4 BORIS FEDOSOV

G-invariant functions modulo the ideal generated by

J.L

coincides with the algebra

C

00

(B) with a Poisson structure defined by the symplectic form

WB.

What happens in deformation quantization? First of all, there exists a canonical

construction of a deformation quantization on a symplectic manifold parametrized

by a closed two-form

i

f2

=

hW

+

Wo

+

hw1

+ ... ,

called the Weyl curvature. Clearly, it may be chosen G-invariant if the form

n

is

invariant. Let *M and *B be the canonical star-products on

M

and

B

with the

Weyl curvatures

.

.

z z

OM=

hWM, f2B

=

hWB

respectively. Then we have a reduction theorem in deformation quantization which

sounds almost identically with the classical one, namely:

THEOREM

2.1. The algebra

(A(B) :=

C

00

(B)[[h]], *B) is isomorphic to the

quantum reduced algebra, which is the quotient of the subalgebra of G-invariants

(A( M),

*

M)

0

modulo the *-ideal generated by the moment map

J.L·

This is a statement of the type "reduction commutes with quantization", known

in geometric quantization as the Guillemin-Sternberg conjecture. While in the geo-

metric quantization framework it is a serious theorem, for deformation quantization

it is much more simple. Besides, it gives a natural correspondence between original

and reduced observables.

3. Non-trivial coefficients

Deformation quantization admits a natural generalization to the case of non-

trivial coefficients at least in the case of a symplectic manifold. For example, the

coefficients may be matrices and they may form a non-trivial bundle over

M,

i.e., we

may consider a deformation quantization with a coefficient bundle K =Hom (E, E)

for some vector bundle E over M. Moreover, in practice we need even more compli-

cated bundles. For instance, for the reduction theorem we need a bundle over the

base

B

whose fibers are quantum observables on

T*G.

The changes are as follows.

The algebra of classical observables is now a non-commutative algebra of sections of

the coefficient bundle

c= (

M, K), and we replace the condition 3 of Definition 1.1

by

ih ..

C1(a,b)

= -

2

w•3

Oiaaib

where 8 now means a covariant derivative in the coefficient bundle.

A canonical construction of deformation quantization with non-trivial coeffi-

cients is quite general. It is meaningful even in infinite-dimensional setting of mod-

uli spaces of connections on a Riemann surface, leading to a quantization procedure

commuting with reduction.

4. Trace and index

Besides the algebra of observables, a mechanical system (classical or quantum)

is characterized by its states: linear functionals on the algebra of observables with

some further additional properties. Usually one imposes a kind of positivity. We

will not go deeper into the subject, instead we consider a trace functional, which

may be introduced without positivity.