invariant forms on representations of compact groups

Let G be a real Lie group. TFAE:

  1. 1.

    Every real representation of G has an invariantMathworldPlanetmath positive definite form, and G has at least one faithful representationMathworldPlanetmath.

  2. 2.

    One faithful representation of G has an invariant positive definite form.

  3. 3.

    G is compactPlanetmathPlanetmath.

Also, any group satisfying these criteria is reductive, and its Lie algebraMathworldPlanetmath is the direct sumMathworldPlanetmathPlanetmathPlanetmath of simple algebras and an abelianMathworldPlanetmath algebraPlanetmathPlanetmath (such an algebra is often called reductive).


(1)(2): Obvious.

(2)(3): Let Ω be the invariant form on a faithful representation V. Let then representationPlanetmathPlanetmath gives an embeddingMathworldPlanetmathPlanetmath ρ:GSO(V,Ω), the group of automorphismsPlanetmathPlanetmathPlanetmath of V preserving Ω. Thus, G is homeomorphic to a closed subgroup of SO(V,Ω). Since this group is compact, G must be compact as well.

(Proof that SO(V,Ω) is compact: By inductionMathworldPlanetmath on dimV. Let vV be an arbitrary vector. Then there is a map, evaluation on v, from SO(V,Ω)SdimV-1V (this is topologically a sphere, since (V,ω) is isometric to dimV with the standard norm). This is a a fiber bundle, and the fiber over any point is a copy of SO(v,Ω), which is compact by the inductive hypothesis. Any fiber bundle over a compact base with compact fiber has compact total space. Thus SO(V,Ω) is compact).

(3)(1): Let V be an arbitrary representation of G. Choose an arbitrary positive definite form Ω on V. Then define


where dg is Haar measure (normalized so that G𝑑g=1). Since K is compact, this gives a well defined form. It is obviously bilinear, bSO(V,Ω)y the linearity of integration, and positive definite since


Furthermore, Ω~ is invariant, since


For representation ρ:TGL(V) of the maximal torus TK, there exists a representation ρ of K, with ρ a T-subrepresentation of ρ. Also, since every conjugacy classMathworldPlanetmathPlanetmath of K intersects any maximal torus, a representation of K is faithful if and only if it restricts to a faithful representation of T. Since any torus has a faithful representation, K must have one as well.

Given that these criteria hold, let V be a representation of G, Ω is positive definite real form, and W a subrepresentation. Now consider


By the positive definiteness of Ω, V=WW. By induction, V is completely reducible.

Applying this to the adjoint representationMathworldPlanetmath of G on 𝔤, its Lie algebra, we find that 𝔤 in the direct sum of simple algebras 𝔤1,,𝔤n, in the sense that 𝔤i has no proper nontrivial ideals, meaning that 𝔤i is simple in the usual sense or it is abelian. ∎

Title invariant forms on representations of compact groups
Canonical name InvariantFormsOnRepresentationsOfCompactGroups
Date of creation 2013-03-22 13:23:40
Last modified on 2013-03-22 13:23:40
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 11
Author bwebste (988)
Entry type Theorem
Classification msc 54-00