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.
Previous Page Next Page