Abstract
Let M = G/K be a compact homogeneous space of a compact semi-simple Lie group G. Let V be a complex homogeneous vector bundle on M. The group G acts naturally on the space of sections T{V) of V. By a theorem of Peter and Weyl, T(V) is a unitary direct sum of finitedimensional representations of G. It is an important problem to decompose T{V) into irreducible (j-modules. By the Frobenius reciprocity theorem, the problem is divided into two parts:
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