Localization of the Parabolic Hecke Algebra at a Strictly Positive Element

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.