  We analyse the `quantization commutes with reduction' problem (first studied
in physics by Dirac, and known in the mathematical literature also as the
Guillemin-Sternberg Conjecture) for the conjugate action of a compact connected
Lie group G on its own cotangent bundle T*G. This example is interesting
because the momentum map is not proper and the ensuing symplectic (or
Marsden-Weinstein quotient) T*G/Ad G is typically singular. In the spirit of
(modern) geometric quantization, our quantization of T*G (with its standard
Kaehler structure) is defined as the kernel of the Dolbeault-Dirac operator
(or, alternatively, the spin Dirac operator) twisted by the pre-quantum line
bundle. We show that this quantization of T*G reproduces the Hilbert space
found earlier by Hall (2002) using geometric quantization based on a
holomorphic polarisation. We then define the quantization of the singular
quotient T*G/Ad G$ as the kernel of the (twisted) Dolbeault-Dirac operator on
the principal stratum, and show that quantization commutes with reduction in
the sense that either way one obtains the same Hilbert space L^2(T)^{W(G,T)}.
