  A Fourier transform S is defined for the quantum double D(G) of a finite
group G. Acting on characters of D(G), S and the central ribbon element of D(G)
generate a unitary matrix representation of the group SL(2,Z). The characters
form a ring over the integers under both the algebra multiplication and its
dual, with the latter encoding the fusion rules of D(G). The Fourier transform
relates the two ring structures. We use this to give a particularly short proof
of the Verlinde formula for the fusion coefficients.
