Chapter 1

Geometry of

Coadjoint Orbits

We start our book with the study of coadjoint orbits. This notion is the

main ingredient of the orbit method. It is also the most important new

mathematical object that has been brought into consideration in connection

with the orbit method.

1. Basic definitions

By a coadjoint orbit we mean an orbit of a Lie group G in the space

g∗

dual to g = Lie(G). The group G acts on

g∗

via the coadjoint representation,

dual to the adjoint one (see the definition below and also Appendix III.1.1).

In this chapter we consider the geometry of coadjoint orbits and discuss

the problem of their classification.

1.1. Coadjoint representation.

Let G be a Lie group. It is useful to have in mind the particular case

when G is a matrix group, i.e. a subgroup and at the same time a smooth

submanifold of GL(n, R).

Let g = Lie(G) be the tangent space Te(G) to G at the unit point e.

The group G acts on itself by inner automorphisms: A(g): x → g

xg−1.

The point e is a fixed point of this action, so we can define the derived map

(

A(g)

)

∗

(e): g → g. This map is usually denoted by Ad(g).

The map g → Ad(g) is called the adjoint representation of G. In the

case of a matrix group G the Lie algebra g is a subspace of Matn(R) and

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http://dx.doi.org/10.1090/gsm/064/01