§4. The moment map 19

4.2. Some particular cases.

For most “classical” (or “natural”) groups the classification of coadjoint

orbits is equivalent to one or another already known problem. In some

cases, especially for infinite-dimensional groups, new interesting geometric

and analytic problems arise. We discuss here only a few examples. Some

others will appear later.

Example 5. Let G = GL(n, R). This group is neither connected

nor simply connected. So, we introduce

G0

= GL+(n, R), the connected

component of unity in G, and denote by G0 the universal cover of

G0.

It

is worthwhile to note that G is homotopically equivalent to its maximal

compact subgroup O(n, R), while

G0

is equivalent to SO(n, R) and

G0

is equivalent to Spin(n, R) for n ≥ 3. This follows from the well-known

unique decomposition g = kp, g ∈ G, k ∈ K, p ∈ P , where G = GL(n, R),

K = O(n, R) and P is the set of symmetric positive definite matrices.

The case n = 2 is a sort of exception. Here

G0

is diffeomorphic to

S1×R3

and G0 is diffeomorphic to

R4.

As we mentioned in Section 1.1, the Lie algebra g = Matn(R) possesses

an Ad(G)-invariant bilinear form

A, B = tr(AB).

Thus, the coadjoint representation is equivalent to the adjoint one. More-

over, because the center acts trivially, the coadjoint action of G0 factors

through

G0

and even through

G0/center

PSL(n, R). Therefore, coad-

joint orbits for

G0

are just

G0-conjugacy

classes in Matn(R).

Since the Lie algebra g = gl(n, R) R ⊕ sl(n, R) has no non-trivial

1-cocycles, the algebraic obstacle is absent. So, all homogeneous symplectic

G-manifolds are coverings of the

G0-conjugacy

classes.

Exercise

4.∗

Show that orbit is homotopic to one of the Stiefel

manifolds O(n1 + · · · + nk)/

(every

O(n1) × · · · × O(nk)

)

.

Hint. Use the information from Appendix I.2.3. ♣

Note that the fundamental group of an orbit is not necessarily commu-

tative, e.g. for n = 3 there are orbits homotopic to

O(3)/

(

O(1) × O(1) × O(1)

)

U(1, H)/{±1, ±i, ±j, ±k}.

The fundamental group of these orbits is the so-called quaternionic group

Q of order 8. ♦

Example 6. Let G = SO(n, R). Here again, the group is not simply

connected and we denote its universal cover by Gn. It is known (and can be