2103.16949

Localization of the Parabolic Hecke Algebra at a Strictly Positive Element

Claudius Heyer

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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.

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Claudius Heyer. 2021-03-31. Localization of the Parabolic Hecke Algebra at a Strictly Positive Element. https://arxiv.org/abs/2103.16949

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