2 1. Geometry of Coadjoint Orbits

the adjoint representation is simply the matrix conjugation:

(1) Ad(g)X = g · X ·

g−1,

X ∈ g, g ∈ G.

The same formula holds in the general case if we accept the matrix notation

introduced in Appendix III.1.1.

Consider now the vector space dual to g. We shall denote it by

g∗.

Recall

that for any linear representation (π, V ) of a group G one can define a dual

representation

(π∗,

V

∗)

in the dual space V

∗:

π∗(g)

:=

π(g−1)∗

where the asterisk in the right-hand side means the dual operator in V

∗

defined by

A∗f,

v := f, Av for any v ∈ V, f ∈ V

∗.

In particular, we have a representation of a Lie group G in

g∗

that is dual

to the adjoint representation in g. This representation is called coadjoint.

Since this notion is very important, and also for brevity, we use the

special

notation1

K(g) for it instead of the full notation

Ad∗(g)

=

Ad(g−1)∗.

So, by definition,

(2) K(g)F, X = F,

Ad(g−1)X

where X ∈ g, F ∈

g∗,

and by F, X we denote the value of a linear func-

tional F on a vector X.

For matrix groups we can use the fact that the space Matn(R) has a

bilinear form

(3) (A, B) = tr (AB),

which is non-degenerate and invariant under conjugation. So, the space

g∗,

dual to the subspace g ⊂ Matn(R), can be identified with the quotient

space

Matn(R)/g⊥.

Here the sign

⊥

means the orthogonal complement with

respect to the form ( , ):

g⊥

= {A ∈ Matn(R) (A, B) = 0 for all B ∈ g}.

In practice the quotient space is often identified with a subspace V ⊂

Matn(R) that is transversal to

g⊥

and has the complementary dimension.

Therefore, we can write Matn(R) = V ⊕

g⊥.

Let pV be the projection of

1In Russian the word “coadjoint” starts with k.