We present a new method of calculating intertwining operators between principal series representations of semisimple Lie groups G. Working in the compact realization we find the eigenvalues of the operators on the K-types, and give several examples. Among the advantages of our method is its applicability to bundlevalued cases.1996 Academic Press, Inc.

The Fourier coefficients of a smooth K-invariant function on a compact symmetric space M = U/K are given by integration of the function against the spherical functions. For functions with support in a neighborhood of the origin, we describe the size of the support by means of the exponential type of a holomorphic extension of the Fourier coefficients.

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In this paper we summarize and give examples of a generalization of the coorbit space theory initiated in the 1980's by H.G. Feichtinger and K.H. Gröchenig. Coorbit theory has been a powerful tool in characterizing Banach spaces of distributions with the use of integrable representations of locally compact groups. Examples are a wavelet characterization of the Besov spaces and a characterization of some Bergman spaces by the discrete series representation of SL 2 (R).We present examples of Banach spaces which could not be covered by the previous theory, and we also provide atomic decompositions for an example related to a non-integrable representation.