Integrability of geodesic flows for metrics on suborbits of the adjoint orbits of compact groups

Let $G/K$ be an orbit of the adjoint representation of a compact connected Lie group $G$, $\sigma$ be an involutive automorphism of $G$ and $\tilde G$ be the Lie group of fixed points of $\sigma$. We find a sufficient condition for the complete integrability of the geodesic flow of the Riemannian metric on $\tilde G/(\tilde G\cap K)$, which is induced by the bi-invariant Riemannian metric on $\tilde G$. The integrals constructed here are real analytic functions, polynomial in momenta. It is checked that this sufficient condition holds when $G$ is the unitary group $U(n)$ and $\sigma$ is its automorphism defined by the complex conjugation.


Introduction
Let G/K be a homogeneous space of a compact Lie group G. We consider the problem of the complete integrability of the geodesic flow of the Riemannian metric on G/K, which is induced by a bi-invariant Riemannian metric on G. This problem was solved for some types of homogeneous manifolds, including symmetric spaces, spherical spaces, Stiefel manifolds, flag manifolds, orbits of the adjoint actions and others (see [Mi], [GS], [My2], [MS], [BJ1], [BJ3], [MP]). Here we consider a new family of homogeneous manifolds -suborbits of orbits of the adjoint actions. This paper is motivated by the paper [DGJ], where were constructed integrable geodesic flows ofG-invariant metrics on the homogeneous spaceG/K = SO(n)/ SO(k 1 ) × SO(k 2 ) × · · · × SO(k r ) , k 1 + k 2 + · · · + k r n. The method of the proof in [DGJ] is based on investigations of bi-Poisson structures on the Lie algebras u(n) and so(n) associated with Lie algebra deformations. We consider the Lie-algebraic aspects of the integrability problem for such homogeneous spaces. Our approach is based on the following observation: the spaceG/K is ã G-suborbit of the adjoint orbit G/K = U (n)/ U (k 1 ) × U (k 2 ) × · · · × U (k r ) of the Lie algebra u(n) of the unitary group, i.e.G/K = Ad(G)(a), where a ∈ u(n), and G/K = Ad(G)(a). Moreover,G is the group of fixed points of the involutive automorphism σ of U (n) induced by the complex conjugation. In other words, the spaceG/K is defined uniquely by the pair (G/K, σ), where G/K = Ad(G)(a) is an arbitrary adjoint orbit of the Lie group G in its Lie algebra g with a ∈ (1−σ * )g, σ * is the tangent automorphism of the Lie algebra g.
Let G be an arbitrary compact connected Lie group G with an involutive automorphism σ : G → G and letG be the set of fixed points of σ. In the article we investigate the integrability of the geodesic flow on the cotangent bundle T * (G/K) defined by aG-invariant metric onG/K, which is induced by a biinvariant Riemannian metric onG. As a homogeneous spaceG/K we consider the homogeneous space associated with the adjoint orbit G/K = Ad(G)(a) of arbitrary point a ∈ (1 − σ * )g, i.e.K =G ∩ K. We found a sufficient purely algebraic condition for the integrability of this geodesic flow on the symplectic manifold T * (G/K) (Theorem 2.17). We prove that this sufficient condition holds when G is the unitary group U (n) and σ is its automorphism defined by the complex conjugation (Theorem 2.18). Our approach is based on the fact thatG/K ⊂ G/K is a totally real (Lagrangian) submanifold of the homogeneous Kähler manifold (the compact orbit) G/K and T (G/K) ⊂ T (G/K) is a totally real submanifold of T (G/K). But to simplify calculations we reformulate this fact in some algebraic terms (not explicitly, since explicit reformulation is very complicated from the point of view of calculations on T (T (G/K))).
One calls a Hamiltonian system on T * M (completely) integrable if it admits a maximal number of independent integrals in involution, i.e. dim M functions commuting with respect to the Poisson bracket on T * M whose differentials are independent in an open dense subset of T * M . By Liouville's theorem the integral curves of an integrable Hamiltonian system under a certain additional compactness hypothesis are quasiperiodic (are the orbits of a constant vector field on an invariant torus).
Let A G be the set of all G-invariant real analytic functions on the cotangent bundle T * M of M = G/K. This space is an algebra with respect to the canonical Poisson bracket on the symplectic manifold T * M . The natural extension of the action of G on M to an action on the symplectic manifold T * M is Hamiltonian with the moment mapping µ can : T * M → g * . The functions of type h•µ can , h : g * → R, are integrals for any G-invariant Hamiltonian flow on T * M , in particular, for the geodesic flow corresponding to any G-invariant Riemannian metric on M . In general, a maximal involutive subset of {h • µ can , h : g * → R} is not a maximal involutive subset of the algebra C ∞ (T * M ). But for the compact Lie group G the problem of constructing of a maximal commutative set of real analytic functions on T * (G/K) is reduced to the problem of a finding of a maximal commutative set of real analytic functions from the set A G (see [My3,§2], [BJ1,Lemma 3], [Pa]). This is true also for the groupG and the corresponding algebra AG ⊂ C ∞ (T * (G/K)).
The algebra of functions A G on T * (G/K), where, recall, G/K is an adjoint orbit, contains some maximal involutive subset F of A G consisting of independent functions ( [MP,Theorem 3.10], see also [BJ2], [BJ3]). The homogeneous spacẽ G/K, as we remarked above, is a submanifold of G/K and therefore T (G/K) is a submanifold of T (G/K). Moreover, T (G/K) is a symplectic submanifold of T (G/K), where the symplectic structures on these spaces are defined via iso-morphisms T (G/K) ≃ T * (G/K) and T (G/K) ≃ T * (G/K) using a standard Ginvariant metric on G/K and its restriction toG/K (see Proposition 1.4). The set F = {f |T (G/K), f ∈ F } of restrictions is an involutive subset of the algebra AG. This involutiveness of the functions fromF is a consequence of the fact that G/G is a symmetric space and follows easily from results published in [MF], [TF]. The following observation is crucial in our approach: if the functions from the set F are independent at some point of the symplectic submanifold T (G/K) ⊂ T (G/K), then the setF is a maximal involutive subset of the algebra AG. Therefore we describe explicitly some open dense subset O of T (G/K), where all functions from the set F are independent (Theorem 2.13) (in the paper [MP] only the existence of such a set was proved). We prove that O ∩ T (G/K) = ∅ if G is the unitary group U (n) and σ is its automorphism defined by the complex conjugation (Theorem 2.18).
1 G-invariant bi-Poisson structures and moment maps 1.1 Some definitions, conventions, and notations All objects in this paper are real analytic, X stands for a connected manifold, E(X) for the space of real analytic functions on X.
We will say that some functions from the set E(X) are independent if their differentials are independent at each point of some open dense subset in X. For any subset F ⊂ E(X) denote by ddim x F the maximal number of independent functions from the set F at a point x ∈ X. Put ddim F def = max x∈X ddim x F . Let η be a Poisson bi-vector on X and let A ⊂ E(X) be a Poisson subalgebra of (E(X), η), i.e. A is a vector space closed under the Poisson bracket {, } : Let B x denote the restriction of η x to this subspace (DA) x . We say that a subset F ⊂ A is a maximal involutive subset of the algebra (A, η) if at each point x of some open Definition 1.1. A pair (η 1 , η 2 ) of linearly independent bi-vector fields (bi-vectors for short) on a manifold X is called Poisson if η t def = t 1 η 1 + t 2 η 2 is a Poisson bivector for any t = (t 1 , t 2 ) ∈ R 2 , i.e. each bi-vector η t determines on X a Poisson structure with the Poisson bracket {, } t : (f 1 , f 2 ) → η t (df 1 , df 2 ); the whole family of Poisson bi-vectors {η t } t∈R 2 is called a bi-Poisson structure.
A bi-Poisson structure {η t } (we shall often skip the parameter space) can be viewed as a two-dimensional vector space of Poisson bi-vectors, the Poisson pair (η 1 , η 2 ) as a basis in this space.
The set X A of all points x ∈ X for which ddim x A = ddim A is an open dense subset in X. But sometimes the exact description of this set is impossible or not constructive. Therefore we will consider some greater open subset R A ⊂ X containing X A and such that there exists a smooth subbundle Suppose that a linear subspace A ⊂ E(X) is a Poisson subalgebra of (E(X), η t ) It is evident that this (micro)definition is independent of the choice of this greater open dense subset R A ⊂ X.
The definitions above are motivated by the following assertion of Bolsinov which is fundamental for our considerations.
Proposition 1.3. [Bo] Let B 1 and B 2 be two linearly independent skew-symmetric bilinear forms on a vector space V . Suppose that the kernel of each form t∈T ker B t is isotropic with respect to any form B t , t ∈ R 2 , i.e. B t (L, L) = 0; (2) the space L is maximal isotropic with respect to any form B t , t ∈ T , i.e.
Let G be a connected Lie group acting on a symplectic manifold (X, ω) and preserving its symplectic structure ω. Let g be the Lie algebra of G. For each vector ξ ∈ g denote by ξ X the fundamental vector field on X generated by the one-parameter diffeomorphism group exp(tξ) ⊂ G. The group G acts on the symplectic manifold (X, ω) in a Hamiltonian fashion if there is a G-equivariant map µ : X → g * such that for each ξ ∈ g the field ξ X is the Hamiltonian vector field with the Hamiltonian function f ξ : X → R, x → µ(x)(ξ), i.e. df ξ = −ω(ξ X , ·). The equivariance property µ(g −1 x)(ξ) = µ(x)(Ad(g)ξ), where g ∈ G, x ∈ X, of the moment map µ implies the identity {f ξ , f ζ } = f [ξ,ζ] , where ξ, ζ ∈ g and {, } is the Poisson bracket associated with ω −1 . In other words, the mapping µ is canonical with respect to the Poisson structure ω −1 on X and the standard linear Poisson structure on g * . Moreover, by definition {f, h • µ} = 0 for any G-invariant function f on X and h ∈ E(g * ).
Consider a connected Riemannian manifold (M, g) and its connected Riemannian submanifold (M ,g), whereg = g|M . The cotangent bundles T * M and T * M are symplectic manifolds with canonical symplectic structures Ω andΩ respectively. Using the metric g (resp.g) we can identify T * M with T M (resp. T * M with TM), denoting by ϕ : T M → T * M (resp.φ : TM → T * M ) the corresponding diffeomorphism. Let p : T M → M (resp.p : TM →M ) be the natural projection and let θ (resp.θ) be the canonical 1-form on T * M (resp. on T * M ).
q M (to simplify notation) and taking into account that π • ϕ = p we obtain that for any Z ∈ T x T M Similarly, we obtain that (φ * θ )x(Z) =gq(x,p * xZ ) for anyq ∈M ,x ∈ TqM ,Z ∈ TxTM .
In other wordsφ * θ = ϕ * θ |TM , because p|TM =p and g|M =g. Now to complete the proof it is sufficient to note that Ω = dθ andΩ = dθ.

Hamiltonian actions on cotangent bundles and maximal involutive sets of functions
Let G be a compact connected Lie group with a closed subgroup K. Denote by g and k the Lie algebras of the Lie groups G and K. Let Ω be the canonical symplectic form on the cotangent bundle X = T * M , where M = G/K. The natural action of G on (X, Ω) is Hamiltonian with the moment map µ can : X → g * . This equivariant moment map µ can has the form µ can (x)(ξ) = θ(ξ X )(x), where θ is the canonical 1-form on X = T * (G/K). Since the canonical form Ω is G-invariant, the set A G X of G-invariant function on X is a subalgebra of (E(X), η can ), η can = Ω −1 . As we remarked above, the moment map µ can is a Poisson map and therefore the set {h • µ can , h ∈ E(g * )} is also a subalgebra of (E(X), η can ). The following assertion is known [My3,§2], [BJ1,Lemma 3], [Pa], but here we formulate it in terms of maximal involutive subsets of Poisson algebras: Proposition 1.5. Suppose that there exist a set of functions F ⊂ A G X which is a maximal involutive subset of the algebra (A G X , η can ). Then there is a set H of polynomial function on g * such that the set F ∪ {h • µ can , h ∈ H} form maximal involutive set of independent functions on T * (G/K).

The integrability of geodesic flows
In this section for any compact Lie algebra a by z(a) we will denote its center and by a s its maximal semisimple ideal, i.e. a = z(a) ⊕ a s ; for any real vector space or Lie algebra a by a C we will denote its complexification.
2.1 Commutator on A K m induced by canonical Poisson structure on T * (G/K) Let M = G/K be a homogeneous space of a compact connected Lie group G with the Lie algebra g. There exists a faithful representation χ of g such that its associated bilinear form Φ χ is negative-degenerate on g (if g is semi-simple we can take the Killing form associated with the adjoint representation of g). Let m = k ⊥ be the orthogonal complement to k with respect to Φ χ . Then The form , = −Φ χ defines a G-invariant metric on G/K. This metric identifies the cotangent bundle T * (G/K) and the tangent bundle T (G/K). Thus we can also talk about the canonical 2-form Ω on the manifold T (G/K). The symplectic form Ω is G-invariant with respect to the natural action of G on T (G/K) (extension of the action of G on G/K).
We can identify the tangent space T o (G/K) at the point o = p(e) with the space m by means of the canonical projection p : G → G/K. Let A G (resp. A K m ) be the set of all G-invariant (resp. Ad(K)-invariant) functions on T (G/K) (resp. on m). There is a one-to-one correspondence between G-orbits in T (G/K) and Ad(K)-orbits in m. Thus we can identify naturally the spaces of functions A G and A K m . For any smooth function f on m write grad m f for the vector field on m such that df x (y) = grad m f (x), y for all y ∈ m.
The Poisson bracket of two functions f 1 , f 2 from the set A K m = A G with respect to the canonical Poisson structure η can (determined by the canonical 2-form Ω) has the form [MP, Lemma 3.1] : (2.2) Now, let us consider an important for our considerations subset of m. For any For any element x ∈ g denote by g x its centralizer in g, i.e. the set of all z ∈ g where q(m) (resp. p(m)) is the minimum of dimensions of the spaces g y (resp.k y ) over all y ∈ m. Put r(m) = q(m) − p(m). Remark that the number p(m) is determined only by ad-representation of k in m.
(see [My3,Prop.10] or [Mi]). Therefore, dim(g we obtain that the number is a maximal number of functions in involution from the set A K m functionally independent at the point x. For arbitrary element x ∈ R(m) we have [My3,Prop.9] [m(x), k x ] = 0. (2.8) Remark 2.1. The subspace m(x) ⊂ m in some sense characterizes the Ad-action of K on m because Ad(K)(m(x)) = m (for any y ∈ m the function f y : k → Ad k(y), x on the compact Lie group K takes its maximum value at some point k 1 ∈ K and therefore Ad k 1 (y)⊥ ad x(k)(see [My2,Lemma 2.1])). By the dimension arguments (see (2.7)) from (2.8) and definition (2.5) we get that for x ∈ R(m) and The compact Lie algebra k is a direct sum k = z(k) ⊕ k s of the center and of the semisimple ideal. The center z(k) of k we will denote simply by z for short. Then we have the following orthogonal splittings with respect to the invariant form , which serve as definition for m z (if z = 0 then m z = m and k s = k).
Consider the set R(m z ) determined by (2.5) for the pair (g, k s ). Then for any In other words, the following lemma is proved: Remark that in the last condition in relation (2.9) describing the set The following proposition is Proposition 2.3 from [MP] adapted to the case of compact Lie algebras.
Proposition 2.3. [MP] Assume that x 0 ∈ R(m). Let g 0 and k 0 be the centralizers of k x0 in g and k respectively.
(3) for any x ∈ m 0 ∩R(m) we have k x = k x0 and the centralizer k x 0 is contained in the center z(g 0 ) of the compact Lie algebra g 0 : k x 0 = z(g 0 ) ∩ k 0 ; (4) any element x ∈ m 0 ∩R(m) is a regular element of the compact Lie algebra g 0 and x ∈ R(m 0 ) (i.e. (m 0 ∩ R(m)) ⊂ R(m 0 )).
Remark 2.4. Recall that a subalgebra a ⊂ g is regular if its normalizer N (a) in g has maximal rank, i.e. rank N (a) = rank g. It is well known that the centralizer g x of the element x ∈ R(m) is a regular subalgebra of g of maximal rank (containing some Cartan subalgebra of g), and, consequently, rank g = dim z(g In particular, the semisimple Lie algebra g x s is a regular subalgebra of g.
Corollary 2.5. The Lie algebra g 0 is a regular subalgebra of g and g The intersection g 0 ∩k x0 s = 0 vanishes because the Lie algebra k x0 s is semisimple. Since we obtain that z(g x ) = g x 0 and rank g 0 = rank g−rank k x0 s . The last equality means that g 0 is a regular subalgebra of g. By dimension arguments k

2.2
The bi-Poisson structure {η t (ω O )}: exact formulas and involutive sets of functions Consider the adjoint action Ad of G on the Lie algebra g. Suppose now in addition that the Lie subgroup K is an isotropy group of some element a ∈ g, i.e. K = {g ∈ G : Ad g(a) = a} and k = g a . Moreover, now by invariance of the form , and ad a(m) ⊂ m.
Using the invariant form , on the Lie algebra g, we identify the dual space g * and g. So Consider on T O two symplectic forms: ω 1 = Ω and ω 2 = Ω + τ * α. Write η 1 = ω −1 1 , η 2 = ω −1 2 for the inverse Poisson bi-vectors. Then the family {η t (ω O ) = η t = t 1 η 1 + t 2 η 2 }, t 1 , t 2 ∈ R, is a bi-Poisson structure [MP,Prop.1.6]. Putting t 2 = λ, t 1 = 1 − λ, λ ∈ R or t 1 = −1, t 2 = 1 we exclude a considering of proportional bi-vectors. The corresponding bi-vectors we denote by η λ , λ ∈ R and η si (the singular bi-vector). The Poisson bracket of two functions f 1 , f 2 from the set A K m = A G ⊂ E(T O) with respect to the Poisson structure η λ , λ ∈ R or η si has the form [MP,Lemma 3.1]: (2.10) Remark that the structure η 0 (λ = 0) is the canonical Poisson structure (see (2.2)). (2.12) Let m C (x) be the complexification of the space m(x), x ∈ R(m). It is easy to see that the kernel of the form B λ x in m C (x) is the subspace of m C (x) given by (2.13) In particular, for λ = 0 (for the canonical Poisson structure on T (G/K)), Since x ∈ R(m), a (real) dimension of the space (g x ) m is equal to the constant r(m) = q(m) − p(m). Therefore a maximal isotropic subspace of the space m(x) with respect to the form B 0 x has dimension 1 2 (r(m) + dim m(x)). It is clear that where ad −1 a def = (ad a|m) −1 . As an immediate consequence of Proposition 1.3 we obtain Denote by I(g) the space of all Ad(G)-invariant polynomials on g. If h ∈ I(g) then it is clear that the function h λ : g → R, h λ (y) = h(y + λa), λ ∈ R, is Ad(K)-invariant on g. Therefore the set F = {h λ |m, h ∈ I(g), λ ∈ R} is a subset of A K m = A G (of G-invariant function on T (G/K)). The following assertion was proved in [MP] (see Proposition 3.6, Lemma 3.3 and Theorem 3.9). But in our paper [MP] the subsets O F and O Kr of m are not described explicitly. Nevertheless, using Theorem 2.7 we prove that each of these sets contains the Proof. Let us consider in the complex spaces (z ⊕ m) C = m C z and m C nonempty Zariski open subsets R(m C z ) and R(m C ) defined as R(m z ) and R(m) in the real case (see (2.5)). For example, where q(m C z ) (resp. p(m C z )) is the minimum of (complex) dimensions of the spaces (g C ) y (resp.(k C ) y ) over all y ∈ m C z . It is clear that q(m C z ) = q(m z ) and p(m C z ) = p(m z ), q(m C ) = q(m) and p(m C ) = p(m) (a complex polynomial function which vanishes on a real form of a complex space vanishes identically on this complex space).
by definitions of the sets R(m) and R(m z ) containing y. Then dim C ((g C ) y+λa ) m C = r(m) = q(m) − p(m) iff dim C (g C ) y+λa = q(m). But q(m) = q(m z ) by (2.9). Now taking into account that p(m C z ) = p(m z ) and q(m C z ) = q(m z ), we complete the proof.
Fix some element x ∈ R(m) ∩ R(z ⊕ m) ∩ Q a (m). If x ∈ O Kr the assertion of the theorem is evident. Suppose that x ∈ O Kr and choose some point is a Zariski open subset of m and x 0 ∈ O Kr 0 , the whole real affine line {y t = x 0 + t(x − x 0 ), t ∈ R} with the exception of a finite set of points with t ∈ T N = {t 1 , .., t N } belongs to O Kr 0 ⊂ O Kr . In other words, at each point y t , t ∈ R \ T N the pair (A G , η t (ω O )) is Kronecker and therefore by Proposition 2.6 dim C ((g C ) yt+λa ) m C = r(m) for all λ ∈ C. Since each such y t ∈ O Kr 0 is an element of R(m) ∩ R(z ⊕ m), by Lemma 2.9 the set R((z ⊕ m) C ) contains each complex affine line l(y t ; a), t ∈ R \ T N .
Consider the complex affine plane π(x 0 , x; a) = {x 0 + λa + µ(x − x 0 ), λ, µ ∈ C} in (z⊕m) C containing the points x 0 and x. But R((z⊕m) C ) is a Zariski open subset, i.e. is defined by a finite family {P 1 , .., P k } of complex polynomial functions. The restriction p j def = P j |π(x 0 , x; a), j = 1, .., k is a polynomial function of the two variables λ, µ ∈ C. Since l(y t ; a) ⊂ R((z ⊕ m) C ), t ∈ R \ T N , then each polynomial p j is constant on such a line l(y t ; a). In other words, p j (λ, µ) µ=t = c j (t) for all t ∈ R \ T N , where c j (t) ∈ C. Since the set t ∈ R \ T N is infinite, the polynomial p j , j = 1, .., l is a function of only one variable µ, i.e. p j (λ, µ) = c j (µ), where c j (µ) is a polynomial.
Suppose that the complex affine line l(x; a) = l(y 1 ; a) ⊂ π(x 0 , x; a) is not a subset of the set R((z ⊕ m) C ). Then all polynomials P j , j = 1, .., k vanish in some point x + λ 0 a of this line, and, consequently, vanish identically on this line: 0 = p j (λ 0 , 1) = c j (1) for all j = 1, .., k. But x ∈ R(z ⊕ m) ⊂ R((z ⊕ m) C ), i.e. P j (x) = p j (0, 1) = c j (1) = 0 for some 1 j k, the contradiction. Thus the line l(x; a) is a subset of the set R((z ⊕ m) C ). But x ∈ Q a (m), i.e. dim C ker B si x = r(m). Now the assertion of theorem follows immediately from Lemma 2.9 and Proposition 2.6. Proof. Our proof of the theorem is based on the proof of Proposition 3.6. in [MP]. Let x ∈ R(m) ∩ R(z ⊕ m) ∩ Q a (m). By Theorem 2.8 the pair (A G , η t (ω O )) is Kronecker at x. Then by Proposition 1.3 the space L x (of maximal rank) corresponding to the canonical Poisson structure η 0 . Here for t ∈ R 2 we consider B t x as a real form on m(x) with C-linear extension described by relations (2.12) But the space L x is generated by a finite subset of spaces from the set {V t x }. Since by the first relation in (2.12) the family V t x depends smoothly on the parameter t ∈ R 2 \ {0}, we can suppose that this finite subset of spaces does not contain the kernel of the singular form B si x . In other words, L x is defined by (2.13) with λ j ∈ R, j = 1, N . Moreover, since x ∈ R(z ⊕ m) and R(z ⊕ m) is a Zariski open subset of z ⊕ m, we can choose these numbers {λ j } such that each x + λ j a ∈ R(z ⊕ m).
Let h ∈ I(g) and y ∈ g. Then [y, grad g h(y)] = 0 by invariance of the form , , i.e grad g h(y) ∈ g y . But since y is a semisimple element of the reductive Lie algebra g C , by [My2,Theorem 2.5] the vectors {grad g h(y), h ∈ I(g)} span the center z(g y ) of g y . But each x + λ j a ∈ R(z ⊕ m). Therefore for y = x + λ j a by (2.6) [g y , g y ] = [k y s , k y s ] ⊂ k, i.e. the maximal semisimple ideal of g y is a subalgebra of k. Since g y = z(g y ) ⊕ [g y , g y ], we have (g y ) m = (z(g y )) m . But by (2.13) V j x = (g x+λj a ) m . Now taking into account that grad m h λj (x) = (grad g h(x + λ j a) m , we obtain that the space L x ⊂ m(x) of dimension 1 2 r(m) + dim m(x) is generated by the vectors {grad m h λj (x), h ∈ I(g), j = 1, N }. The proof is completed.
The following corollary is evident.
x, [y, L x ] = 0 and y ∈ m(x) implies y ∈ L x .

Adjoint orbits and involutive automorphisms
Let σ be an involutive automorphism of g and let g =g ⊕ g ′ be the decomposition of g into the eigenspaces of σ for the eigenvalues +1 and −1 respectively: Denote byG the closed connected subgroup of G with the Lie algebrag. Fix some element a ∈ g ′ (σ(a) = −a) and consider the orbitÕ = Ad(G)(a) =G/K in g through this element a. It is clear thatÕ is a submanifold (G-suborbit) of the G-orbit O = G/K of a andK =G ∩ K.
Since σ(a) = −a, the algebra k = g a is σ-invariant. Suppose that the form , = −Φ χ is also σ-invariant (if g is semi-simple its Killing form is invariant with respect of an arbitrary automorphism of g). Then σ(m) = m and we have in addition to (2.1) the following orthogonal decompositions of algebras g,g, k with respect to the form , wherek,m are subspaces ofg, k ′ , m ′ are subspaces of g ′ . In particular, (k,k) is a symmetric pair of compact Lie algebras, i.e.k is the fixed point set of the involutive automorphism σ|k.
Since ker ad a = k and m = k ⊥ in g, then ad a(m) = m and the operator ad a|m : m → m is invertible. Moreover, for m ′ ⊂ m andm ⊂ m we have and therefore dimm = dim m ′ . Since ker(ad a|m) = 0, we have ad a(m ′ ) =m, and ad a(m) = m ′ . (2.19) Let AG (resp. AK m ) be the set of allG-invariant (resp. Ad(K)-invariant) functions on T (G/K) (resp. onm). The Poisson bracket of two functionsf 1 ,f 2 ∈ AK m with respect to the canonical Poisson structureη can (determined by the canonical 2-formΩ on TÕ) has the form (see (2.2)) : (2.20) Since σ(a) = −a the center z of the reductive Lie algebra k = g a is σ-invariant, i.e. z =z ⊕ z ′ , wherez = z ∩g, z ′ = z ∩ g ′ . It is clear that a ∈ z ′ . Then for each element b ∈ z ′ we can consider the endomorphism ϕ a,b : g → g on g putting ϕ a,b (x) = ad −1 a ([b, x]) for x ∈ m and ϕ a,b (z) = z for z ∈ k, where, recall, It is clear that the endomorphism ϕ a,b is symmetric and the group Ad(K) commutes elementwise with ϕ a,b on m. Therefore the operator ϕ a,b |m is also symmetric and the group Ad(K) commutes elementwise with ϕ a,b onm. So the functioñ (as a function on T (G/K) from the set AG = AK m ) is a Hamiltonian function of the geodesic flow of some pseudo-Riemannian metric onG/K if ϕ a,b |m is non-degenerate.
Consider the space I(g) of all Ad(G)-invariant polynomials on g. As we remarked in the previous subsection for each h ∈ I(g) the function h λ : g → R, h λ (y) = h(y + λa), λ ∈ R, is an Ad(K)-invariant function on g. Therefore the set F = {h λ |m, h ∈ I(g), λ ∈ R} is a subset of A K m = A G (of G-invariant function on T (G/K)) and the setF = {h λ |m, h ∈ I(g), λ ∈ R} is a subset of AK m = AG (of G-invariant function on T (G/K)). Put H λ = h λ |m andH λ = h λ |m. The following lemma follows easily from the results of [MF] (see also [TF,Ch.6,16.Lemma] or [DGJ,sec.3]).
Lemma 2.12. [DGJ] For any functions h 1 , h 2 , h ∈ I(g) and arbitrary parameters λ 1 , λ 2 , λ ∈ R we have {H λ1 1 ,H λ2 2 } can = 0 and {H λ ,H a,b } can = 0. Proof. Mainly to fix notations we shall prove this lemma here. Since σ is an automorphism of g and Ad(G) is a normal subgroup of Aut(g), we have f = h • σ ∈ I(g) if h ∈ I(g). But for any x ∈g, (2.21) because σ(a) = −a and σ(x) = x. The five functions h λ1 and h a,b commute pairwise on g ≃ g * with respect to the standard (linear) Lie-Poisson bracket on g [MF]. This means that for any pair of functions F 1 , F 2 from this set we have for all x ∈m ⊂ g.
Then by (2.21) As follows from the lemma above the setF is an involutive subset of (AK m ,η can ).
Let us define the numbers q(m), p(m), r(m) and the subset R(m) ⊂m similarly to the numbers q(m), p(m), r(m) and the subset R(m) ⊂ m but for the pair of algebras (g,k) (see (2.5)).
Theorem 2.13. Suppose that the Zariski open subsetÕ ofm is nonempty. Then the setF is a maximal involutive subset of the algebra (AK m ,η can ). For each point x from the nonempty Zariski open subsetÕ∩R(m) ⊂m there are 1 2 r(m)+dimm(x) functions from the setF functionally independent at x.
It is evident thatL x = (L x )m, where ()m denotes the projection ontom along m ′ in m =m ⊕ m ′ . Moreover, since x ∈m and k =k ⊕ k ′ , we have ad x(k) = ad x(k) ⊕ ad x(k ′ ), where ad x(k) ⊂m and ad x(k ′ ) ⊂ m ′ , and therefore (2.23) By Lemma 2.12 the spaceL(x) is an isotropic subspace ofm(x) with respect to the formB x : (y 1 , y 2 ) → x, [y 1 , y 2 ] onm(x) associated with Poisson bracket (2.20). So that to prove the theorem it is sufficient to show that this subspace is maximal isotropic.
To this end suppose that x, [y,L x ] = 0 for some y ∈m(x). Then [x, y],L x = 0 by invariance of the form , . Taking into account that [x,m(x)] ⊂m by definition (2.3) andm⊥m ′ , we obtain that But y ∈ m(x) by (2.23). Also by Corollary 2.11 the space L x is a maximal isotropic in m(x) and, consequently, y ∈ L x . Then y ∈L x , because y ∈m and L x ∩m ⊂ (L x )m =L x . In other words,L x is a maximal isotropic subspace inm(x) with respect to the formB x .
As follows from Theorem 2.13 we have to establish when the setÕ (2.22) is nonempty. It is clear thatÕ is nonempty iff R(m) ∩ R(z ⊕ m) ∩m = ∅ and Q a (m) ∩m = ∅. Therefore we consider these two Zariski open subsets in more detail.
Suppose that the set R(m)∩R(z⊕ m)∩m is nonempty and that x 0 is a common element of this set and the set R(m). Let k 0 be the centralizer of the Lie algebra k x0 in k, putk 0 = k 0 ∩k. Since σ(x 0 ) = x 0 , then σ(k x0 ) = k x0 and, consequently, σ(k 0 ) = k 0 . We have the following splitting of k 0 = k ′ 0 ⊕k 0 , [k ′ 0 , k ′ 0 ] ⊂k 0 , associated with σ. Let K 0 andK 0 be the connected Lie subgroups of K with the Lie algebras k 0 andk 0 respectively. These subgroups are closed in K and K 0 /K 0 is a symmetric space. Let V be a some vector subspace of the space m. Put Proposition 2.14. Suppose that R(m) ∩ R(z ⊕ m) ∩m = ∅ and choose arbitrary point (2.25) Proof. By Remark 2.1 Ad(K) m(x 0 ) = m. Since Ad k(a) = a for all k ∈ K, we obtain that m a (m) = m a (m(x 0 )) = r(m). Similarly, since Ad(K) m(x 0 ) =m and K = K ∩G ⊂ K, then m a (m) = m a m(x 0 ) . (2.26) We will use the moment map theory to calculate the number m a (m) (this method was proposed in [Pa]). For our aim it is convenient to use the moment map constructed in our previous paper [MP]. So here we briefly describe main properties of this moment map [MP,Remark 3.2].
Since the endomorphism ad a|m : m → m is skew-symmetric (with respect to the form , ), the form β is also skew-symmetric. Identifying the tangent space T x m with m for each x ∈ m, we can consider β as a symplectic form on m. Since the Ad-action of K on m preserves the form β, this action of K is Hamiltonian with the K-equivariant moment map µ β : m → k * , µ β (x)(ζ) = − 1 2 ad −1 a (x), [ζ, x] , ∀ζ ∈ k (see [MP,Remark 3.2]). In particular, for each ζ ∈ k the vector field ζ X (x) = [ζ, x] ∈ T x m is the Hamiltonian vector field of the function f ζ (x) = µ β (x)(ζ) on the manifold X = m.
Let x ∈ m, W x ⊂ T x m be the tangent space to the K-orbit in m and let W β⊥ x be the (skew)orthogonal complement to W x in T x m with respect to the form β. It is easy to see that W x = ad x(k) and W β⊥ x = ad a(m(x)), i.e.
dim W x ∩ W β⊥ x = dim ad a m(x) ∩ ad x(k) .
Identifying the space k with its dual k * using the form , , we obtain that (2.28) Fix some element x 0 ∈ R(m) ∩ R(z ⊕ m) ∩ R(m). Since x 0 ∈m ⊂ m, we have thatm(x 0 ) ⊂ m(x 0 ) (see definition (2.3) or proof of (2.23)). Let g 0 and k 0 be the centralizers of k x0 in g and k respectively. Let m 0 = k ⊥ 0 ⊂ g 0 . Then by Proposition 2.3 the centralizer k x0 0 of x 0 in k 0 is the center z(k x0 ) of the Lie algebra k x0 and a subalgebra of the center z(g 0 ) of the Lie algebra g 0 : (2.30) But as we remarked above, σ(k x0 ) = k x0 because σ(x 0 ) = x 0 and σ(k) = k. It is clear also that the spaces g 0 , k 0 and m 0 are σ-invariant, i.e. (2.31) It is evident that a ∈ g 0 and ker(ad a|g 0 ) = k 0 , ad a(m 0 ) = m 0 and the operator ad a|m 0 is invertible. Then by (2.27) µ β (m 0 ) ⊂ (g 0 ) k = k 0 . We can prove also that using the same arguments as in the proof of relations (2.19). Taking into account that dim k x0 is the minimum of dimensions of centralizers k y , y ∈ m and x 0 ∈m(x 0 ) ⊂m 0 ⊂m, using (2.28) and (2.31), we conclude that As we remarked above µ β (m 0 ) ⊂ k 0 and the operator ad a|m 0 is invertible. Therefore it is naturally to consider on the vector space m 0 the non-degenerate skew-symmetric form β 0 (y 1 , y 2 ) = y 1 , ad −1 a (y 2 ) , where y 1 , y 2 ∈ m 0 . It is clear that the pair (m 0 , β 0 ) is a symplectic submanifold of the symplectic manifold (m, β). Since the Ad-action of K 0 on m 0 preserves the form β 0 , this action of K 0 is Hamiltonian with the K 0 -equivariant moment map µ β0 : 0 is a subalgebra of the center z(g 0 ) of the Lie algebra g 0 and is σ-invariant (see (2.30)). In particular, k x0 0 = z(k x0 ) is subalgebra of the center z(k 0 ) of the Lie algebra k 0 . Therefore, the orthogonal complement to k x0 0 in k 0 is a compact σ-invariant Lie subalgebra k e of k 0 and k 0 = k e ⊕ k x0 0 . Let us prove that µ β0 (m 0 ) is subset of the Lie algebra k e ("effective part"). Indeed, as we remarked above [k contains an open subset in the space k ′ e def = (1−σ)k e = k e ∩k ′ 0 . It is easy to calculate that for any tangent vector y ∈ m 0 = T x0 m 0 Taking into account relations (2.32) and (2.18) and the inclusion x 0 ∈m 0 ⊂g, we obtain that The image µ β0 * (T x0 m 0 ) ⊂ k 0 of the tangent map of the moment map µ β0 at x 0 coincides with the annihilator in k * 0 ≃ k 0 of the Lie algebra k x0 0 of the isotropy group {k ∈ K 0 : Ad k(x 0 ) = x 0 } of x 0 ∈ m 0 [GS]. Since this annihilator coincides with k e , then k e = D x0 (m 0 ). Since the Lie algebra k e is σ-invariant, then k ′ (2.33) It is not evident that min α∈k ′ e dim k α = min α∈k ′ 0 dim k α , where, recall, k 0 = k e ⊕ k x0 0 . We will prove this fact, using the moment map µ β0 . To this end first of all remark Choose an arbitrary element x ∈ m 0 ∩ R(m). By item (2)  Let W x ⊂ T x m 0 be the tangent space to the K 0 -orbit in m 0 and let W β0⊥ x be the (skew)orthogonal complement to W x in T x m 0 with respect to the form β 0 . It is easy to see that W Now we can apply the method used above to prove expression (2.28) changing the moment map µ β by µ β0 . By the K 0 -equivariance of the moment map µ β0 , [GS]. In other words, Since dim k x0 Since for any element α ∈ k ′ e ⊂ k 0 its centralizer k α contains the algebra k α e ⊕ k x0 0 ⊕ k x0 s , we obtain, comparing the dimensions in (2.33) and (2.35), that k α = k α e ⊕k x0 0 ⊕k x0 s ⊂ k for almost all α ∈ k ′ e . Taking into account that k x0 0 is a subalgebra of the center of k e ⊕ k x0 0 ⊕ k x0 s , we can replace the space k ′ e in expressions (2.33) and (2.35) by the space k ′ 0 ⊂ k ′ e ⊕ k x0 0 . As a result we obtain (2.25). Remark also that the algebra k e ⊕ k x0 0 ⊕ k x0 s is the normalizer N (k x0 ) of k x0 in k. As an immediate consequence of the proof we have Corollary 2.15. For all elements α from some nonempty Zariski open subset of the space k ′ 0 ⊂ k 0 the centralizer k α belongs to the subalgebra k 0 + k x0 = N (k x0 ) ⊂ k. Corollary 2.16. Suppose that R(m) ∩ R(z ⊕ m) ∩m = ∅ and choose arbitrary point The set Q a (m) ∩m is nonempty, i.e. m a (m) = r(m), iff one of the following equivalent conditions holds: (1) for some element α ∈ k ′ 0 : dim k α = dim g x0 ; (2) the subspace k ′ 0 contains regular elements of the Lie algebra k 0 . Proof. To prove item (1) it is sufficient to remark that by definition m a (m) m a (m) = r(m), r(m) = dim g x0 − dim k x0 , and by (2.25) for all γ from some nonempty Zariski open subset of k ′ 0 we have dim k γ = m a (m) + dim k x0 = (m a (m) − r(m)) + dim g x0 .
By Corollary 2.5 r(m) = rank g 0 − dim k x 0 and x 0 is a regular element of the Lie algebra g 0 , i.e. g x0 0 is a Cartan subalgebra of g 0 . As above for all γ from some nonempty Zariski open subset of k ′ 0 we have dim k γ 0 = m a (m) + dim k x0 0 = (m a (m) − r(m)) + dim g x0 0 . By definition the algebra g 0 contains the element a ∈ g ′ (σ(a) = −a) and k = g a . Therefore k 0 = k∩g 0 = g a 0 , i.e. the Lie algebra k 0 is a a subalgebra of g 0 of maximal rank: rank k 0 = rank g 0 . Therefore, dim k α 0 = dim g x0 0 for some α ∈ k ′ 0 iff k α 0 is a Cartan subalgebra of k 0 .

Integrable geodesic flows
Here we will use notations of Subsections 2.2 and 2.3. Consider the suborbit O = Ad(G) · a ≃G/K of the adjoint orbit O = Ad(G) · a ≃ G/K in the compact Lie algebra g.
Theorem 2.17. Suppose that the Zariski open subset O = R(m) ∩ R(z ⊕ m) ∩m ofm is nonempty and the subspace k ′ 0 contains regular elements of the Lie algebra k 0 . Here k 0 is the centralizer of k x0 for arbitrary x 0 ∈ O, k ′ 0 = (1 − σ)k 0 . Then there exists a maximal involutive set of independent real analytic functions on (T (G/K),Ω). These functions are integrals for 1) the geodesic flow determined by the Riemannian metric , onG/K; 2) the Hamiltonian flow with the Hamiltonian functionH a,b on T (G/K).
Proof. By Corollary 2.16 and by Theorem 2.13 there exists m = 1 2 (r(m)+ddim AG) independent involutive functions from the set AG. These functions form a maximal involutive subset of independent functions in the algebra AG = AK m with respect to the canonical Poisson structure on T (G/K). Moreover, by Lemma 2.12 these functions are integrals of 1) the geodesic flow on T (G/K) determined by the form , ; 2) the Hamiltonian flow with the Hamiltonian functionH a,b on T (G/K). Now the assertion of the theorem follows immediately from Proposition 1.5.
2.5 Integrable geodesic flows on SO(n)/ SO(n 1 )×· · ·×SO(n p ) In this subsection we show that the conditions of Theorem 2.17 hold for the homogeneous space SO(n)/ SO(n 1 ) × · · · × SO(n p ) . Consider the symmetric space G/K = U (n)/SO(n), where n 4, with the involution σ on the Lie algebra of skew-hermitian matrices g = u(n) defined by the complex conjugation. Then the Lie algebrag = (1+σ)g is the Lie algebra so(n) of all real skew-symmetric n × n matrices. The space g ′ = (1 − σ)g coincides with the set i sym(n), where sym(n) is the space of all real symmetric n × n matrices.
Putting X, Y = −TrXY (using the trace-form) we define an invariant scalar product on g. To describe the space m = k ⊥ consider any matrix X ∈ g as a block-matrix consisting of rectangle elements X k,l , which are rectangle complex n k × n l matrices, 1 k, l p. It is clear that (X k,l ) t = −X l,k and therefore any element of u(n) is defined by its blocks X k,l with k l. As a space the Lie algebra g = u(n) is a direct sum of its block-type subspaces V k,l , 1 k l p. In this notation the Lie subalgebra k is the direct sum p j=1 V j,j and m = 1 k<l p V k,l . We will denote the corresponding to X k,l element of the space V k,l by ϕ(X k,l ).
Suppose now that p 3 and n p n 1 + .. + n p−1 . We claim that for some element x 0 ∈m its centralizer k x0 is the one-dimensional center z(g) of g = u(n). To simplify our calculations remark that each space V k,k ⊕ V l,l ⊕ V k,l , k < l is a Lie subalgebra of u(n) isomorphic to u(n k + n l ), V k,k ⊕ V l,l ≃ u(n k ) ⊕ u(n l ) and [V k,l , V k,l ] ⊂ V k,k ⊕ V l,l . But U (n k + n l )/ U (n k ) × U (n l ) is a Hermitian symmetric space and therefore we can use our calculation for the case p = 2. Since each subspace V k,l is k-module, we will construct the element x 0 selecting step by step its V k,l -entries. For our aim it is enough to consider the submodule p−1 j=1 V j,j+1 ⊕ p−2 j=1 V j,p of m. Choosing in each k-module V j,j+1 of the first component the "diagonal" element ϕ(X j,j+1 * ) as above, we obtain that their common isotropy algebra is a direct sum h * ⊕ b * , where b * ≃ u(n p − n p−1 ) and h * is of commutative algebra of dimension n p−1 , consisting of diagonal matrices diag(ix 1 , .., ix n1 , ix 1 , .., ix n2 , .., ix 1 , .., ix np−1 , ix 1 , .., ix np−1 , 0, .., 0), ∀x j ∈ R. Remark that the maximal semisimple ideal of b * coincides with the maximal semisimple ideal of the centralizer of h * in k. Now we consider V = p−2 j=1 V j,p as a h * ⊕ b *module (not as k-module) of complex dimension N × n p , N = n 1 + .. + n p−2 . Then V is direct sum of h * ⊕ b * -modules V (1) ⊕ V (2) , V (1) ⊥V (2) , where V (1) (of complex dimension N × n p−1 ) is trivial b * -module, and therefore the isotropy subalgebra h * * in the diagonal subalgebra h * of a real generic point from V (1) is one-dimensional (consisting of elements of h * with x 1 = x 2 = .. = x np−1 = λ). Considering the module V (2) as the space of complex N × (n p − n p−1 ) matrices with elements B, the ad-representation of h * * ⊕ b * in V (2) is described as follows: (iλ, A) · B = iλB − BA, A ∈ u(n p − n p−1 ). Since the number of rows N in B is greater then n p − n p−1 by our assumption, then for any real B of maximal rank iλB − BA = 0 iff A = iλ (is a scalar matrix). Therefore k x0 0 = z(g) for some real matrix x 0 ∈ m and k 0 = k. This element belong to the set R(m) ∩ R(z ⊕ m). Since the space k ′ 0 = k ′ = isym(n 1 ) ⊕ .. ⊕ isym(n p ) contains a regular elements of u(n 1 ) ⊕ .. ⊕ u(n p ), the conditions of Theorem 2.17 hold.
Suppose now that p 3 and n p > n 1 + .. + n p−1 . In this case, since the last component k p ≃ u(n p ) of k is dominant in k, the calculation problem can be reduced to the previous case with n p = n 1 + .. + n p−1 . To this end we consider the representation of the Lie group K p ⊂ K with the Lie algebra k p ≃ u(n p ) in the k-submodule V = p−1 j=1 V j,p of m. Identifying V with the space of complex N × n p , N = n 1 + .. + n p−1 matrices with elements B, the Ad-action of K p in V is described as follows: k · B = Bk −1 , k ∈ K p = U (n p ). Since the number of rows N in B is less then its number of columns n p , then the Ad(K p )-orbit of B in V contains an matrix in which last n p − N columns vanish. In other words, each element of m is Ad(K) conjugated to some element of the first component g 2N of the Lie algebra u(2N ) ⊕ u(n p − N ) ⊂ u(n). Taking into account that the pair g 2N and k 2N = g 2N ∩ g a has the properties considered above, the space m ∩ g 2N contains a real matrix x 0 with k x0 2N ≃ R. Then k x0 ≃ R ⊕ u(n 2 − N ) because x 0 is regular element of g 2N by property (2.6) and therefore g x0 ≃ R 2N ⊕ u(n 2 − N ). It can be checked easily that the conditions of Theorem 2.17 hold. The following theorem is proved: Theorem 2.18. There exists a maximal involutive set of independent real analytic functions on (T (G/K),Ω), whereG = SO(n) andK = SO(n 1 ) × · · · × SO(n p ) with the standard embedding ofK ⊂G. These functions are integrals for 1) the geodesic flow determined by the Riemannian metric , onG/K; 2) the Hamiltonian flow with the Hamiltonian functionH a,b on T (G/K).