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This paper contains the lectures given by the author at the Winter School on ``Geometry and Physics'' in Srn\'{\i} 2001. These lectures are based on two recent works of the author with A. Kor\'anyi and on a forthcoming paper with K. Johnson and A. Kor\'anyi. In the paper results are presented concerning equivariant differential operators on homogeneous spaces (section 1), first order equivariant differential operators on boundaries of symmetric spaces (section 2), the Poisson transform (section 3) and complex hyperbolic space (section 4).
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