§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).
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