Darboux coordinates on k-orbits of lie algebras

DARBOUX COORDINATES ON K-ORBITS

OF LIE ALGEBRAS

Nguyen Viet Hai

Faculty of Mathematics, Haiphong University

Abstract. We prove that the existence of the normal polarization associated with a linear

functional on the Lie algebra is necessary and sufficient for the linear transition to local

canonical Darboux coordinates (p, q) on the coadjoint orbit.

1 Introduction

The method of orbits discovered in the pioneering works of Kirillov (see [K]) is a universal

base for performing harmonic analysis on homogeneous spaces and for constructing new

methods of integrating linear differential equations. Here we describle co-adjoint Orbits O

(the K-orbit) of a Lie algebra via linear algebraic methods. We deduce that in Darboux

coordinates (p, q) every element F ∈ g = Lie G can be considered as a function F˜ on O,

linear on pa’s-coordinates, i.e.

F˜ =

n

i=1

α

a

i

(q)pa + χi(q). (1)

Our main result is Theorem 3.2 in which we show that the existence of a normal polariza￾tion associated with a linear functional ξ is necessary and sufficient for the existence of local

canonical Darboux coordinates (p, q) on the K-orbit Oξ such that the transition to these coordi￾nates is linear in the “momenta” as equation (1). For the good strata, namely families of with

some good enough parameter space, of coadjoint orbits, there exist always continuous fields

of polarizations (in the sense of the representation theory), satisfying Pukanski conditions: for

each orbit Oξ and any point ξ in it, the affine subspace, orthogonal to some polarizations with

respect to the symplectic form is included in orbits themselves, i.e.

ξ + H⊥ ⊂ Oξ, dim H = n −

1

2

dim Oξ.

In the next section, we construct K-orbits via linear algebraic methods. In Section 3 we

consider Darboux coordinates on K-orbits of Lie algebras and give the proof of Theorem 3.2.

2 The description of K-orbits via linear algebraic methods

Let G be a real connected n-dimensional Lie group and G be its Lie algebra. The action

of the adjoint representation Ad∗

of the Lie group defines a fibration of the dual space G

∗

into even-dimensional orbits (the K-orbits). The maximum dimension of a K-orbit is n − r,

where r is the index (ind G) of the Lie algebra defined as the dimension of the annihilator of

1