§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
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