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. | 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, . F˜ = n αia (q)pa + χi (q). (1) i=1 Our main result is Theorem in which we show that the existence of a normal polarization 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 coordinates 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, . ξ + H ⊥ ⊂ Oξ , dim H = n − 1 dim Oξ . 2 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 . 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 .
