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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|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|orbit method]] (see [[#References|[2]]]). Each  $  G $-
orbit  $  \Omega $
in the coadjoint representation carries a canonical  $  G $-
invariant [[Symplectic structure|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 [[#References|[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|Lie algebra, local]]), in the space of smooth functions on  $  \mathfrak g  ^ {*} $(
see [[#References|[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 constant]]s of  $  \mathfrak g $.

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