§1. Basic definitions 3

Matn(R) onto V parallel to

g⊥.

Then the coadjoint representation K can

be written in a simple form

(4) K(g): F → pV

(gFg−1).

Remark 1. If we could choose V invariant under Ad (G) (which we

can always assume for g semisimple or reductive), then we can omit the

projection pV in (4). ♥

Example 1. Denote by G the group of all (non-strictly) upper trian-

gular matrices g ∈ GL(n, R), i.e. such that gij = 0 for i j. Then the

Lie algebra g consists of all upper triangular matrices from Matn(R). The

space

g⊥

is the space of strictly upper triangular matrices X satisfying the

condition xij = 0 for i ≥ j.

We can take for V the space of all lower triangular matrices.

The projection pV in this case sends any matrix to its “lower part” (i.e.

replaces all entries above the main diagonal by zeros). Hence, the coadjoint

representation takes the form

K(g) : F → (g

Fg−1)lower

part.

Although this example has been known for a long time and has been

thoroughly studied by many authors, we still do not know how to classify

the coadjoint orbits for general n. ♦

Example 2. Let G = SO(n, R). Then g consists of all skew-symmetric

matrices X =

−Xt

from Matn(R).

Here we can put V = g and omit the projection pV in (4) (cf. Remark

1):

K(g)X = g · X ·

g−1.

Thus, the coadjoint representation is equivalent to the adjoint one and co-

incides with the standard action of the orthogonal group on the space of

antisymmetric bilinear forms. It is well known that a coadjoint orbit passing

through X is determined by the spectrum of X, which can be any multiset

in iR, symmetric with respect to the complex conjugation. Another conve-

nient set of parameters labelling the orbits is the collection of real numbers

{tr

X2,

tr

X4,

. . . , tr

X2k}

where k = [

n

2

]. ♦

We also give the formula for the infinitesimal version of the coadjoint

action, i.e. for the corresponding representation K∗ of the Lie algebra g in

g∗:

(5) K∗(X)F, Y = F, −ad(X) Y = F, [Y, X] .