ADJOINT ACTION

A CONTRIBUTION TO ENC. OF MATH., KLUWER

Toshiyuki KOBAYASHI

University of Tokyo

Adjoint action of a Lie group G -The linear action on the Lie algebra 𝔤 of G, denoted by Ad: G → GL(𝔤), that is defined as follows: Each element g of G induces an inner automorphism of the Lie group G by the formula Int(g): G → G, x ↦ gxg^{−1}. Its differential, Ad(g): 𝔤 → 𝔤, gives an automorphism of the Lie algebra 𝔤. The resulting linear representation Ad: G → GL(𝔤) is called the adjoint representation of a Lie group G.

The kernel of the adjoint representation, Ker(Ad), contains the center Z_{G} of G, and coincides with Z_{G} if G is connected. The image Ad(G) is called the adjoint group, which is a Lie subgroup of Aut(𝔤), the group of all automorphisms of the Lie algebra 𝔤.

The differential of the adjoint representation Ad: G → GL(𝔤) gives rise to a linear representation ad: 𝔤 → End(𝔤) of the Lie algebra 𝔤. It is given by the formula

(ad X)(Y) = [X, Y], X, Y ∈ 𝔤

and is called the adjoint representation of a Lie algebra 𝔤. The kernel, Ker(ad), coincides with the center of the Lie algebra 𝔤. The image, ad(𝔤), forms a subalgebra of Der(𝔤), the Lie algebra of all derivations of 𝔤. If 𝔤 is a semisimple Lie algebra (see Lie algebra, semisimple), then Ker(ad) = {0} and ad(𝔤) = Der(𝔤). An opposite extremal case is when G is abelian where Ker(ad) = 𝔤 and ad(𝔤) = {0}.

If G ⊂ GL(V) is a linear group acting on a vector space V, then we can regard 𝔤 ⊂ End(V) and the adjoint representation is written in terms of matrix computation:

Ad(g)Y = gY g^{−1}, (ad X)Y = XY − Y X, (g ∈ G, X, Y ∈ 𝔤).

The adjoint orbit through X ∈ 𝔤 (see Orbit) is defined to be Ad(G)X = {Ad(g)X : g ∈ G}, which is a submanifold of 𝔤.

Adjoint orbits for reductive Lie groups G have been particularly studied. The adjoint orbit Ad(G)X is called a semisimple orbit (respectively, a nilpotent orbit) if ad X is a semisimple (respectively, nilpotent) endomorphism of 𝔤. The set N := {X ∈ 𝔤 : ad X is a nilpotent endomorphism of 𝔤} is an algebraic variety in 𝔤, called the nilpotent variety. It is the union of a finite number of nilpotent orbits. On the other hand, Ad(G)X is a closed set in 𝔤 if and only if it is a semisimple orbit. A semisimple orbit Ad(G)X is called an elliptic orbit (respectively, hyperbolic orbit) if all eigenvalues of ad X are purely imaginary (respectively, real). Any

Typeset by AMS-TEX

elliptic orbit carries a G-invariant complex structure, while any hyperbolic obit carries a G-invariant paracomplex structure.

If G is compact, then all adjoint orbits are elliptic. For example, if G = U(n) (unitary group), then each adjoint orbit is biholomorphic to a generalized flag variety U(n)/(U(n_{1}) × ⋯ × U(n_{k})), for some partition (n_{1}, …, n_{k}) of n, and vise versa.

We write 𝔤/ Ad for the set of all adjoint orbits. The classical theory of the Jordan normal form of matrices (as well as the theory of other normal forms of matrices) can be interpreted as the classification of 𝔤/ Ad, for G = GL(n, ℂ). If G is a connected compact Lie group, then 𝔤/ Ad is bijective to 𝔞/W, the set of all orbits in 𝔞 of the Weyl group W, where 𝔞 is a maximal abelian subspace of 𝔤. This reduction is important in the Cartan-Weyl theory of the classification of irreducible representations as well as their characters for a compact Lie group.

Bi-invariant tensors on a Lie group G are described in terms of invariants under the adjoint action. For example, the left invariant measure on a connected Lie group G is also right invariant, if and only if det Ad(g) = 1 for any g ∈ G. Such a Lie group is called unimodular (cf. Haar measure on a locally compact group). This is the case if G is nilpotent or reductive.

We write 𝔤^{∗} for the dual vector space of 𝔤. The contragredient representation Ad^{∗} : G → GL(𝔤^{∗}) of the adjoint representation (Ad, 𝔤) is called the coadjoint representation. There is a close connection between irreducible unitary representations of G and coadjoint orbits (see Orbit method).

References

[1] N. Bourbaki, Elements of mathematics, Lie groups and Lie algebras, Chapter 2; 3, Addison-Wesley, 1975; (original in French).

[2] , Séminaire “Sophus Lie” de l’Ecole Normale Supérieure, 1954/1955. Théorie des algèbres de Lie. Topologie des groupes de Lie, 1955.

[3] T. Kobayashi, Harmonic analysis on homogeneous manifolds of reductive type and unitary representation theory, Translations, Series II, vol. 183, Amer. Math. Soc., 1998, pp. 1–31.

[4] F. Warner, Foundations of differentiable manifolds and Lie groups, GTM, vol. 94, 1983.

[5] G. Warner, Harmonic Analysis on Semisimple Lie groups I, Grundlehren, vol. 188, 1972.

Graduate School of Mathematical Sciences, University of Tokyo

Komaba, Meguro, Tokyo 153-8914 Japan

E-mail address: toshi@ms.u-tokyo.ac.jp