  This paper develops two theories, the geometric theory of derived
Grassmannians (and flag schemes) and the algebraic theory of derived Schur (and
Weyl) functors, and establishes their connection, a derived generalization of
the Borel-Weil-Bott theorem. More specifically:
  (1) The theory of derived Grassmannians and flag schemes is the natural
extension of the theory of derived projectivizations [arXiv:2202.11636] and
generalizes Grothendieck's theory of Grassmannians and flag schemes of sheaves
to the case of complexes. We establish their fundamental properties and study
various natural morphisms among them.
  (2) The theory of derived Schur and Weyl functors extends the classical
theory of Schur and Weyl module functors studied in
$\mathrm{GL}_n(\mathbb{Z})$-representation theory to the case of complexes. We
show that these functors have excellent functorial properties and satisfy
derived generalizations of classical formulae such as Cauchy's decomposition
formula, direct-sum decomposition formula and Littlewood-Richardson rule. We
also generalize various results from the case of derived symmetric powers to
derived Schur functors, such as Illusie-Lurie's d\'ecalage isomorphism.
  (3) These two theories are connected by a derived version of the
Borel-Weil-Bott theorem, which generalizes the classical Borel-Weil-Bott
theorem and calculates the derived pushforwards of tautological perfect
complexes on derived flag schemes in terms of derived Schur functors when the
complexes have perfect-amplitude $\le 1$ and positive ranks.
