I will discuss a joint work with Arthur Forey and Emmanuel Kowalski in which we obtain an equidistribution theorem for discrete Fourier transforms of trace functions of perverse sheaves on a commutative algebraic group over a finite field. The proof relies on a generic vanishing theorem for twists of perverse sheaves, which allows for the construction of a tannakian category with convolution as tensor operation. If time permits, I will also explain how to compute the groups governing the equidistribution in a few interesting examples.