Coadjoint representation explained

In mathematics, the coadjoint representation

of a Lie group

is the dual of the adjoint representation. If

ak{g}

denotes the Lie algebra of

, the corresponding action of

ak{g}^*

, the dual space to

ak{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

The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups

a basic role in their representation theory is played by coadjoint orbits.In the Kirillov method of orbits, representations of

are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of

, which again may be complicated, while the orbits are relatively tractable.

Formal definition

Let

be a Lie group and

ak{g}

be its Lie algebra. Let

Ad:G → Aut(ak{g})

denote the adjoint representation of

. Then the coadjoint representation

Ad^*:G → GL(ak{g}^*)

is defined by

\langle

	*
Ad
	g

\mu,Y\rangle=\langle\mu,

	-1
Ad
	g

Y\rangle=\langle\mu,

Ad
	g^-1

Y\rangle

for

g\inG,Y\inak{g},\mu\inak{g}^*,

where

\langle\mu,Y\rangle

denotes the value of the linear functional

\mu

on the vector

Let

ad^*

denote the representation of the Lie algebra

ak{g}

ak{g}^*

induced by the coadjoint representation of the Lie group

. Then the infinitesimal version of the defining equation for

Ad^*

reads:

\langle

	*
ad
	X

\mu,Y\rangle=\langle\mu,-ad_XY\rangle=-\langle\mu,[X,Y]\rangle

for

X,Y\inak{g},\mu\inak{g}^*

where

is the adjoint representation of the Lie algebra

ak{g}

Coadjoint orbit

A coadjoint orbit

l{O}_\mu

for

\mu

in the dual space

ak{g}^*

ak{g}

may be defined either extrinsically, as the actual orbit

	*
Ad
	G

\mu

inside

ak{g}^*

, or intrinsically as the homogeneous space

G/G_\mu

where

G_\mu

is the stabilizer of

\mu

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

ak{g}^*

and carry a natural symplectic structure. On each orbit

l{O}_\mu

, there is a closed non-degenerate

-invariant 2-form

\omega\in

	2(l{O}
\Omega
	\mu)

inherited from

ak{g}

in the following manner:

	*
\omega
	X

\nu,

	*
ad
	Y

\nu):=\langle\nu,[X,Y]\rangle,\nu\inl{O}_\mu,X,Y\inak{g}

The well-definedness, non-degeneracy, and

-invariance of

\omega

follow from the following facts:

(i) The tangent space

T_\nul{O}_\mu=\{

	*
-ad
	X

\nu:X\inak{g}\}

may be identified with

ak{g}/ak{g}_\nu

, where

ak{g}_\nu

is the Lie algebra of

G_\nu

(ii) The kernel of the map

X\mapsto\langle\nu,[X, ⋅ ]\rangle

is exactly

ak{g}_\nu

(iii) The bilinear form

\langle\nu,[ ⋅ , ⋅ ]\rangle

ak{g}

is invariant under

G_\nu

\omega

is also closed. The canonical 2-form

\omega

is sometimes referred to as the Kirillov-Kostant-Souriau symplectic form or KKS form on the coadjoint orbit.

Properties of coadjoint orbits

The coadjoint action on a coadjoint orbit

(l{O}_\mu,\omega)

is a Hamiltonian

-action with momentum map given by the inclusion

l{O}_\mu\hookrightarrowak{g}^*

References

Kirillov, A.A., Lectures on the Orbit Method, Graduate Studies in Mathematics, Vol. 64, American Mathematical Society,,

Coadjoint representation explained

Formal definition

Coadjoint orbit

Properties of coadjoint orbits

See also

References