Working Seminar on Representation Theory

A result of Gelfand and Graev shows that the supercuspidal representations of $\mathrm{SL}_2$ are neatly parameterized by characters of elliptic tori, and that the stable character data for all such representations may be collected into a single function by means of a Fourier Transform. Using recent advances in the computation of characters of supercuspidal representations, we prove analogous results for the group $\mathrm{SL}_n$.