Coadjoint semi-direct orbits and Lagrangian families with respect to Hermitian form

Órbitas coadjuntas semi-directas y familias Lagrangianas con respecto a la forma Hermitiana

The symplectic structures and Lagrangian submanifolds of coadjoint orbits were stud-ied and developed by renowned mathematicians such as Kirillov, Arnold, Kostant, and Souriau from the early to the mid-1960s, although they had its roots in the work of Lie, Borel, and Weyl. Alternatively, there are several theories and applications to physics using general reduction theory, as in [2], [9], [10], and [11], among others. We study some applications of the semisimple Lie theory to symplectic geometry, in particular to find Lagrangian submanifolds on adjoint orbits. In this paper, we follow the next construction: let g be a non-compact semisimple Lie algebra with Cartan decomposition g = k ⊕s and Iwasawa decomposition g = ŧ ⊕ a ⊕ n with a ⊂ s maximal Abelian. In the under-lying vector space g, there is another Lie algebra structure ŧ_ad = ŧ ×_ad s given by the semi-direct product defined by the adjoint representation of ŧ in s, which is viewed as an Abelian Lie algebra. Let G = Aut₀g be the adjoint group of g (identity component of the automorphism group) and put K = exp ŧ ⊂ G. The semi-direct product K_ad = K ×_ad s obtained by the adjoint representation of K in s has Lie algebra ŧ _ad = ŧ ×_ad s, that orbit was studied in [1]. Then, we consider coadjoint orbits for both Lie algebras g and ŧ_ad . These orbits are submanifolds of g^∗ that we identify with g via the Cartan-Killing form of g, so that the orbits are seen as submanifolds of g. These are just the adjoint orbits for the Lie algebra g while for ŧ_ad they are the orbits in g of the representation of K_ad obtained by transposing its coadjoint representation. The orbits through H ∈ g are denoted by ad (G) · H and K_ad· H, respectively.

In Section 2, our goal is to generalize that construction, to get a wider variety of La-grangian submanifolds of any adjoint semisimple orbit. For this, we are going to change the usual structure of semisimple Lie algebras, i.e., with a new Lie bracket given by a convenient semi-direct product. This construction was inspired by [8], where the author defines a semi-direct product using a closed subgroup of a semi-simple Lie group and the vector space g (seeing g = Lie(G) as a vector space), but focused on solving some applications of control theory. In this way, the first part of this chapter is focused on the general construction of coadjoint orbits of this semi-direct structure. After that, we adapt those general results to the mentioned semi-direct product given by a Cartan decomposition.

In Section 3, we build some families of Lagrangian submanifolds on ad(G) · H with respect to the Hermitian symplectic form Ω^τ , characterized by Cartan involutions on g. The idea and future goal of this result is: Classify the families of Lagrangian submanifolds determined by Cartan involutions.

The construction presented in this section is a more general version of the one given in [1], some proofs will have a lot of similarities, but here we will not be able to use some special structure such as compactness. In Section 2.1, we will see under what conditions the construction is identical to the one cited above.

Let G be a connected Lie group with Lie algebra g and take a representation ρ: G → Gl (V ) on a vector space V (with dim V < ∞). The infinitesimal representation of g on gl(V ) is also going to be denoted by ρ. The vector space V can be seen as an Abelian Lie group (or Abelian Lie algebra). In this way, we can take the semi-direct product G ×_ρV which is a Lie group whose underlying manifold is the cartesian product G × V. This group is going to be denoted by G_ρ and its Lie algebra g_ρ is the semi-direct product

Our first purpose is to describe the coadjoint orbit on the dual g^∗_ρ of g_ρ . To begin with, let’s see how to determine the ρ-adjoint representation ad_ρ (X, v), where (X, v) ∈ g ×_ρV . Thus, take a basis of g × V denoted by ℬ = ℬ _g∪ ℬ_V with ℬ _g = {X₁, . . . , X_n} and ℬ_V = {v₁, . . . , v_d} basis of g and V , respectively. On this basis, the matrix of ad_ρ (X, v) is given by

where ad (X) is the adjoint representation of g while for each v ∈ V , A (v) is the linear map g → V defined by

The dual space g^∗_ρ can be identified with g^∗⊕ V^∗ , where g^∗ is immersed on (g × V )^∗ by extensions of linear functionals on g to g×V by the zero functional on V (in the same way, V^∗ is immersed on (g × V )^∗ ). Therefore, the dual basis of B is B^∗ = B_g^∗∪ B_V^∗ , where B_g^∗ and B_V^∗ are the dual basis of B_g and B_V , respectively. Then, the coadjoint representation ad^∗_ρ (X, v), for (X, v) ∈ g_ρ , with respect to B^∗ , is transposed with a negative sign on the off-diagonal term of (1), that is

In this matrix, ad^∗ is the coadjoint representation of g, ρ^∗ is the dual representation of ρ, that is

and A(v)^∗ : V^∗→ g^∗ is the transpose of A(v) for v ∈ V , which by the above equation can be seen as follows:

The adjoint representation ad_ρ and coadjoint representation ad^∗_ρ of G_ρ are obtained by exponentials of representations in g_ρ . In particular, the following matrices are obtained (on the basis B and B^∗ ):

On the other hand, for g ∈ G the restriction of ad_ρ (g) to V coincides with ρ (g) and the restriction of ad^∗_ρ (g) to V^∗ coincides with ρ^∗ (g), where we are seeing V and V^∗ as subspaces of g_ρ = g ⊕ V and g^∗_ρ = g^∗⊕ V^∗ , respectively.

To describe the map A(v)^∗ , it is convenient to define the momentum map of the repre-sentation ρ.

Definition 2.1. The momentum map of the representation ρ is the map

given by

Then

because we have the following identifications

Lemma 2.2.The momentum map is G-equivariant, with respect to the representation ρ ⊗ ρ^∗and the coadjoint representation, i.e., for g ∈ G, v ∈ V , and α ∈ V^∗

Proof. Let g ∈ G, note that the restrictions of ad_ρ (g) and ad^∗_ρ (g) to V and V^∗ coincide with ρ(g) and ρ^∗ (g), respectively. Then, for X ∈ g

Since µ is bilinear, for any fixed α ∈ V^∗ , the map µ_α : V → g^∗ given by µ_α (v) = µ (v ⊗ α) is a linear map and consequently, its image µ_α (V ) is a subspace of g^∗ . Let α ∈ V^∗ , the coadjoint orbit of G_ρ through α will be denoted by

The following proposition shows that the coadjoint orbit for α ∈ V^∗ is the union of subspaces µ_β (V ), with β ∈ ρ^∗ (G) α.

Proposition 2.3.For α ∈ V^∗, the coadjoint orbit can be written as

and identifying g^∗× V^∗with g^∗⊕ V*

where β + µ_β (V ) is an affine subspace of g^∗⊕ V^∗.

Proof. Firstly, if g ∈ G we can identify ad^∗_ρ (g) with ρ^∗ (g) in the subspace V^∗⊂ g^∗× V^∗ . Therefore, ρ^∗ (G) α ⊂ ad^∗_ρ (G_ρ ) α, and as we saw before

which shows that if β ∈ V^∗⊂ g^∗⊕ V^∗ = g^∗× V^∗ , then

which in terms of the momentum map is

Then, varying v ∈ V , we can see that the affine subspace β + µ_β (V ) is contained in the coadjoint orbit of β, for β ∈ V^∗ . As ρ^∗ (G) · α ⊂ ad^∗_ρ (G_ρ ) · α, we conclude that

Conversely, if g ∈ G and β ∈ V^∗

where the last equality is a consequence of the fact that µ is equivariant. For h ∈ G_ρ , there are g ∈ G and v ∈ V , such that

As ad^∗_ρe^t(0,v)α ∈ α + µ_α (V ), then ad^∗_ρ (h) α ∈ ρ^∗ (g) α + µ_ρ∗_(g)α (V ).

The action of G_ρ is obviously transitive on G_ρ· α, then it is an homogeneous space given by

the isotropy subgroup at α ∈ V^∗⊂ g_ρ , with Lie algebra

Then in terms of the basis B^∗

therefore,

Thus

And

As T_α (G_ρ· α) ≈ g_ρ/z_ρ (α), for any (X, v) ∈ g_ρ , there induced by () at ξ = β + µ_β (w) ∈ G_ρ· α (with β = ρ^∗ (g)α, g ∈ G, and w ∈ V ) given by

Hence

By Proposition 2.3, the coadjoint orbit G_ρ·α is the union of vector spaces and fibers over ρ^∗ (G) x of the representation ρ^∗ . This union is disjoint because given ξ ∈ (β + µ_β (V )) ∩ (γ + µ_γ (V )) then

with X, Y ∈ g. Since the sum g^∗_ρ = g^∗⊕ V^∗ is direct, it follows that β = γ and X = Y . Therefore there is a fibration

such that an element ξ = β + X ∈ β + µ_β (V ) associates β ∈ ρ^∗ (G) α, and its fibers are vector spaces. The following proposition shows that this fibration can be identified with the cotangent space of ρ^∗ (G) α.

Let ϕ be the map

such that, for β ∈ ρ^∗ (G) α

This implies that the restriction of ϕ to a fiber β+µ_β (V ) is given by a linear isomorphism

Theorem 2.4.The map ϕ is an isomorphism of vector bundles.

If ω is the KKS symplectic form on G_ρ· α and ω is the canonical symplectic form on T^∗ (ρ (G) α), then ϕ is a symplectic isomorphism of vector bundles, i.e., ϕ^∗ $\tilde{\omega }$ = ω.

The proof of this result has several steps, essentially they are as follows:

• The restriction of ϕ to a fiber w + µ_w (V ) is given by the isomorphism:

To see that: take ξ ∈ G_ρ· α, there is a unique β ∈ ρ^∗ (G) α, such that ξ ∈ µ_β (V ), then there is v ∈ V with ξ = β + µ (v ⊗ β). The vector v ∈ V defines a linear functional f_v on V^∗ , and of course their respective restriction to T_β (ρ^∗ (G) α), therefore f_v∈ T_β^∗ (ρ^∗ (G) α). Set

A map ϕ is a linear injective map and the linear map µ (v ∧ w) →f_v is surjective.

• In the coadjoint orbit G_ρ· α we can define the Konstant-Kirillov-Souriau (KKS) symplectic form, denoted by ω and defined as

where is the Hamiltonian vector field of the function H_(X,v) : M → R given by

Furthermore, as is known, for the cotangent bundle T^∗ (ρ^∗ (G) α) we can define the canonical symplectic form $\tilde{\omega }$.

• The best way to relate these symplectic forms is through the action of the semi-direct product G_ρ = G × V on the cotangent bundle of ρ^∗ (G) α. This action is described in Proposition 3.11 (in a general case), the action of G_ρ on T^∗ (ρ (G) α) is Hamiltonian and then it defines a moment map

The construction of m shows that it is the inverse of ϕ. Moreover, m is equivariant, that is, it interchanges the actions on T^∗ (ρ (G) α) and the adjoint orbit, which implies that m is a symplectic morphism.

2.1. Compact case

Let U be a compact connected Lie group with Lie algebra u and take a representation : ρ U → Gl(V ), where V admits a U-invariant inner product h·, ·i when V is a real vector space (a Hermitian inner product when V is a complex vector space).

We will denote by U_ρ the semi-direct Lie group U ×_ρV , with Lie algebra u_ρ = u ×_ρV . The inner product allows us to identify V with V^∗ by

and we can also identify ρ with ρ^∗ by

Now, analogously to the discussion for the general case, we can characterize the coadjoint orbit of U_ρ in terms of the momentum map µ: V ⊗ V → u^∗ given by

By construction, ρ(u) is an isometry for all u ∈ U with respect to the fixed U-invariant inner product, then ρ(X) is a skew-symmetric linear map with respect to {.,.} for all X ∈ u, and we have

that is, µ is skew-symmetric. Therefore, the momentum map µ is defined in the exterior product ∧²V = V ∧ V . Furthermore, the compact Lie algebra u admits an ad-invariant inner product such that we can identify u^∗ with u, then

Similarly, the dual u^∗× V^∗ of u_ρ = u × V is identified by its inner product which is a direct sum of ad-invariant inner products of u and V . In that identification, the coadjoint representation of u can be seen as the adjoint representation of u because its inner product is u-invariant, but the inner product of V is not invariant under the adjoint representation of V , then the coadjoint representation of that Abelian algebra is the transpose of its adjoint representation. This means that the coadjoint representation of u × V is written in u × V as type matrices on orthonormal bases:

where for each v ∈ V , A (v) : V → u can be identified by A (v) (w) = µ (v ∧ w).

Then the representations ad_ρ and ad^∗_ρ of U_ρ are obtained by exponentials of representa-tions in u_ρ , take v ∈ V ⊂ u × V and by Proposition 2.3

2.2. Examples

We will see some examples of semi-direct coadjoint orbits to compare them with the usual orbits. To begin with, take ρ the canonical representation of u = so (n) in V = (ℝⁿ, h·, ·i). The momentum map with values in u is given by

and as we know the invariant inner product on so(n) is

To describe the orbit, take the isomorphism I: ∧²V → so (n) given by

which satisfies

If A ∈ so(n) we have

Let {e₁, . . . , e_n} be an orthonormal basis of V , then

Therefore identifying V^∗ with V by h·, ·i, and so(n)^∗ with so(n) by , the momentum map is µ(v ∧ w) = I(v ∧ w), that is

For simplicity of notation, we will denote I(w ∧ v) for v, w ∈ V as v ∧ w. If v and w are n × 1 column vectors, we have

which is an n × n matrix.

As we saw above, the coadjoint representation of so(n) ×_ρ ℝ is given by

where for each v ∈ ℝⁿ , A (v) : ℝⁿ→ so (n) is the map

The representation so (n)_ρ ℝⁿ defines a representation of the semi-direct product U_ρ = SO(n) ρ ℝⁿ on so (n) ℝⁿ by exponentials (here so (n) ℝⁿ is the space where the U_ρ -orbit is being identified). As discussed earlier a U_ρ -orbit of v 2 ℝⁿ so (n) ℝⁿ is given by

In this case, the orbits of SO (n) in ℝⁿ are the (n − 1)-dimensional spheres centered at the origin.

Example 2.5.For n = 2, we have that so (2) x_ρ ℝ²is isomorphic with ℝ³and for all w 2 ℝ²the image A(w) ℝ² = so (2), therefore the coadjoint semi-direct orbits are the circular cylinders with the axis on the line generated by so (2) in so (2) ℝ² ≈ ℝ³.

Let h be a subalgebra of so(n). We can induce the canonical representation of h in ℝⁿ as a restriction on so(n), then

because the inner product of ℝⁿ is invariant by h. The trace form −tr AB provides (by restriction) an inner invariant product in h, that allows us to identify h with h^∗ .

Let p: so(n) → h be the orthogonal projection in relation to the trace form. By the identification above of h^∗ and h we can define the h-momentum map

where µ is the momentum map of so(n). Then

For u(n), we can take a canonical representation in ℂⁿ = ℝ²ⁿ and see u(n) as an immersed subalgebra of so(2n) by matrices 2n × 2n of the form

Then h = u(n) and p: so(2n) → u(n) is the orthogonal projection with respect to the trace form. Hence the momentum map is

In this section, we will see the results of Section 2 in the structure of any semisimple non-compact Lie algebra, determined by a given Cartan decomposition. This structure was studied and described in [1], where the authors made the following: Let g be a non-compact semisimple Lie algebra with Cartan decomposition g = ŧ ⊕ s. As [ŧ, s] ⊂ s, the subalgebra ŧ can be represented on s by the adjoint representation. Then, we can define the semi-direct product ŧ_ad = ŧ × s, where s can be seen as an Abelian algebra. This is a new Lie algebra structure on the same vector space g where the brackets [X, Y ] are the same when X or Y are in k, but the bracket changes when X, Y ∈ s. The identification between ŧ_ad = ŧ × s and its dual ŧ^∗_ad = ŧ^∗× s^∗ is given by the inner product B_θ (X, Y ) = −hX, θY i, where h·, ·i is the Cartan-Killing form of g and θ is a Cartan involution. If A ∈ ŧ, then ad (A) is anti-symmetric with respect to B_θ , while ad (X) is symmetric for X ∈ s. The moment map is given by

the second part of that equality is

because [X, Y ] ∈ ŧ. Therefore the moment map of the adjoint representation of ŧ on s is

where [·, ·] is the usual bracket of g. Therefore, the coadjoint representation of the semi-direct product ŧ × s is given by (in an orthonormal basis)

where for each Y ∈ s, A (Y ) : s → k is the map A (Y ) (Z) = [Y, Z].

Let G be a connected semisimple Lie group with Lie algebra g and take K ⊂ G the subgroup given by K = {exp k ŧ }. The semi-direct product of K and s will be denoted by

The coadjoint orbit of x̅ ∈ s ⊂ ŧ ×s is the union of the fibers A (Y ) (s) with Y belonging to e the K-coadjoint orbit of X in s. As A (Y ) (Z) = [Y, Z], then A (Y ) (s) = ad (Y ) (s) where ad is the adjoint representation in g. To detail the coadjoint orbits of the semi-direct product, take a maximal Abelian subalgebra a ⊂ s. The ad (K)-orbits in s are passing through a are thus the flags on g. Take a positive Weyl chamber a⁺⊂ a. If H ∈ cl (a ⁺) then the orbit ad (K) H is the flag manifold _H . By Proposition 2.4, the K_ad -orbit in H ∈ cl(a⁺) is diffeomorphic to the cotangent bundle of _H , thus the K_ad -orbit itself is the union of the fibers ad (Y ) (s), with Y ∈_H . In conclusión

In this union, the fiber over H is H + ad (H) (s) with ad (H) (s) ⊂ ŧ. With the notations above this subspace of k is given by

3.3. Hermitian symplectic form and Lagrangian submanifolds

Let g_C be a complex semisimple Lie algebra and u its compact real form with Cartan involution τ, such that

is a Hermitian form of g_C, where h·, ·i_C is the complex Cartan-Killing form of g_C.

Remark 3.1. To avoid confusion, we have that g^ℂ is the complexification of g (or realifi-cation for g^ℝ), the complexification will be denoted at the top. While g_ℂ will simply be to indicate that it is complex (or real for gℝ) , this is will be denoted at the bottom.

The imaginary part of ℋ_τ will be denoted by Ω^τ , which is

is a symplectic form on g_C (see [13]) and will be called the symplectic Hermitian form determined by τ.

Let G be a connected Lie group with Lie algebra g_C. For that, let g_C = u ⊕ iu be a Cartan decomposition with Cartan involution τ, for g a semisimple complex Lie algebra. If U ⊂ G is the compact subgroup with Lie algebra u. Then, we will denote by U_ad its respective semi-direct product (described in Section 3 for the general case).

If H ∈ s = iu, then its semi-direct orbit is denoted by U_ad· H, given by

Remark 3.2. Without loss generality, we can choose H ∈ a or H ∈ cl(a⁺), where a ⊂ s is the maximal Abelian subalgebra of g, or a⁺ their respective positive Weyl chamber (see [12] and [13]).

In [1] it was proved that the form Ω^τ of g restricted to U_ad· H is a symplectic form, for H ∈ cl(a⁺) and the following theorem:

Theorem 3.3.The manifolds (U_ad· H, Ω^τ ) and (ad(G) · H, Ω^τ ) are symplectomorphic.

3.4. Lagrangian families

Let g be a real semisimple non-compact Lie algebra, such that is a real form of g_C, and u a compact real form of g_C with Cartan involution τ (i.e., g and u are real forms of g_ℂ). Then

is a Cartan decomposition of g.

Lemma 3.4.The restriction of H_τto g is real.

Proof. For X, Y ∈ g, there are X₁, Y₁∈ g ∩ u and X₂, Y₂∈ g ∩ iu such that X = X₁ + X₂ and Y = Y₁ + Y₂. Then

As we have that

where ·{.,.}·i_ℂ is the Cartan-Killing form of g_ℂ, then

However

as H_τ is an Hermitian form, we have that , thus,

and by equation (11), we have that

but X₁, Y₁, iX₂, iY₂∈ u, and the restriction of {.,.}·i_ℂ to u is negative-definite, we can conclude that H_τ|_g is real.

Corollary 3.5. Ω^τ|_g≡ 0.

Moreover, let G^ℂ be a Lie group with Lie algebra g_ℂ. Therefore, given any submanifold M of U_ad· H, M must be contained on g. In fact, M is an isotropic submanifold of U_ad· H. By Theorem 3.3 the same applies to any submanifold M of ad_r (G^ℂ) · H.

Remark 3.6. To avoid confusion, in this subsection, we say G^ℂ to specify that this is a complex Lie group with Lie algebra g_ℂ.

Now, our purpose is to apply the last results for a non-trivial immersion on the coadjoint semi-direct orbit to find some Lagrangian submanifolds. With the Cartan decomposition of g given in (10), then

Take K = {exp kŧ}, then K_ad· H is an immersed submanifold on U_ad· H, for H ∈ a. Moreover,

where ŧ_ad can be identified with g as a vector space and by Corollary 3.5, the restriction of H_τ to g is real, thus,

Therefore, K_ad· H is an isotropic submanifold of U_ad· H, we want to see that K_ad· H is a Lagrangian submanifold of U_ad· H, as we can see in the following example.

Example 3.7.For g = sl(2, ℝ), k = so(2), and u = su(2). Given

we have that K_ad· H (cylinder) is a 2-dimensional isotropic submanifold of U_ad· H, a 4-dimensional manifold.

Hence, the cylinder K_ad· H is a Lagrangian submanifold of U_ad· H.

Let σ be an anti-linear involutive conjugation on g_C, such that g is the subspace of fixed points of σ, that is

If we have that A: = {X ∈ U_ad· H: σ(X) = X} coincides with K_ad· H, then we can conclude that K_ad· H is a Lagrangian submanifold of U_ad· H, with respect to the Hermitian symplectic form, for H ∈ a.

As K_ad·H is contained on g and it is a submanifold of U_ad·H, we have that K_ad·H ⊆ A. For the opposite inclusion, by equation (8) we have that

then given an element x ∈ U_ad· H implies that

As u = k ⊕ is, we have the following possibilities:

• Take X ∈ ŧ, then e^tX∈ U

if Z 2 ŧ, we have that σ(Z) = Z and if Z 2 is, we have that σ(Z) = -Z, then σ(x) = x if and only if Z 2 is.

Thus, x is a fixed point if and only if x 2 K_adH.

• Take X 2 is, then e^tX∈ U

for Z ∈ u, we have that σ(x) =6 x, then in this case it is impossible to have fixed points.

• For any other possible choice of X ∈ u, we do not have fixed points because it would be a combination of the cases above.

Therefore, A = K_ad· H and K_ad· H is the set of fixed points of σ, its dimension is half the dimension of U_ad· H. Hence,

Proposition 3.8.For H ∈ a, the coadjoint orbit K_ad· H is a Lagrangian submanifold of U_ad· H, with respect to the Hermitian symplectic form.

By Theorem 3.3, the K_ad -coadjoint orbit is symplectomorphic to G-adjoint orbit and U_ad -coadjoint orbit is symplectomorphic to G^ℂ -adjoint orbit, with respect to Ω^τ . Then, we can conclude that

Corollary 3.9.For H ∈ a, the orbit ad(G) ·H is a Lagrangian submanifold of ad(G^ℂ) ·H, with respect to the Hermitian symplectic form.

Furthermore, the coadjoint orbit U_ad· H is invariant by the automorphism of u, because any automorphism of u leaves invariant its Cartan subalgebra (see [12] or [14]). Given k ∈ Aut(k) we know that the k-action on g leaves invariant the Cartan decomposition of g, its maximal Abelian subalgebra and u (because k is contained in u). If exp is the exponential between the Lie algebra u and the Lie group Aut(u), then for any X ∈ is we have that g^tX = exp(tX) · g is a real form of g^ℂ with Cartan decomposition g^tX = ŧ^tX⊕ s^tX . Take G^tX a Lie group with Lie algebra g^tX⊂ u, then we can conclude that.

Theorem 3.10.For X ∈ is ⊂ u, there are a Lagrangian family of submanifolds {M_tX} on ad(G^ℂ )·H with respect to the Hermitian symplectic form. For t ∈ I, M_tX = ad(G^tX)·.

In fact, the family of Lagrangian submanifolds is determined by g, and given by the is-conjugated real forms of g.