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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
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