Coadjoint representation

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In mathematics, the coadjoint representation ρ of a Lie group G is the dual of the adjoint representation. Therefore, if g denotes the Lie algebra of G, it is the action of G on the dual space to g. More geometrically, G acts by conjugation on its cotangent space at the identity element e, and this linear representation is ρ. Another geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on G.

The importance of the adjoint 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. A coadjoint orbit O(x) for x in the dual space g* of g may be defined either extrinsically, as the actual orbit G.x inside g*, or intrinsically as the homogeneous space G/H where H is the stabilizer of x; this distinction is worth making since the embedding of the orbit may be complicated. The coadjoint orbits are all symplectic manifolds with a natural 2-form inherited from g.

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.