A unitary "quantization commutes with reduction" map for the adjoint action of a compact Lie group

Let $K$ be a simply connected compact Lie group and $T^{\ast }(K)$ its cotangent bundle. We consider the problem of “quantization commutes with reduction” for the adjoint action of $K$ on $T^{\ast }(K)\mathrm{.}$ We quantize both $T^{\ast }(K)$ 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 $T^{\ast }(K)$ to the quantization of the reduced phase space. We show that this map is a constant multiple of a unitary map.