In this paper, we study the irreducible decomposition of a (ℂ[X];G)-module M for a quasi-affine spherical variety X of a connected reductive algebraic group G over ℂ. We show that for sufficiently large parameters, the decomposition of M with respect to G is reduced to the decomposition of the ‘fiber’ M/m(x0)M with respect to some reductive subgroup L of G. In particular, we obtain a method to compute the maximum value of multiplicities in M. Our main result is a generalization of earlier work by F. Satō in . We apply this result to branching laws of holomorphic discrete series representations with respect to symmetric pairs of holomorphic type. We give a necessary and sufficient condition for multiplicity-freeness of the branching laws.
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