§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
= GL+(n, R), the connected
component of unity in G, and denote by G0 the universal cover of
is worthwhile to note that G is homotopically equivalent to its maximal
compact subgroup O(n, R), while
is equivalent to SO(n, R) and
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
is diffeomorphic to
and G0 is diffeomorphic to
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
and even through
PSL(n, R). Therefore, coad-
joint orbits for
are just
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
Show that orbit is homotopic to one of the Stiefel
manifolds O(n1 + · · · + nk)/
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(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
Previous Page Next Page