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Let $\mathbf{G}$ be a connected reductive group defined over a locallycompact non-archimedean field $F$, let $\mathbf{P}$ be a parabolic subgroupwith 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 theparabolic 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 functorMod-$\mathcal{H}_R(M) \to$ 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 applicationwe give a new proof of the transitivity of parabolic induction.

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