Özet:
This thesis is concerned with the spherical analysis of two di erent Olshanski pairs, one of which is related to Heisenberg groups, and the other to the automorphism groups of homogeneous trees. The spherical functions of positive type on the in nite dimensional Heisenberg group H(1) which are invariant under the natural action of the in nite dimensional unitary group U(1) are determined. On the other hand, we consider an Olshanski pair which is constructed from the stabilizers of the horicycles of homogeneous trees of nite degree, where the horicycles form a partition of the set of vertices of the tree, and then we nd all spherical functions of this pair. Finally, we give realizations of the corresponding irreducible unitary representations.
