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 $\mathrm{L}(U,-)$. 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.

52 pages. Comments welcome! v2: Strengthened Corollary 3.4.20 and removed the now obsolete construction of a character; v3: several other small changes. This version may differ slightly from published version

References in corpus (2)

• Grothendieck-Neeman duality and the Wirthmüller isomorphism
• Parabolic induction and extensions