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LIE GROUP ACTIONS 1. Smooth Actions Let M Be a Smooth Manifold

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LIE GROUP ACTIONS 1. Smooth Actions Let M Be a Smooth Manifold LECTURE 14: LIE GROUP ACTIONS 1. Smooth actions Let M be a smooth manifold, Diff(M) the group of diffeomorphisms on M. Definition 1.1. An action of a Lie group G on M is a homomorphism of groups τ : G ! Diff(M). In other words, for any g 2 G, τ(g) is a diffeomorphism from M to M such that τ(g1g2) = τ(g1) ◦ τ(g2): The action τ of G on M is smooth if the evaluation map ev : G × M ! M; (g; m) 7! τ(g)(m) is smooth. We will denote τ(g)(m) by g · m. Remark. What we defined above is the left action. One can also define a right action to be an anti-homomorphismτ ^ : G ! Diff(M), i.e. such that τ^(g1g2) =τ ^(g2) ◦ τ^(g1): −1 Any left action τ can be converted to a right actionτ ^ by requiringτ ^g(m) = τ(g )m: Example. S1 acts on R2 by counterclockwise rotation. Example. Any linear group in GL(n; R) acts on Rn as linear transformations. Example. If X is a complete vector field on M, then ρ : R ! Diff(M); t 7! ρt = exp(tX) is a smooth action of R on M. Conversely, every smooth action of R on M is given by this way: one just take X(p) to be X(p) =γ _ p(t), where γp(t) := ρt(p). Example. Any Lie group G acts on itself by many ways, e.g. by left multiplication, by right multiplication and by conjugation. For example, the conjugation action of G on G is given by −1 g 2 G c(g): G ! G; x 7! gxg : More generally, any Lie subgroup H of G can act on G by left multiplication, right multiplication and conjugation. Example. Any Lie group G acts on its Lie algebra g = TeG by the adjoint action: g 2 G Adg = (dc(g))e : g ! g: For example, one can show that the adjoint action of GL(n; R) on gl(n; R) is given by −1 A AdA : gl(n; R) ! gl(n; R);X 7! AXA : Similarly one can define the coadjoint action of G on g∗. 1 2 LECTURE 14: LIE GROUP ACTIONS Now suppose Lie group G acts smoothly on M. Definition 1.2. For any X 2 g, the induced vector field XM on M associated to X is d XM (m) = exp(tX) · m: dt t=0 Lemma 1.3. For each m 2 M and X 2 g, XM (m) = (devm)e(X); where evm is the restricted evaluation map evm : G ! M; g 7! g · m: Proof. For any smooth function f we have d (devm)e(X)f = Xe(f ◦ evm) = f(exp(tX) · m) = XM (m)(f): dt t=0 Remark. So from any smooth Lie group action of G on M we get a map 1 dτ : g ! Γ (M);X 7! XM : This can be viewed as the differential of the map τ : G ! Diff(M). One can prove that dτ is a Lie algebra anti-homomorphism. It is called the infinitesimal action of g on M. (There is a natural way to regard Γ1(M), an infinitely dimensional Lie algebra, as the \Lie algebra" of Diff(M).) Lemma 1.4. The integral curve of XM starting at m 2 M is γm(t) = exp(tX) · m: Proof. By definition, we have γm(0) = m, and d d γ_ m(t) = (exp tX · m) = (exp sX ◦ exp tX · m) = XM (γm(t)): dt ds s=0 As a consequence, we see that the induced vector field is \natural": Corollary 1.5. Let τ : G ! Diff(M) be a smooth action. Then for any X 2 g, τ(exp tX) = exp(tXM ); where ρt = exp(tXM ) is the flow on M generated by XM . LECTURE 14: LIE GROUP ACTIONS 3 2. Orbits and the quotient space Definition 2.1. Let τ : G ! Diff(M) be a smooth action. (1) The orbit of G through m 2 M is G · m = fg · m j g 2 Gg ⊂ M: (2) The stabilizer (also called the isotropic subgroup) of m 2 M is the subgroup Gm = fg 2 G j g · m = mg < G: Proposition 2.2. Let τ : G ! Diff(M) be a smooth action, m 2 M. Then (1) The orbit G · m is an immersed submanifold whose tangent space at m is Tm(G · m) = fXM (m) j X 2 gg: (2) The stabilizer Gm is a Lie subgroup of G, with Lie algebra gm = fX 2 g j XM (m) = 0g: Proof. (1) By definition, evm ◦ Lg = τg ◦ evm: Taking derivative at h 2 G, we get (devm)gh ◦ (dLg)h = (dτg)h·m ◦ (devm)h: Since (dLg)h and (dτg)h·m are bijective, the rank of (devm)gh equals the rank of (devm)h for any g and h. It follows that the map evm is of constant rank. By the constant rank theorem, its image, evm(G) = G · m, is an immersed submanifold of M. The tangent space of G · m at m is the image under devm of TeG = g. But for any X 2 g, we have (devm)e(X) = XM (m): So the conclusion follows. (2) Consider the map evm : G ! M; g 7! g · m; −1 then Gm = evm (m), so it is a closed set in G. It is a subgroup since τ is a group homomorphism. It follows from Cartan's theorem that Gm is a Lie subgroup of G. We have seen that the Lie algebra of the subgroup Gm is gm = fX 2 g j exp(tX) 2 Gm; 8t 2 Rg: It follows that exp(tX) · m = m for X 2 gm. Taking derivative at t = 0, we get gm ⊂ fX 2 g j XM (m) = 0g: Conversely, if XM (m) = 0, then γ(t) ≡ m, t 2 R, is an integral curve of the vector field XM passing m. It follows that exp(tX) · m = γ(t) = m, i.e. exp(tX) 2 Gm for all t 2 R. So X 2 gm. 4 LECTURE 14: LIE GROUP ACTIONS Obviously if m and m0 lie in the same orbit, then G · m = G · m0. We will denote the set of orbits by M=G. For example, if the action is transitive, i.e. there is only one orbit M = G · m, then M=G contains only one point. In general, M=G contains many points. We will equip with M=G the quotient topology. This topology might be very bad in general, e.g. non-Hausdorff. Example. Consider the natural action of R>0 on R by multiplications, then there are three orbits, f+; 0; −}. The open sets with respect to the quotient topology are f+g; {−}; f+:−}; f+; 0; −} and the empty set ;. So the quotient is not Hausdorff. Finally we list some conceptions to guarantee that each G · m is an embedded submanifold, and to guarantee that M=G is a smooth manifold. Definition 2.3. Let Lie group G acts smoothly on M. (1) The G-action is proper if the action map α : G × M ! M × M; (g; m) 7! (g · m; m) is proper, i.e. the pre-image of any compact set is compact. (2) The G-action is free if Gm = feg for all m 2 M. Remarks. (1) If the G-action on M is proper, then the evaluation map evm : G ! M; g 7! g · m is proper. In particular, for each m 2 M, Gm is compact. (2) One can prove that if G is compact, then any smooth G-action is proper. Example. Let H ⊂ G be a closed subgroup. Then the left action of H on G, τh : G ! G; g 7! hg is free. It is also proper since for any compact subset K ⊂ G×G, the preimage α−1(K) is contained in the image of K under the smooth map −1 f : G × G ! G × G; (g1; g2) 7! (g1g2 ; g2); which has to be compact. The action is not transitive unless H = G. The major theorem that we will not have time to prove is Theorem 2.4. Suppose G acts on M smoothly, then (1) If the action is proper, then (a) Each orbit G · m is an embedded closed submanifold of M. (b) The orbit space M=G is Hausdorff. (2) If the action is proper and free, then (a) The orbit space M=G is a smooth manifold. (b) The quotient map π : M ! M=G is a submersion. LECTURE 14: LIE GROUP ACTIONS 5 (3) If the action is transitive, then for each m 2 M, the map F : G=Gm ! M; gGm 7! g · m is a diffeomorphism. In particular, we see that if the G-action on M is transitive, then for any m 2 M, M ' G=Gm: Such a manifold is called a homogeneous space. Obviously any homogeneous space is of the form G=H for some Lie group G and some closed Lie subgroup H. Example. According to Gram-Schmidt, the natural action of O(n) on Sn−1 is transitive. It follows that Sn−1 is a homogeneous space. Moreover, if we choose m to be the \north n−1 pole" of S , then one can check that the isotropy group Gm is O(n − 1). It follows Sn−1 ' O(n)=O(n − 1): Example. Let k < n. Consider n Grk(n) = fk dimensional linear subspaces of R g: k Then O(n) acts transitively on Grk(n), and the isotropy group of the standard R inside Rn is O(k) × O(k − 1).
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