Let G be a real reductive Lie group in Harish-Chanclra's class, and a an involution on G commuting with a Cartan involution 0. Let H be an open subgroup of the subgroup G" of ovfixed points in G. Then G/H is a reductive symmetric space. Let q0 (resp. po)be the — 1-eigenspace of the involution ). Then we have a decomposition go = ho?qo = t0 ? p0 of the Lie algebra g0 of G (I)o (resp. Io) is the Lie algebra of H (resp. K)). G/H is called split if a maximal abelian subspace of q0 n Po is also maximal abelian in q0. Let SGIH) be the Frechetspace of smooth functions on G/H. The Poisson transform maps certain parabolically induced representations equivarianly into S'(G/H). I am interested in describing the image of the Poisson transform for split spaces G/H.
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