International Congress of Mathematicians

550 Patrick Delormeeasily that the factor in front of (A — A') -1 vanishes for A = A'. Hence we get| Gp[p(l,A) | 2 = | Cp|p( —1,A) | 2 , A £ ia*. This is one of the Maass-Selberg relations^.[D2], Theorem 2, and the work with J. Carmona [CD2], Theorem 2 for thegeneral case, see [BI], [B2] for the case where P is minimal). These relations implythatthe G-functions attached to normalized Eisenstein integrals are unitary, whendefined, for A purely imaginary. Hence they are locally bounded . This implies thatthey are holomorphic around the imaginary axis. This implies in particular someholomorphy property of the constant term of normalized Eisentein integrals. Fromthis, with the help of [BCD], one deduces:Theorem 6 (Regularity theorem for normalized Eisenstein integrals, [CD2], [BS1]for P minimal): The normalized Eisenstein integrals are holomorphic in a neighbourhoodof the imaginary axis.6. Fourier transform and wave packetsTheorem 7 ([CD2], [BS1] for F minimal): For f £ e(G/H,r), one has $ P f £§(ia*) ®A 2 (M/M fl H,TM), where Ipf is characterized by:((3%fi)(X),fi)= [ (f(x),E°(P,fi,X)(xj)dx, X£ia*, fi £ A 2 (M/M n H,T M ),JG/Hhere S (io*) is the usual Schwartz space.This theorem follows from the sharp estimates of Eisenstein integrals.Theorem 8 ([BCD]): If ^ is an element of$(ia*) A 2 (M/M n H,T M ), one has3° P £e(G/H,T), where :fp(x) :-- f E°(P,