A Fourier transform for a finite group G is an isomorphism from the complex group algebra CG to a direct product of complex matrix algebras, which are determined beforehand by the structure of G. Given such an isomorphism, naive application of that isomorphism to an arbitrary element of CG takes time proportional to |G|2. A fast Fourier transform for some (family of) groups is an algorithm which computes the Fourier transform of a group G of the family in less than O(|G|2) time, generally O(|G| log |G|) or O(|G|(log |G|)2). I describe the construction of a fast Fourier transform for the special linear groups SL(q) with q = 2n.