DISTINGUISHED REPRESENTATIONS AND A FOURIER SUMMATION FORMULA

A new ''Fourier'' summation formula is developed in the context of both GL(n, E) and the quasi-split unitary group U(n, E/F) associated with a quadratic extension E/F of global fields. It is used to reduce to a local technical conjecture concerning matching ''Fourier'' orbital integrals, the following precise form of the conjecture of [F1]. The stable (if n is odd) and the unstable (if n is even) base-change lifting is a surjection from (a) the set of irreducible automorphic discrete-series non-degenerate representations pi of the group of adele points on U(n, E/F), to (b) the set of automorphic irreducible non-degenerate representations pi' of GL(n, A(E)) normalizedly induced from a representation rho1 x ... x rho(a) of a parabolic subgroup of type (n1,..., n(a)), where the pi are pairwise inequivalent distinguished cuspidal nondegenerate representations of GL(n(i), A(E)).