Coadjoint representation

The representation of a Lie group contragredient to the adjoint representation Ad of (cf. Adjoint representation of a Lie group). The coadjoint representation acts on the dual of the Lie algebra of the group .

If is a real matrix group, i.e. a subgroup of , then is a subspace of the space of real matrices of order . Let be the orthogonal complement of relative to the bilinear form

let be some subspace of complementary to , and let be the projection onto parallel to . Then is identified with and the coadjoint representation is given by the formula

The corresponding representation of the Lie algebra is also called the coadjoint representation. In the case above it is defined by

The coadjoint representation plays a fundamental role in the orbit method (see [2]). Each -orbit in the coadjoint representation carries a canonical -invariant symplectic structure. In other words, on each orbit there is a uniquely defined non-degenerate -invariant closed differential -form (whence it follows that all -orbits in the coadjoint representation are even-dimensional). To obtain an explicit expression for one proceeds as follows. Let , let be the orbit through the point and let be tangent vectors to at . There exist and in such that

Then

For every , the vector field is Hamiltonian with respect to ; as its generator (generating function) one can take itself, considered as a linear function on .

The stabilizer of a point with orbit of maximal dimension in the coadjoint representation is commutative [1]. The Poisson bracket arising on each orbit generates a single Berezin bracket, which defines the structure of a local Lie algebra (cf. Lie algebra, local), in the space of smooth functions on (see [3]). The coordinate expression for the Berezin bracket is

where are the structure constants of .

References

[1]	P. Bernal, et al., "Représentations des groupes de Lie résolubles" , Dunod (1972)
[2]	A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)
[3]	A.A. Kirillov, "Local Lie algebras" Russian Math. Surveys , 31 : 4 (1976) pp. 55–75 Uspekhi Mat. Nauk , 31 : 4 (1976) pp. 57–76