Characteristic classes and Hilbert-Poincar\'e series for perverse sheaves on abelian varieties

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.