26 1. Geometry of Coadjoint Orbits

Or, more geometrically, the functions in question are constant along

leaves of D0 and holomorphic along leaves of E0/D0.

Remark 5. The subbundles P ⊂ T

CM

for which E = P + P is not

integrable are rather interesting, but until now have not been used in repre-

sentation theory. In this case CP

∞(M)

is still a subalgebra in

C∞(M).

The

nature of this subalgebra can be illustrated in the following simple

example.7

Let M =

R3

with coordinates x, y, t, and let P be spanned by a single

complex vector field ξ = ∂x + i∂y + (y − ix)∂t. In terms of the complex

coordinate z = x + iy it can be rewritten as ξ = ∂z − iz∂t. Then P is

spanned by ξ = ∂z + iz∂t, and E = C · ξ ⊕ C · ξ is not integrable since

[ξ, ξ] = 2i∂t / ∈ E.

The equation ξ f = 0 is well known and rather famous in analysis. The

point is that the corresponding non-homogeneous equation ξ f = g has no

solution for most functions g.

Some of the solutions to the equation ξ f = 0 have a transparent inter-

pretation. Consider the domain

D = {(z, w) ∈

C2

Im w ≥

|z|2}.

The boundary ∂D is diffeomorphic to

R3

and is naturally parametrized by

coordinates z and t = Re w. It turns out that the boundary values of

holomorphic functions in D satisfy the equation ξ f = 0. However, they do

not exhaust all the solutions which can be non-analytic in a real sense. ♥

5.2. Invariant polarizations on homogeneous symplectic mani-

folds.

In the representation theory of Lie groups one is interested mainly in

G-invariant polarizations of homogeneous symplectic G-manifolds.

We know also that the latter are essentially coadjoint orbits. In this

situation the geometric and analytic problems can be reduced to pure alge-

braic ones. Let G be a connected Lie group, and let Ω ⊂

g∗

be a coadjoint

orbit of G. Choose a point F ∈ Ω and denote by Stab(F ) the stabilizer of

F in G and by stab(F ) its Lie algebra.

Definition 5. We say that a subalgebra h ⊂ g is subordinate to a func-

tional F ∈

g∗

if the following equivalent conditions are satisfied:

(i) F

[h, h]

= 0;

(i ) the map X → F, X is a 1-dimensional representation of h.

Note that the codimension of h in g is at least

1

2

rk BF .

7In

this example M is not a symplectic manifold but a so-called contact manifold. In a sense,

contact manifolds are odd-dimensional analogues of symplectic manifolds.