Coadjoint representation

The representation of a Lie group $ G $ contragredient to the adjoint representation Ad of $ G $( cf. Adjoint representation of a Lie group). The coadjoint representation acts on the dual $ \mathfrak g ^ {*} $ of the Lie algebra $ \mathfrak g $ of the group $ G $.

If $ G $ is a real matrix group, i.e. a subgroup of $ \mathop{\rm GL} ( n, \mathbf R ) $, then $ \mathfrak g $ is a subspace of the space $ \mathop{\rm Mat} _ {n} ( \mathbf R ) $ of real matrices of order $ n $. Let $ \mathfrak g ^ \perp $ be the orthogonal complement of $ \mathfrak g $ relative to the bilinear form

$$ ( X, Y) \rightarrow \mathop{\rm tr} XY \ \ \mathop{\rm in} \ \mathop{\rm Mat} _ {n} ( \mathbf R ), $$

let $ V $ be some subspace of $ \mathop{\rm Mat} _ {n} ( \mathbf R ) $ complementary to $ \mathfrak g ^ \perp $, and let $ P $ be the projection onto $ V $ parallel to $ \mathfrak g ^ \perp $. Then $ \mathfrak g ^ {*} $ is identified with $ V $ and the coadjoint representation is given by the formula

$$ K ( g) X = \ P ( gXg ^ {-} 1 ),\ \ g \in G,\ \ X \in V. $$

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

$$ K ( X) Y = \ P ( XY - YX),\ \ X \in \mathfrak g ,\ \ Y \in V. $$

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

$$ \xi = K ( X) F,\ \ \eta = K ( Y) F. $$

Then

$$ B _ \Omega ( \xi , \eta ) = \ \langle F, [ X, Y] \rangle. $$

For every $ X \in \mathfrak g $, the vector field $ \xi _ {X} ( F) = K ( X) F $ is Hamiltonian with respect to $ B _ \Omega $; as its generator (generating function) one can take $ X $ itself, considered as a linear function on $ \mathfrak g ^ {*} $.

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 $ \mathfrak g ^ {*} $( see [3]). The coordinate expression for the Berezin bracket is

$$ \{ f _ {1} , f _ {2} \} = \ \sum _ {i, j, k } c _ {ij} ^ {k} x _ {k} \frac{\partial f _ {1} }{\partial x _ {i} } \frac{\partial f _ {2} }{\partial x _ {j} } , $$

where $ c _ {ij} ^ {k} $ are the structure constants of $ \mathfrak g $.

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