Quantization Commutes with Reduction on Compact Lie Groups under the Adjoint Action

Over the last half-century mathematicians and physicists alike have done quite a bit of work on the problem of quantization commutes with reduction and its generalizations. It turns out that in general quantization commutes with reduction, but only weakly; that is the map between the first-quantized-then-reduced space and the first-reduced-then-quantized space is only a vector isomorphism, not necessarily unitary and therefore respecting the physically relevant inner product. In this dissertation we use the techniques developed by Hall and Kirwin to show that if our starting manifold is a compact, simply connected Lie group under the adjoint action, then the map we get is, in fact, unitary.