Abstract
Let X = ,G/K be a symmetric space and let Dk(G ) denote the ring of left-invariant differential operators on G which are right-invariant under K. A spherical function of X is a K-biinvariant eigenfunction ,p(g) of the ring Dk(G ) with the normalization ~o(e) = 1 [1, 2, 31. For noncompact symmetric spaces, i.e., for a noncompact connected real semisimple Lie group G with finite center and for its maximal compact subgroup K, the following Harish-Chandra formula gives all spherical functions on G/K as integrals over K:
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