The identification between the Mumford algebro- geometric quotient and the symplectic quotient is a 'classical' one. It has a quantum counterpart relating the G-invariant algebra Aa to the quantization of the symplectic quotient, which we shall now explain.
In order to quantize a symplectic manifold X the symplectic form a) (divided by 2ir) has to be integral, so that ico is the curvature of a line-bundle if. One way to quantize X is (if possible) to pick a complex Kahler structure (so that w is the (1,1) form defined by the Kahler metric). This makes if into
we see that Af can be interpreted as the quantum Hilbert space of the symplectic quotient X//G at level k (i.e. replacing L by Lk). Note however that the algebra structure of Aa is not a symplectic invariant of X//G. In fact the algebra determines the complex manifold structure of X//G as the maximal ideal space.
The non-compact linear case X = C" leads of course to infinite-dimensional Hilbert spaces and possibly non-compact symplectic quotients X//G. The flatness here follows from that for C", by just restricting to the G-invariant part.
One important warning should be made at this stage. Although the G-invariant part of the space W of holomorphic sections of if on X can be identified with the holomorphic sections of L on X//G the inner product cannot easily be seen on X//G. By definition the norm of a section s of if is defined by integration over X. A G-invariant section is determined by its restriction to /i. '(0), since this meets the generic Gc- orbit. However, the norm involves a double-integral, first over the Gc-orbit and then over X // G. The integration over the Gc-orbit (which contributes the volume of the orbit) cannot be seen on the quotient space.
In the next chapter we shall meet an infinite-dimensional version of the story described in this chapter. This is the version required for the Jones-Witten theory and it has features which are not present in the finite-dimensional case. In particular it starts from a linear case (as for C") but the symplectic quotient is compact. The present chapter provides a general background and introduction to this infinite- dimensional case.