In mathematics, the coadjoint representation
K
of a Lie group
G
is the dual of the adjoint representation. If
g
denotes the Lie algebra of
G
, the corresponding action of
G
on
g
∗
, the dual space to
g
, is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on
G
.
The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups
G
a basic role in their representation theory is played by coadjoint orbit. In the Kirillov method of orbits, representations of
G
are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of
G
, which again may be complicated, while the orbits are relatively tractable.
Let
G
be a Lie group and
g
be its Lie algebra. Let
A
d
:
G
→
A
u
t
(
g
)
denote the adjoint representation of
G
. Then the coadjoint representation
K
:
G
→
A
u
t
(
g
∗
)
is defined as
A
d
∗
(
g
)
:=
A
d
(
g
−
1
)
∗
. More explicitly,
⟨
K
(
g
)
F
,
Y
⟩
=
⟨
F
,
A
d
(
g
−
1
)
Y
⟩
for
g
∈
G
,
Y
∈
g
,
F
∈
g
∗
,
where
⟨
F
,
Y
⟩
denotes the value of a linear functional
F
on a vector
Y
.
Let
K
∗
denote the representation of the Lie algebra
g
on
g
∗
induced by the coadjoint representation of the Lie group
G
. Then for
X
∈
g
,
K
∗
(
X
)
=
−
a
d
(
X
)
∗
where
a
d
is the adjoint representation of the Lie algebra
g
. One may make this observation from the infinitesimal version of the defining equation for
K
above, which is as follows :
⟨
K
∗
(
X
)
F
,
Y
⟩
=
⟨
F
,
−
a
d
(
X
)
Y
⟩
for
X
,
Y
∈
g
,
F
∈
g
∗
. .
A coadjoint orbit
Ω
:=
O
(
F
)
for
F
in the dual space
g
∗
of
g
may be defined either extrinsically, as the actual orbit
K
(
G
)
(
F
)
inside
g
∗
, or intrinsically as the homogeneous space
G
/
S
t
a
b
(
F
)
where
S
t
a
b
(
F
)
is the stabilizer of
F
with respect to the coadjoint action; this distinction is worth making since the embedding of the orbit may be complicated.
The coadjoint orbits are submanifolds of
g
∗
and carry a natural symplectic structure. On each orbit
Ω
, there is a closed non-degenerate
G
-invariant 2-form
σ
Ω
inherited from
g
in the following manner. Let
B
F
be an antisymmetric bilinear form on
g
defined by,
B
F
(
X
,
Y
)
:=
⟨
F
,
[
X
,
Y
]
⟩
,
X
,
Y
∈
g
Then one may define
σ
Ω
∈
H
o
m
(
Λ
2
(
Ω
)
,
R
)
by
σ
Ω
(
F
)
(
K
∗
(
X
)
(
F
)
,
K
∗
(
Y
)
(
F
)
)
:=
B
F
(
X
,
Y
)
.
The well-definedness, non-degeneracy, and
G
-invariance of
σ
Ω
follow from the following facts:
(i) The tangent space
T
F
(
Ω
)
may be identified with
g
/
s
t
a
b
(
F
)
, where
s
t
a
b
(
F
)
is the Lie algebra of
S
t
a
b
(
F
)
.
(ii) The kernel of
B
F
is exactly
s
t
a
b
(
F
)
.
(iii)
B
F
is invariant under
S
t
a
b
(
F
)
.
σ
Ω
is also closed. The canonical 2-form
σ
Ω
is sometimes referred to as the Kirillov-Kostant-Souriau symplectic form on the coadjoint orbit.
The coadjoint action on a coadjoint orbit
(
Ω
,
σ
Ω
)
is a Hamiltonian
G
-action with moment map given by
Ω
↪
g
∗
.
