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

Abstract

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 ${\Theta }_M^P:{\mathcal{H}}_R(P)\to {\mathcal{H}}_R(M)$ and ${\Xi }_G^P:{\mathcal{H}}_R(P)\to {\mathcal{H}}_R(G)$ 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 ${\Theta }_M^P$ and extension of scalars along ${\Xi }_G^P$) coincides with the parabolic induction due to Ollivier--Vignéras. The maps ${\Theta }_M^P$ and ${\Xi }_G^P$ 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.

36 pages, published version. Added Lemma 3.3 and the example after Proposition 3.4; fixed some misprints. Comments welcome!