International Congress of Mathematicians

Harmonie Analysis on Real Reductive Symmetric Spaces 547Some spaces of r-spherical functions on G/H play a crucial role in the theory,namely:(a) G(G/H, T) : the Schwartz space of r-spherical functions on G/H which arerapidly decreasing as well as their derivatives by elements of the enveloping algebraU(g) of g (see [B2]).(b) A(G/H, T): the space of smooth r-spherical functions on G/H which areB(G/H) finite. Here A is used to evoke automorphic forms.(c) Atem,p(G/H, T): the space of elements of A(G/H, r) which have temperedgrowth as well as their derivatives by elements of U(g) ([D2]). Integration offunctions on G/H defines a pairing between At emp (G/H, r) and G(G/H, r).(d) A2(G/H, T): the space of square integrable elements of A(G/H, r). Thisis a subspace of the three proceeding spaces.One has:Theorem 1 ( [D2] ): The space ./^(G/ff, r) is finite dimensional.This is deduced from the theory of discrete series for G/H initiated by M.Flensted-Jensen [F-J] and achieved by T. Oshima and T. Matsuki, using the Flensted-Jensen duality [OM]. One has also to use the behaviour of the discrete series undercertain translation functors, studied by D. Vogan [V] and a result of H. Schlichtkrull[S] on the minimal if-types of certain discrete series.The next result follows from the work of J. Bernstein [Be] on the support ofthe Plancherel measure.Theorem 2 ([CD1], Appendice C): Every function in G(G/H, r) can be canonicallydesintegrated as an integral of elements of At emp (G/H, r).This information appeared to be crucial at the end of our proof.3. The continuous spectrum: Eisenstein integralsLet P = MAN the Langlands ^-decomposition of a -¥ E(P,fi,X) admits a meromorphic continuationin X £ o£- This meromorphic continuation, denoted in the same way,multiplied by a suitable product, p v , of functions of type X H> (a, A) + c, where a isa root of a and c £ C, is holomorphic around ia*.