THE PLANCHEREL FORMULA FOR SL(2) OVER A LOCAL FIELD

More than two decades ago, in his classical paper on the irreducible unitary representations of the Lorentz group, V. Bargmann initiated the concrete study of Fourier analysis on real Lie groups and obtained the analogue of the classical Fourier expansion theorem in the case of the Lorentz group. Since then the general theory for real semisimple Lie groups has been extensively developed, chiefly through the work of Harish-Chandra. More generally, one may consider groups defined by algebraic equations over locally compact fields, in particular local fields, and ask for an explicit Fourier expansion formula. In the present article the authors obtain this formula for the group SL(2).