The Left Adjoint of Derived Parabolic Induction

TL;DR

This paper establishes the existence of a left adjoint to the derived parabolic induction functor in the context of smooth mod p representations of p-adic groups, and explores its properties and applications.

Contribution

It introduces and analyzes the left adjoint of the derived parabolic induction functor, connecting it to the mod p Satake homomorphism and its explicit descriptions.

Findings

The functor $ ext{L}(U,-)$ preserves bounded complexes.

It maintains global admissibility of representations.

The derived Satake homomorphism encodes Herzig's mod p Satake maps.

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 $\mathrm{L}(U,-)$. We study the cohomology functors $\mathrm{H}^i\circ \mathrm{L}(U,-)$ in some detail and deduce that $\mathrm{L}(U,-)$ preserves bounded complexes and global admissibility in the sense of Schneider--Sorensen. Using $\mathrm{L}(U,-)$ we define a derived Satake homomorphism und prove that it encodes the mod $p$ Satake homomorphisms defined explicitly by Herzig.