A Gelfand-Graev Formula and Stable Transfer Factors for SL_n(F)

A result of Gelfand and Graev shows that the supercuspidal representations of SL_2(F) for a nonarchimedean local field F 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. For the group SL_n(F) we state a conjectural formula for the character values of supercuspidal representations arising from unramified tori of SL_n(F) and use this formula to prove analogous to those of Gelfand and Graev in this case.