Projections Separating Spectra for L2 on Pseudounitary Groups U(p, q)

The spectrum of L-2 on a pseudo-unitary group U(p, q) (we assume p >= q) naturally splits into q + 1 types. We give explicit formulas for the orthogonal projections in L-2 to subspaces with uniform spectra (this is an old question formulated by Gelfand and Gindikin). We also write two finer separations of L-2. In the first case, pieces are parametrized by r = 0, 1,..., q and representations of the discrete series of U(p-r, q-r), where r = 0,..., q. In the second case, pieces are parametrized by all discrete parameters of the tempered spectrum of U(p, q).