Abstract

Generalized spherical functions are defined to be joint eigenfunctions of some invariant differential operators with fixed K-type on both sides on a semisimple Lie group. One of our main results is the same dimension formula for all spaces of generalized spherical functions, which is independent of eigenvalues. The other main result is proof of the existence of a global basis for all spaces of generalized spherical functions, which is holomorphic in the parameter for the eigenvalues. As an application, we calculate the distribution characters of K-bi-finite eigenfunctions of the center of enveloping algebra for split groups.

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