Let G be a compact connected semisimple Lie group and let H\subset G be a
closed connected subgroup such that rank(G)=rank(H) and G/H is a symmetric
space. Given an irreducible representation of H, we define a Dirac operator D
and determine the representations of G in the kernel of D. Moreover, we show
that any irreducible representation of G can be constructed in this way. Our
approach is similar to that of Parthasarathy.