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Let G be a connected, real, semisimple Lie group contained in its complexification G C , and let K be a maximal compact subgroup of G. We construct a K C -G double coset domain in G C , and we show that the action of G on the K-finite vectors of any irreducible unitary representation of G has a holomorphic extension to this domain. For the resultant holomorphic extension of K-finite matrix coefficients we obtain estimates of the singularities at the boundary, as well as majorant/minorant estimates along the boundary. We obtain L ∞ bounds on holomorphically extended automorphic functions on G/K in terms of Sobolev norms, and we use these to estimate the Fourier coefficients of combinations of automorphic functions in a number of cases, e.g. of triple products of Maaß forms.

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Abstract. Let G be an algebraic real reductive group and Z a real spherical G-variety, that is, it admits an open orbit for a minimal parabolic subgroup P . xWe prove a local structure theorem for Z. In the simplest case where Z is homogeneous, the theorem provides an isomorphism of the open P -orbit with a bundle Q× L S. Here Q is a parabolic subgroup with Levi decomposition L ⋉ U , and S is a homogeneous space for a quotient D = L/L n of L, where L n ⊆ L is normal, such that D is compact modulo center.

Abstract. Let Z be an algebraic homogeneous space Z = G/H attached to real reductive Lie group G. We assume that Z is real spherical, i.e., minimal parabolic subgroups have open orbits on Z. For such spaces we investigate their large scale geometry and provide a polar decomposition. This is obtained from the existence of simple compactifications of Z which is established in this paper.