4.4 Co-adjoint orbits

We conclude this chapter with some additional remarks about the generalized symplectic quotients where MA is a co-adjoint orbit, i.e. a G-orbit in Lie(G)*. According to a well-known general result of Kirillov these co-adjoint orbits are the homogeneous symplectic manifolds of G. They are of the form G/H where H is the centralizer of a torus in G. Moreover they have natural complex Kahler structures which for 'integral' A are projective algebraic. Their quantizations give the irreducible representations of G and so the set of integral orbits may be identified with G. If A* denotes the representation dual to A then one can verify that

 

 

The moment map for MA is just its natural embedding in Lie(G)*. Now compare the moment maps

 

Thus the quantization of YK should be the G-invariant part of the quantization of X x MK. But this is just the A -covariant part of the quantization X of X, i.e. HomG (A, X).

Thus, analysing the moment maps over the different integral orbits of L(G)* is the classical counterpart of decomposing the quantization of X as a G-module.