  The convolution powers of a perverse sheaf on an abelian variety define an
interesting family of branched local systems whose geometry is still poorly
understood. We show that the generating series for their generic rank is a
rational function of a very simple shape and that a similar result holds for
the symmetric convolution powers. We also give formulae for other Schur
functors in terms of characteristic classes on the dual abelian variety, and as
an example we discuss the case of Prym-Tjurin varieties.
