Derived parabolic induction

TL;DR

This paper develops a derived version of the classical parabolic induction functor within the context of the Langlands program, utilizing dg categories and six-functor formalism to extend representation theory tools.

Contribution

It constructs a derived parabolic induction functor on dg Hecke algebras, extending classical methods to the derived setting in representation theory.

Findings

Established the derived parabolic induction functor on dg Hecke algebras.

Connected the derived category of smooth G-representations with dg algebra frameworks.

Discussed adjoint functors and formal properties of the derived induction.

Abstract

The classical parabolic induction functor is a fundamental tool on the representation theoretic side of the Langlands program. In this article, we study its derived version. It was shown by the second author that the derived category of smooth $G$-representations over $k$, $G$ a $p$-adic reductive group and $k$ a field of characteristic $p$, is equivalent to the derived category of a certain differential graded $k$-algebra $H_G^\bullet$, whose zeroth cohomology is a classical Hecke algebra. This equivalence predicts the existence of a derived parabolic induction functor on the dg Hecke algebra side, which we construct in this paper. This relies on the theory of six-functor formalisms for differential graded categories developed by O.\ Schn\"urer. We also discuss the adjoint functors of derived parabolic induction.