Abstract
Let G be a connected real semisimple Lie group having a finite center and a compact Cartan subgroup T with Lie algebra to. Let ω be a G x T-invariant symplectic form on X = G x to. We incorporate Dirac cohomology into the geometric quantization of (X, ω) and study the resulting multiplicity-free unitary G x T -representation on a Hilbert space H(X, ω). We also perform symplectic reduction of (X, ω) and show that our quantization method satisfies the principle "quantization commutes with reduction". As an application we construct various models of discrete series.