Harmonic analysis on noncompact Riemannian symmetric spaces is in a sense equivalent to the theory of the horospherical transform. There are no horospheres on compact symmetric spaces, but we define a complex version of horospherical transform which plays a similar role for the harmonic analysis on them.
Notes on Stein-Sahi representations and some problems of non $L^2$ harmonic analysis
Harmonic analysis on spherical homogeneous spaces with solvable stabilizer
Harmonic analysis on the fourfold cover of the space of ordered triangles I: the invariant differentials
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