§5. Polarizations 23

Here as usual the sign denotes the sum over cyclic permutations of three

variables.

We identify

g∗

with

g∗

⊕ R and denote its general element by (F, α).

The coadjoint action of G reduces to an action of G because the central

subgroup A acts trivially.

Lemma 7. The coadjoint actions of G on

g∗

and on

g∗

are related by the

formula

(29) K(g) (F, α) = (K(g)F + α · S(g), α)

where S is a 1-cocycle on the group G with values in

g∗,

i.e. a solution to

the cocycle equation

(30) S(g1g2) = S(g1) + K(g1)S(g2).

Proof. Since the extension g is central, the action of G on the quotient

space

g∗/g∗

is trivial. Therefore, K(g) preserves hyperplanes α = const and

on the hyperplane α = 0 coincides with the ordinary coadjoint action of

G on

g∗.

Hence, K(g) has the form (29) for some map S : G →

g∗.

The

cocycle property (30) follows directly from multiplicativity of the map K.

Exercise 5. Show that for any connected Lie group G the map S in (29)

can be reconstructed from the cocycle c(X, Y ) entering in (27) as follows.

For any g ∈ G the cocycles c(X, Y ) and c (X, Y ) = c(Ad g X, Ad g Y )

are

equivalent.6

Thus, we can write

(31) c(Ad g X, Ad g Y ) = c(X, Y ) + Φ(g), [X, Y ] .

From this we derive that

Ad g(X, a) = (Ad g X, a + Φ(g), X )

and, consequently, (29) follows with S(g) =

Φ(g−1).

5. Polarizations

5.1. Elements of symplectic geometry.

We shall use here the general facts about symplectic manifolds from

Appendix II.3: the notions of skew gradient, Poisson brackets, etc.

In the general scheme of geometric quantization (which is a quantum

mechanical counterpart of the construction of unirreps from coadjoint orbits)

the notion of a polarization plays an important role.

6The infinitesimal version of this statement follows directly from (28).