20 1. Geometry of Coadjoint Orbits
derived from results in Appendix I.2.3) that
π1
(
SO(n, R)
)
=
Z for n = 2,
Z/2Z for n 3.
For small values of n the group Gn is isomorphic to one of the following
classical groups:
n 2 3 4 5 6
Gn R SU(2) SU(2) × SU(2) SU(2, H) SU(4)
For general n 3 the group Gn is the so-called spinor group Spin n. The
most natural realization of this group can be obtained using the differential
operators with polynomial coefficients on a supermanifold
R0|n
(see, e.g.,
[QFS], vol. I).
The point is that the operations Mk of left multiplication by an odd
coordinate ξk and the operations Dk of left differentiation with respect to
ξk satisfy the canonical anticommutation relations (CAR in short):
MiMj + MjMi = DiDj + DjDi = 0; MiDj + DjMi = δi,j · 1.
The Lie algebra g = so(n, R) is the set Asymn(R) of all antisymmetric
matrices X satisfying
Xt
= −X. The coadjoint action of G factors through
G.
The restriction of the bilinear form above to g is non-degenerate and
Ad(G)-invariant. So, the coadjoint representation is again equivalent to
the adjoint one. The description of coadjoint orbits here is the problem of
classification of antisymmetric matrices up to orthogonal conjugacy. In this
case the orbits are simply connected and have the form
Ω = SO(2n1 + · · · + 2nk + m)/U(n1) × · · · × U(nk) × SO(m).

Example 7. Let G = Sp(2n, R). It consists of matrices g satisfying
gtJng
= Jn where Jn =
0n −1n
1n 0n
.
The Lie algebra g consists of Jn-symmetric matrices X satisfying
XtJn
+ JnX = 0, or S = JnX is symmetric :
St
= S.
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