The Principal Series for a Reductive Symmetric Space, Ii. Eisenstein Integrals. :)i ( 2 a Qc ; X 2 G)

Contents 0 Introduction 2 1 Notations and preliminaries 5 2 Invariant diierential operators 9 3 Deenition of the Eisenstein integral 13 4 Relation with the principal series 14 5 Finite dimensional class (1,1) representations 20 6 Functions of S-polynomial growth 23 7 S-genericity 25 8 Projection along innnitesimal characters 25 9 Estimates for j 30 10 Initial estimates for Eisenstein integrals 38 11 Families of spherical modules 39 12 Asymptotics of eigenfunctions 49 13 Properties of the coeecients 57 1 14 Expansions for Eisenstein integrals 64 15 The c-functions 67 16 A normalized Eisenstein integral 72 17 Schwartz functions 75 18 Uniform temperedness of eigenfunctions 81 19 The Fourier transform 92 20 Appendix: spectral projections 95 0 Introduction In this paper we develop a theory of Eisenstein integrals related to the principal series for a reductive symmetric space G=H: Here G is a real reductive group of Harish-Chandra's class, an involution of G and H an open subgroup of the group G of xed points for : The group G itself is a symmetric space for the leftright action of G G : we refer to this setting as the group case. Up to a normalization, our Eisenstein integrals generalize those of Harish-Chandra 18] associated with a minimal parabolic subgroup in the group case. In 4] we studied the principal series for G=H and their H-xed generalized vectors, motivated by the expectation that they constitute the building blocks for an explicit Plancherel decomposition of L 2 mc (G=H); the most continuous part of L 2 (G=H): Let K be a-stable maximal compact subgroup of G: Then on the level of left K-nite functions the decomposition should be described in terms of matrix coeecients of K-nite and H-xed vectors, i.e. in terms of Eisenstein integrals. In the present paper we concentrate on the Eisenstein integrals, and their asymptotic behaviour towards innnity. The main results are: (1) a unitarity result for c-functions, (2) uniform tempered estimates for the Eisenstein integral, and (related to this) (3) a functional equation for H-xed generalized vectors. These results will be applied in a forthcoming joint paper with H. Schlichtkrull 8] where the decomposition of L 2 mc (G=H) will be given. We shall now describe the results of this paper in more detail (for unspeciied notations see Section 1). The principal series for G=H is a series of parabolically induced representations ;; = Ind G P …