4 1. Geometry of Coadjoint Orbits

For matrix groups it takes the form

(5 ) K∗(X)F = pV ([X, F ]) for X ∈ g, F ∈ V

g∗.

Remark 2. The notions of coadjoint representation and coadjoint orbit

can be defined beyond the realm of Lie groups in the ordinary sense.

Three particular cases are of special interest: infinite-dimensional groups,

algebraic groups over arbitrary fields and quantum groups (which are not

groups at all). In all three cases the ideology of the orbit method seems to

be very useful and often suggests the right formulations of important results.

We discuss below some examples of such results, although this subject

is outside the main scope of the book. ♥

1.2. Canonical form σΩ.

One feature of coadjoint orbits is eye-catching when you consider a few

examples: they always have an even dimension. This is not accidental, but

has a deep geometric reason.

All coadjoint orbits are symplectic manifolds. Moreover, each coad-

joint orbit possesses a canonical G-invariant symplectic structure. This

means that on each orbit Ω ⊂

g∗

there is a canonically defined closed non-

degenerate G-invariant differential 2-form σΩ.

In the next sections we give several explanations of this phenomenon

and here just give the definition of σΩ.

We use the fact that a G-invariant differential form ω on a homogeneous

G-manifold M = G/H is uniquely determined by its value at the initial

point m0 and this value can be any H-invariant antisymmetric polylinear

form on the tangent space Tm0 M.

Thus, to define σΩ it is enough to specify its value at some point F ∈ Ω,

which must be an antisymmetric bilinear form on TF Ω invariant under the

action of the group Stab F , the stabilizer of F .

Let stab(F ) be the Lie algebra of Stab(F ). We can consider the group

G as a fiber bundle over the base Ω G/Stab(F ) with projection

pF : G → Ω, pF (g) = K(g)F.

It is clear that the fiber above the point F is exactly Stab(F ). Consider the

exact sequence of vector spaces

0 → stab(F ) → g

(pF )∗

−→ TF (Ω) → 0

that comes from the above interpretation of G as a fiber bundle over Ω. It

allows us to identify the tangent space TF (Ω) with the quotient g/stab(F ).