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The Left Adjoint of Derived Parabolic Induction

Claudius Heyer

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.

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 representations of a -adic reductive group, admits a left adjoint . We study the cohomology functors in some detail and deduce that preserves bounded complexes and global admissibility in the sense of Schneider--Sorensen. Using we define a derived Satake homomorphism und prove that it encodes the mod Satake homomorphisms defined explicitly by Herzig.
Paper Structure (34 sections, 95 theorems, 177 equations, 1 table)

This paper contains 34 sections, 95 theorems, 177 equations, 1 table.

Table of Contents

  1. Introduction
  2. The setting
  3. Main results
  4. Structure of the paper
  5. Acknowledgments
  6. Notation and conventions
  7. Foundations
  8. About adjoint functors
  9. Smooth representations
  10. General properties of smooth representations
  11. Inflation
  12. Symmetric monoidal structure
  13. The derived category
  14. Derived functors
  15. Brown representability
  16. ...and 19 more sections

Key Result

Theorem A

The functor $\mathrm{R}\mathop{\mathrm{Inf}}\nolimits^T_S\colon \mathrm{D}(T)\longrightarrow \mathrm{D}(S)$ admits a left adjoint $\mathrm{L}_U$. In particular, $\mathrm{R} i_M^G$ admits a left adjoint $\mathrm{L}(U,{\raisebox{-1.5pt}{$-$} })$.

Theorems & Definitions (209)

  • Theorem A: Theorem \ref{['thm:Inf-prod-general']}
  • Theorem B: Corollary \ref{['cor:LU-explicit']}
  • Theorem C: Theorem \ref{['thm:globally-admissible']}
  • Theorem D: Theorem \ref{['thm:Satake']}
  • Definition
  • Remark 2.1.2
  • Lemma 2.1.3: KS
  • Lemma 2.1.4
  • proof
  • Lemma 2.1.5: KS
  • ...and 199 more