1. Smooth actions Let M be a smooth , Diff(M) the group of diffeomorphisms on M. Definition 1.1. An action of a G on M is a homomorphism of groups τ : G → Diff(M). In other words, for any g ∈ 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 ˆ : G → Diff(M), i.e. such that

τˆ(g1g2) =τ ˆ(g2) ◦ τˆ(g1). −1 Any left action τ can be converted to a right actionτ ˆ by requiringτ ˆg(m) = τ(g )m. Example. S1 acts on R2 by counterclockwise . 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 ∈ G c(g): G → G, x 7→ gxg . More generally, any Lie 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 ∈ 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∗.


Now suppose Lie group G acts smoothly on M.

Definition 1.2. For any X ∈ 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 ∈ M and X ∈ 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 of G on M we get a map

∞ 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 Γ∞(M), an infinitely dimensional Lie algebra, as the “Lie algebra” of Diff(M).)

Lemma 1.4. The integral curve of XM starting at m ∈ 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 ∈ 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 Definition 2.1. Let τ : G → Diff(M) be a smooth action. (1) The of G through m ∈ M is G · m = {g · m | g ∈ G} ⊂ M. (2) The stabilizer (also called the isotropic subgroup) of m ∈ M is the subgroup

Gm = {g ∈ G | g · m = m} < G. Proposition 2.2. Let τ : G → Diff(M) be a smooth action, m ∈ M. Then (1) The orbit G · m is an immersed submanifold whose tangent space at m is

Tm(G · m) = {XM (m) | X ∈ g}.

(2) The stabilizer Gm is a Lie subgroup of G, with Lie algebra

gm = {X ∈ g | XM (m) = 0}.

Proof. (1) By definition,

evm ◦ Lg = τg ◦ evm. Taking derivative at h ∈ 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 ∈ 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 in G. It is a subgroup since τ is a . 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 = {X ∈ g | exp(tX) ∈ Gm, ∀t ∈ R}.

It follows that exp(tX) · m = m for X ∈ gm. Taking derivative at t = 0, we get

gm ⊂ {X ∈ g | XM (m) = 0}.

Conversely, if XM (m) = 0, then γ(t) ≡ m, t ∈ 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) ∈ Gm for all t ∈ R. So X ∈ 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 . 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, {+, 0, −}. The open sets with respect to the quotient topology are {+}, {−}, {+.−}, {+, 0, −} and the ∅. 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 = {e} for all m ∈ 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 ∈ 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 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 ∈ 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 ∈ M,

M ' G/Gm. Such a manifold is called a . 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 group Gm is O(n − 1). It follows Sn−1 ' O(n)/O(n − 1). Example. Let k < n. Consider n Grk(n) = {k dimensional linear subspaces of R }. 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). It follows

Grk(n) ' O(n)/(O(k) × O(n − k)) n−1 The manifold Grk(n) is called a manifold. Note that Gr1(n) = RP is the real .