Differential Geometry

Coadjoint orbits are examples of symplectic manifolds. This post aims at defining coadjoint orbits with a review on Lie group action.

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January 12, 2021

In this post, we will denote by $$G$$ a Lie group. A very good exposition on Lie group action can be found in Section 9.3 in .

## 1 Lie Group Actions

Let $$X$$ be a smooth manifold. A Lie group action of $$G$$ on $$X$$ is defined to be a smooth group morphism $$\Psi: G\to \Diff(X)$$ in the sense that $$\psi: G\times X\to X$$, $$\psi(g, x)=\Psi_g(x)$$ is a smooth map, where $$\Diff(X)$$ is the group of diffeomorphisms of $$X$$ and $$\Psi_g:=\Psi(g)$$.

Generally, a group action on a space is a group homomorphism from a given group to the group of transformations of the space.

For a point $$x$$ in $$X$$, consider the evaluation map: \begin{aligned} \omicron_x: G&\to X\\ g&\mapsto g\cdot x:=\Psi_g(x). \end{aligned} This map $$\omicron_x$$ is known as the orbit map which is smooth by the definition of smooth action.

The $$G$$-orbit (or simply the orbit) of $$x$$ in $$X$$ is the image of $$\omicron_x$$: $\Orb_x=\omicron_x(G)=\{g\cdot x\vert g\in G \},$ where $$g\cdot x:=\Psi_g(x)$$.

The stabilizer of $$x$$ in $$X$$ is the preimage of $$\omicron$$ $G_x=\omicron_x^{-1}(x)=\{g\vert g\cdot x=x\}\subset G.$

Since the action $$\Psi$$ is smooth, the stabilizer $$G_x=\omicron_x^{-1}(x)$$ is a closed submanifold of $$G$$. Moreover, $$G_x$$ is a subgroup of $$G$$. Hence $$G_x$$ is a Lie subgroup by Cartan’s Theorem.

Note by $$L_g: G\to G$$, $$h\mapsto gh$$ the left multiplication. Then \begin{aligned} (\omicron_x\circ L_g)(h)=&\omicron_x(gh)=\Psi_{gh}(x)\\ =&(\Psi_g\circ\Psi_h)(x)=\Psi_g(\Psi_h(x))\\ =&\Psi_g(\omicron_x(h))=(\Psi_g\circ\omicron_x)(h). \end{aligned}

Therefore, $$(\d\omicron_x)_{gh}\circ (\d L_g)_h=(\d \Psi_g)_{h\cdot x}\circ(\d\omicron_x)_h$$ for any $$g, h$$ in $$G$$. Since $$L_g$$ and $$\Psi_g$$ are both diffeomorphisms, the equality implies that $$\d\omicron_x$$ is a function on $$G$$ of constant rank. By the constant rank theorem, the orbit $$\Orb_x$$ is an immersed submanifold. The manifold structure on $$\Orb_x$$ is induced by the orbit map $$\omicron_x$$. In general, the orbit $$\Orb_x$$ is not a submanifold of $$X$$, in the subset topology.

However, if $$G$$ is compact, then $$\Orb_x$$ is a submanifold of $$X$$ (Theorem 2.3 in ).

Denote by $$G/G_x$$ the set of cosets. Then $$G/G_x$$ is a smooth manifold. The smooth map $$G\to G/G_x$$ is a locally trivial fiber bundle (Theorem I.4.8. in ). Consequently, the tangent space of $$T_e(G/G_x)$$ can be identified with $$\Lg/\Lg_x=T_e(G)/T_e(G_x)$$.

Induced by the orbit map $$\omicron_x$$, we get a bijection \begin{aligned} G/G_x&\to \Orb_x\\ gG_x&\mapsto g\cdot x. \end{aligned} The $$G$$-action on $$X$$ induces a $$G$$-action on $$\Orb_x$$. Moreover, the $$G$$-action on $$\Orb_x$$ is transitive with the stabilizer $$G_x$$. Hence $$G/G_x$$ is diffeomorphic to $$\Orb_x$$ (Theorem 3.3 in ).

### 1.1 Conjugation Action

Consider the map $$\Psi: G\to\Aut(G)$$ sending an element $$g$$ to the inner automorphism $$\Psi_g$$, i.e. $$\Psi_g(h)=ghg^{-1}$$. Then $$\Psi$$ is a group homomorphism. Indeed, for any $$x, g, h\in G$$, we have $\Psi(gh)(x)=(gh)x(gh)^{-1}=g(hxh^{-1})g^{-1}=\Psi_g\Psi_h(x).$ Therefore, $$\Psi(gh)=\Psi_g\Psi_h$$.

Moreover, since $$G$$ is a Lie group, the morphism $$\Psi$$ is smooth. Indeed, $$\Psi$$ can be viewed as the composition of multiplication and inverse operations: $\begin{array}{cccccc} G\times G &\to & G\times G & \to & G\times G & \to & G\\ (g, x) &\mapsto & (gx, g) & \mapsto & (gx, g^{-1}) & \mapsto & gxg^{-1} \end{array}$

This action is known as the conjugation action.

## 2 Representations of a Lie Group

A representation of a Lie group $$G$$ is a continuous group action on a vector space $$V$$: a group homomorphism $\Pi: G\to \operatorname{GL}(V)$ that is continuous in the sense that $$G\times V\to V$$, $$(g, v)\mapsto \Pi(g)(v)$$ is continuous.

### 2.1 Adjoint Representation of a Lie Group

Let $$G$$ be a Lie group and $$e$$ be the identity element of $$G$$. The tangent space $$T_eG$$ can be identified with the vector space of left invariant vector fields on $$G$$ and admits a Lie algebra structure defined by $$[X, Y]=XY-YX$$. This Lie algebra, denoted as $$\Lg$$, is called the Lie algebra associated to $$G$$.

For each $$g$$ in $$G$$, we define $$\Ad_g=(\d\Psi _{g})_{e}: \Lg \to \Lg$$ to be the derivative of the inner automorphism $$\Psi_g: G\to G$$ at the origin. Let $$\exp(tX)$$ be the unique integral curve associated to $$X$$. Then \begin{aligned} \Ad_g(X)=&(\d\Psi_g)_e(X)\\ =&\frac{\d}{\d\, t}\bigg\vert_{t=0}(\Psi_g(\exp(tX)))\\ =&\frac{\d}{\d\, t}\bigg\vert_{t=0}(g\exp(tX)g^{-1}). \end{aligned}

As the exponential map $$\exp: \Lg\to G$$ commutes with Lie group homomorphism and its derivative, we see that $\exp(\Ad_g(X))=g\exp(X)g^{-1}$

Because the pushforward of a diffeomorphism commutes with Lie brackets (see, for example, Corollary 8.31 in (Lee 2012)), and inner automorphisms and left multiplications of Lie groups are all diffeomorphisms. Then $\Ad_g([X, Y])=[\Ad_g(X), \Ad_g(Y)]$ which means that $$\Ad_g$$ is a Lie algebra automorphism of $$\Lg$$.

Consider the map \begin{aligned} \Ad: G &\to \Aut(\Lg)\\ g &\mapsto \Ad_g. \end{aligned} For any $$g$$ and $$h$$ in $$G$$, we have \begin{aligned} \Ad(gh)=&\Ad_{gh}=(\d\Psi_{gh})_e\\ =&(\d(\Psi_g\circ\Psi_h))_e=(\d\Psi_g)_{\Psi_h(e)}\circ(\d\Psi_h)_e\\ =&(\d\Psi_g)_{e}\circ(\d\Psi_h)_e=\Ad(g)\Ad(h). \end{aligned} Hence, $$\Ad$$ is a group representation called the adjoint representation of $$G$$.

### 2.2 Coadjoint Representation of a Lie Group

Let $$G$$ be a Lie group, $$\Lg$$ the associated Lie algebra and $$\Lg^*$$ the dual vector space of the Lie algebra $$\Lg$$. The coadjoint representation of $$G$$, $$\Ad^*:G \to \Aut(\Lg^*)$$, is defined by the identity $<\Ad_{g}^{*}\,\mu ,Y> =<\mu ,\Ad_{g^{-1}}Y>$ for all $$g\in G$$, $$Y\in\Lg$$, and $$\mu \in \Lg^*$$.

## 3 Infinitesimal Actions

Differentiating a group action at the origin induces an infinitesimal action on Lie algebras. In particular, the infinitesimal actions of adjoint and coadjoint representation of a connected Lie group $$G$$ are nothing but the adjoint and coadjoint representation of the Lie algebra $$\Lg$$.

### 3.1 Adjoint Representation of a Lie Algebra

Consider the differential $$(\d\Ad)_e$$ of the adjoint representation of $$G$$. For any $$X$$ and $$Y$$ in $$\Lg$$, we have \begin{aligned} &(\d\Ad)_e(X)(Y)\\ =&\frac{\d}{\d s}\bigg\vert_{s=0}(\Ad_{\exp(sX)}(Y))\\ =&\frac{\d}{\d s}\bigg\vert_{s=0}\left(\frac{\d}{\d t}\bigg\vert_{t=0}\exp(sX)\exp(tY)\exp(-sX)\right)\\ =&\frac{\d}{\d s}\bigg\vert_{s=0}\left(\exp(sX)Y\exp(-sX)\right)\\ =&(\exp(sX)XY\exp(-sX)-\exp(sX)Y\exp(-sX))X\big\vert_{s=0}\\ =&[X, Y]. \end{aligned}

We denote $$(\d\Ad)_e$$ as $$\ad$$ and $$\ad(X)$$ as $$\ad_X$$. It can be checked that $$\ad([X, Y])=[\ad(X),\ad(Y)]$$. Indeed, for any $$Z$$, we have \begin{aligned} \ad([X, Y])(Z)=&[[X, Y], Z]\\ =&[X, [Y, Z]]-[Y, [X, Z]]\\ =&\ad_X\ad_Y(Z)-\ad_Y\ad_X(Z). \end{aligned} Therefore, $$\ad:\Lg\to \Lgl(\Lg)$$ defines a Lie algebra representation which is called the adjoint representation of the Lie algebra $$\Lg$$.

### 3.2 Coadjoint Representation of a Lie Algebra

From the definition of $$\Ad^*$$, one can check that the differential $$\ad^*=(\d\Ad^*)_e: \Lg \to \Lgl(\Lg^*)$$ is a representation of $$\Lg$$ in $$\Lg^*$$, which is called the coadjoint representation of $$\Lg^*$$.

Define $$\ad_X^*:=\ad^*(X)$$. We see that $\ad_X^*(\mu)=-<\mu, [X, -]>,$ for any $$X$$ in $$\Lg$$ and $$\mu$$ in $$\Lg^*$$.

### 3.3 Tangent Space of a Stabilizer of the Coadjoint Representation

For an element $$\mu$$ in $$\Lg$$, the stabilizer $$G_\mu$$ of the coadjoint representation $$\Ad^*$$ of $$G$$ is $G_\mu=\{g\in G\mid \Ad_g^*(\mu)=\mu\}.$

We know that $$G_\mu\subset G$$ is a Lie subgroup. Let $$\Lg_\mu$$ be the Lie algebra associated to $$G_\mu$$. Then $\Lg_\mu=\{X\in \Lg\mid \Ad^*_{\exp(tX)}\mu=\mu\}.$ For a proof, see Proposition 9.1.13 in .

Differentiate the identity at $$t=0$$, we find that $\Lg_\mu=\{X\vert \ad_X^*\mu=0\}.$

Let $$G$$ be a Lie group and $$\Lg$$ be the associated Lie algebra. Given an element $$\mu$$ in the dual $$\Lg^*$$ of the Lie algebra $$\Lg$$, we denote by $$\O_\mu$$ the $$G$$-orbit of $$\mu$$ in $$\Lg^*$$.

On a coadjoint orbit, there is a naturally defined symplectic structure. Consider the antisymmetric bilinear form $$\omega_\mu$$ defined by $\omega_\mu(\ad_X^*\mu, \ad_Y^*\mu)=-<\mu, [X, Y]>,$ where $$X$$ and $$Y$$ are vector field in $$\Lg$$.

Theorem 1 Let $$G$$ be a Lie group and $$\O_\mu\subset\Lg^*$$ be a coadjoint orbit. Then the antisymmetric bilinear form $$\omega_\mu$$ is a $$\Ad_{g^{-1}}^*$$-invariant symplectic form on $$\O_\mu$$.

For a proof of the theorem, see for example Theorem 14.3.1 in .

Corollary 1 The coadjoint orbit of a finite-dimensional Lie groups is of even dimension.

Via the diffeomorphism $$\rho: G/G_\mu\to \O_\mu$$, the manifold $$G/G_\mu$$ admits symplectic structure that pulls back $$\omega_\mu$$.

## References

Audin, Michèle. 2012. Torus Actions on Symplectic Manifolds. Softcover reprint of the original 2nd ed. 2004 edition. Basel: Birkhäuser.
Gorbatsevich, V. V., A. L. Onishchik, and E. B. Vinberg. 1993. Lie Groups and Lie Algebras I: Foundations of Lie Theory Lie Transformation Groups. Edited by A. L. Onishchik. Encyclopaedia of Mathematical Sciences, Lie Groups and Lie Algebras. Berlin Heidelberg: Springer-Verlag. https://doi.org/10.1007/978-3-642-57999-8.
Lee, John M. 2012. Introduction to Smooth Manifolds. Vol. 218. Graduate Texts in Mathematics. New York, NY: Springer New York. https://doi.org/10.1007/978-1-4419-9982-5.
Marsden, Jerrold E., and Tudor S. Ratiu. 1999. Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. 2nd edition. New York: Springer.