Semi-homogeneous vector bundles over abelian varieties

A vector bundle $E$ over an abelian variety is semi-homogeneous if every translate of $E$ is isomorphic to the tensor product of $E$ with a line bundle. The vector bundle $E$ is simple if $\mathrm{End}(E)$ is one-dimensional. Atiyah proved that every simple vector bundle over an elliptic curve is semi-homogeneous. This is no longer true over abelian varieties of dimension > 1. Nevertheless, semi-homogeneous vector bundles are particularly important. For example, if an equivalence $F\colon D(A) \rightarrow D(B)$ between the derived categories of two abelian varieties $A$ and $B$ maps points to objects of non-zero rank, then up to a shift the kernel of $F$ is a simple semi-homogeneous vector bundle over the product $A \times B$ (a result of Orlov). We will present Mukai's 1978 theorem that a simple semi-homogeneous vector bundle $E$ over an abelian variety $A$ is determined, up to translation, by its slope $c_1(E)/\mathrm{rank}(E)$. Furthermore, every class in the rational Neron-Severi vector space $\mathrm{NS}(A)_{\mathbb Q}$ is the slope of a simple semi-homogeneous vector bundle.