The objects of the talk are the translation-invariant vector bundles over an abelian variety. We will present a representation-theoretic description of these vector bundles, which displays a remarkable analogy with finite-dimensional representations of a compact connected Lie group: the weight lattice is replaced with the dual abelian variety, the Weyl group with the Galois group of the ground field...

Let $f : X \to Y$ be a fibration between two projective manifolds. The Iitaka's conjecture predicts that the Kodaira dimension of $X$ is larger than the sum of the Kodaira dimension of $X$ and the Kodaira dimension of the generic fiber. We explain a proof of the Iitaka conjecture for algebraic fiber spaces over abelian varieties or projective surfaces.
It is a joint work with Mihai Paun.