The left adjoint of derived parabolic induction

Abstract

We prove that the derived parabolic induction functor, defined on the unbounded derived category of smooth mod p representations of a p-adic reductive group, admits a left adjoint textrm{L}(U,-). We study the cohomology functors textrm{H}^icirc textrm{L}(U,-) in some detail and deduce that textrm{L}(U,-) preserves bounded complexes and global admissibility in the sense of Schneider–Sorensen. Using textrm{L}(U,-) we define a derived Satake homomorphism and prove that it encodes the mod p Satake homomorphisms defined explicitly by Herzig.