Derived parabolic induction

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