The Geometrical Lemma for Smooth Representations in Natural Characteristic

TL;DR

This paper extends the classical Geometrical Lemma to the derived category of smooth mod p representations of p-adic reductive groups, enabling new computations of extension groups and cohomology functors in this setting.

Contribution

It establishes the Geometrical Lemma for derived categories of mod p representations and computes higher extension groups and cohomology functors, advancing the understanding of smooth mod p representations.

Findings

Computed higher extension groups between parabolically induced representations.

Determined the cohomology of the left adjoint of derived parabolic induction.

Extended the Geometrical Lemma to the derived category setting.

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.