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The Geometrical Lemma for Smooth Representations in Natural Characteristic

Claudius Heyer

Abstract

The Geometrical Lemma is a classical result in the theory of (complex) smooth representations of $p$-adic reductive groups, which helps to analyze the parabolic restriction of a parabolically induced representation by providing a filtration whose graded pieces are (smaller) parabolic inductions of parabolic restrictions. In this article, we establish the Geometrical Lemma for the derived category of smooth mod $p$ representations of a $p$-adic reductive group. As an important application we compute higher extension groups between parabolically induced representations, which in a slightly different context had been achieved by Hauseux assuming a conjecture of Emerton concerning the higher ordinary parts functor. We also compute the (cohomology functors of the) left adjoint of derived parabolic induction on principal series and generalized Steinberg representations.

The Geometrical Lemma for Smooth Representations in Natural Characteristic

Abstract

The Geometrical Lemma is a classical result in the theory of (complex) smooth representations of -adic reductive groups, which helps to analyze the parabolic restriction of a parabolically induced representation by providing a filtration whose graded pieces are (smaller) parabolic inductions of parabolic restrictions. In this article, we establish the Geometrical Lemma for the derived category of smooth mod representations of a -adic reductive group. As an important application we compute higher extension groups between parabolically induced representations, which in a slightly different context had been achieved by Hauseux assuming a conjecture of Emerton concerning the higher ordinary parts functor. We also compute the (cohomology functors of the) left adjoint of derived parabolic induction on principal series and generalized Steinberg representations.
Paper Structure (18 sections, 37 theorems, 108 equations)

This paper contains 18 sections, 37 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. History and motivation
  3. Main results
  4. Organization of the paper
  5. Acknowledgments
  6. Preliminaries
  7. Notation and conventions
  8. Compact induction and derived coinvariants
  9. The duality character
  10. The Geometrical Lemma
  11. Setup
  12. Filtrations
  13. The Geometrical Lemma
  14. Applications
  15. Setup and notation
  16. ...and 3 more sections

Key Result

Theorem A

The functor $\mathrm{L}(N,\raisebox{-2pt}{$-$})\circ \mathrm{R}\!\mathop{\mathrm{\mathit{i}}}\nolimits_P^G \colon \mathrm{D}(M)\to \mathrm{D}(L)$ admits a filtration of length $\lvert\mathcal{N}_{P,Q}\rvert$ with graded pieces of the form for $n\in \mathcal{N}_{P,Q}$, where $\omega_n \in \mathrm{D}(n^{-1}Mn\cap L)$ is a character in cohomological degree $-\dim(n^{-1}\overline{U}n\cap N)$ and $\ov

Theorems & Definitions (83)

  • Theorem A: Corollary \ref{['cor:geometrical']}
  • Theorem B: Theorems \ref{['thm:PS']} and \ref{['thm:Ext']}
  • Lemma 1: Projection formula
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 73 more