  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.
