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.