Derived parabolic induction

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$G$-representations over k$k$, G$G$ a p$p$-adic reductive group and k$k$ a field of characteristic p$p$, is equivalent to the derived category of a certain differential graded k$k$-algebra HG center dot$H_G<^>\bullet$, 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. Schnurer. We also discuss the adjoint functors of derived parabolic induction.

Automated summary

This article studies the derived version of the classical parabolic induction functor, and proves that the derived category of smooth G-G representations is equivalent to a certain differential graded k-algebra. It constructs a derived parabolic induction functor on the dg Hecke algebra side and discusses its adjoint functors.