Let K be a simply connected compact Lie group and T * (K) its cotangent bundle. We consider the problem of "quantization commutes with reduction" for the adjoint action of K on T * (K). We quantize both T * (K) and the reduced phase space using geometric quantization with half-forms. We then construct a geometrically natural map from the space of invariant elements in the quantization of T * (K) to the quantization of the reduced phase space. We show that this map is a constant multiple of a unitary map.
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