  Suppose a finite dimensional semisimple Lie algebra $\mathfrak g$ acts by
derivations on a finite dimensional associative or Lie algebra $A$ over a field
of characteristic $0$. We prove the $\mathfrak g$-invariant analogs of
Wedderburn - Mal'cev and Levi theorems, and the analog of Amitsur's conjecture
on asymptotic behavior for codimensions of polynomial identities with
derivations of $A$. It turns out that for associative algebras the differential
PI-exponent coincides with the ordinary one. Also we prove the analog of
Amitsur's conjecture for finite dimensional associative algebras with an action
of a reductive affine algebraic group by automorphisms and anti-automorphisms
or graded by an arbitrary Abelian group. In addition, we provide criteria for
$G$-, $H$- and graded simplicity in terms of codimensions.
