Parabolic Induction via the Parabolic pro-$p$ Iwahori--Hecke Algebra

Let $\mathbf{G}$ be a connected reductive group defined over a locally compact non-archimedean field $F$, let $\mathbf{P}$ be a parabolic subgroup with Levi $\mathbf{M}$ and compatible with a pro-$p$ Iwahori subgroup of $G := \mathbf{G}(F)$. Let $R$ be a commutative unital ring. We introduce the parabolic pro-$p$ Iwahori--Hecke $R$-algebra $\mathcal{H}_R(P)$ of $P := \mathbf{P}(F)$ and construct two $R$-algebra morphisms $\Theta^P_M\colon \mathcal{H}_R(P)\to \mathcal{H}_R(M)$ and $\Xi^P_G\colon \mathcal{H}_R(P) \to \mathcal{H}_R(G)$ into the pro-$p$ Iwahori--Hecke $R$-algebra of $M := \mathbf{M}(F)$ and $G$, respectively. We prove that the resulting functor Mod-$\mathcal{H}_R(M) \to$ Mod-$\mathcal{H}_R(G)$ from the category of right $\mathcal{H}_R(M)$-modules to the category of right $\mathcal{H}_R(G)$-modules (obtained by pulling back via $\Theta^P_M$ and extension of scalars along $\Xi^P_G$) coincides with the parabolic induction due to Ollivier--Vign\'eras. The maps $\Theta^P_M$ and $\Xi^P_G$ factor through a common subalgebra $\mathcal{H}_R(M,G)$ of $\mathcal{H}_R(G)$ which is very similar to $\mathcal{H}_R(M)$. Studying these algebras $\mathcal{H}_R(M,G)$ for varying $(M,G)$ we prove a transitivity property for tensor products. As an application we give a new proof of the transitivity of parabolic induction.