  We show that the hypercohomology of most character twists of perverse sheaves
on a complex abelian variety vanishes in all non-zero degrees. As a consequence
we obtain a vanishing theorem for constructible sheaves and a relative
vanishing theorem for a homomorphism between abelian varieties. Our proof
relies on a Tannakian description for convolution products of perverse sheaves,
and with future applications in mind we discuss the basic properties of the
arising Tannaka groups.
