Parabolic induction via the parabolic pro-𝑝 Iwahori–Hecke algebra

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