A CANONICAL DISTRIBUTION ON ISOPARAMETRIC SUBMANIFOLDS I

We show that on every compact, connected homogeneous isoparametric submanifold M of codimension h ≥ 2 in a Euclidean space, there exists a canonical distribution which is bracket generating of step 2. An interesting consequence of this fact is also indicated. In this first part we consider only the case in which the system of restricted roots is reduced, reserving for a second part the case of non-reduced restricted roots.


Introduction
This paper, of which we present here its first part, is devoted to indicating some properties (that we think have not been previously studied) of compact, connected homogeneous isoparametric submanifolds of Euclidean spaces of codimension h ≥ 2.
By a celebrated theorem due to G. Thorbergsson [7], all compact, connected, isoparametric submanifolds of Euclidean spaces of codimension h ≥ 3 are homogeneous. On the other hand, in codimension h = 2 there are infinitely many non-homogeneous examples and only a finite number of homogeneous ones. Homogeneous isoparametric submanifolds M n of R n+h are obtained as principal orbits of the tangential representation (at a basic point) of a compact (or non compact dual) symmetric space. A way to obtain these submanifolds explicitly is to consider a real simple noncompact Lie algebra g 0 with Cartan decomposition g 0 = k 0 ⊕ p 0 . Then k 0 is a maximal compactly embedded subalgebra of g 0 [4,Pr. 7.4,p. 184]. Let K be the analytic subgroup K of Int(g 0 ) corresponding to the subalgebra ad g0 (k 0 ) of ad g0 (g 0 ) which is compact. The principal orbits of the representation of K on p 0 are isoparametric submanifolds M n of R n+h = p 0 . The central objective of this work is to present the following result: Theorem 1.1. On any compact, connected, homogeneous isoparametric submanifold (for a real simple noncompact Lie algebra g 0 ) there exist a smooth completely non-integrable (i.e., bracket generating) step 2 distribution D ⊂ T (M n ), canonically associated to the manifold.
The proof of Theorem 1.1 is naturally divided into two parts by the nature of the system of restricted roots associated to the above Cartan decomposition of the corresponding real simple noncompact Lie algebra g 0 . The system of restricted roots can be either the system of roots of a complex simple Lie algebra (reduced case) or (BC q ), in the non-reduced case [4, 3.25, p. 475]. In this first part we take care of the proof of Theorem 1.1 when the system of restricted roots is the system of roots of a complex simple Lie algebra (i.e., it is reduced), and reserve for part 2 of this paper the proof for (BC q ). Nevertheless, in the present part (up to Section 6) we introduce the facts and tools needed for both parts. Corollary 1.2 is somehow related to one of the results of the important paper [3]. We refer to Theorem D of that paper, which motivated Theorem 1.1. We do not include a proof of Corollary 1.2 since it is a well known consequence of the fact that the distribution D is completely non-integrable of step 2.
The rest of the paper contains the description of the distribution D and proof of the fact that it is bracket generating of step 2. This is organized as follows. In the next section we collect the necessary facts from Lie theory. The majority of them are taken from the very interesting paper [5], from which we adopt much of the notation and the important Proposition 2.1. The paper [5] has been very useful to help us carry on the somewhat involved computations required. Other more standard facts are recalled from the usual sources (such as [4,2,6]). In Section 3 we define our submanifolds and their tangent and normal spaces. Section 4 introduces the basis from [5] and other notations required. In Section 5 we present the distribution and study its local fields and their covariant derivatives and brackets. In Section 6 we indicate some necessary lemmata about properties of the roots of g C 0 and their restricted counterpart. Finally, Section 7 contains the proof of Theorem 1.1, where the computations of products and brackets performed in the Appendix are extensively used.

Facts from Lie theory
We shall use some of the notation and a result (Proposition 2.1 below) from the paper [5], which plays an important part here.
Let g be a complex simple Lie algebra, h ⊂ g a Cartan subalgebra, and B the Killing form of g. Let Φ(g, h) ⊂ h * (dual space) be the root system. Given α ∈ Φ(g, h), let g α ⊂ g be its root space. Basic properties are in [4,Th. 4.2]; in particular, if α = ±β, g α is orthogonal to g β by B. Let t α ∈ h be the root vector corresponding to α defined by be a set of simple roots; we keep ∆(g, h) and the "order" generated by it in Φ(g, h), fixed. Set h α = 2tα B(tα,tα) , and h j = h αj , 1 ≤ j ≤ n. A Chevalley basis of (g, h) is a basis C = {x α , h j : α ∈ Φ(g, h), 1 ≤ j ≤ n} of g with the following properties: The fundamental importance of this basis is that the structure constants are integers. In fact their properties are: and h α is a Z linear combination of {h 1 , . . . , h n }. (d) c α,β = ±(r + 1), where r is the largest integer such that β − rα ∈ Φ(g, h).
Let g 0 be a real simple Lie algebra, g C 0 its complexification, and σ the corresponding conjugation with respect to g 0 . The Killing forms of both algebras agree on g 0 [4, 6.1, p. 180]; we shall use B for this form in both of them. If g 0 is not complex and one of the algebras (g 0 or g C 0 ) is simple then so is the other one. A decomposition of g 0 into a direct sum as g 0 = k 0 ⊕ p 0 (k 0 subalgebra of g 0 and p 0 a subspace) is called a Cartan decomposition of g 0 if there exists a compact real form u 0 of g C 0 such that the following conditions are satisfied: σ (u 0 ) ⊂ u 0 , k 0 = g 0 ∩ u 0 , and p 0 = g 0 ∩ (iu 0 ). Every semisimple Lie algebra g 0 over R has a Cartan decomposition, unique up to conjugation by an inner automorphism of g 0 . Clearly, u 0 = k 0 ⊕ ip 0 . Let τ be the conjugation of g C 0 = u C 0 with respect to u 0 . Then σ and τ commute on g C 0 . Therefore θ = στ = τ σ is an automorphism of g C 0 . Since θ leaves g 0 invariant, it is an automorphism called in [5] a Cartan involution, and k 0 is a maximal compactly embedded subalgebra of g 0 [4, Pr. 7.4, p. 184].
Let g 0 be a real simple Lie algebra and g 0 = k 0 ⊕ p 0 a Cartan decomposition with Cartan involution θ. Let h 0 ⊂ g 0 be a θ stable Cartan subalgebra such that a 0 = h 0 ∩ p 0 is a subspace of maximal dimension in p 0 . Let us consider the complexification (g, h) of (g 0 , h 0 ) ( h C 0 = h is a Cartan subalgebra of g C 0 = g) and, as above, let Φ(g, h) ⊂ h * be the root system. We have, of course, h 0 = (h 0 ∩ k 0 )⊕(h 0 ∩ p 0 ) and the roots of Φ(g, h) take imaginary values on (h 0 ∩ k 0 )

The submanifolds
Let us continue considering our real simple Lie algebra g 0 with Cartan decomposition g 0 = k 0 ⊕ p 0 and involution θ. Since k 0 is a maximal compactly embedded subalgebra of g 0 , the analytic subgroup K of Int(g 0 ) corresponding to the subalgebra ad g (k 0 ) of ad g (g 0 ) is compact. We take in g 0 the inner product (2.1). Then the group K acts on (p 0 , B θ ) by the adjoint representation (i.e., by isometries) and we consider the principal orbits of this action usually called isoparametric submanifolds. Then we fix a regular element E ∈ a 0 ⊂ p 0 and call It goes without saying that the split (normal) real form g S of g is included in our considerations. But we need to include also the so called manifolds of complete flags of compact connected simple Lie groups. To that end recall that our compact Lie algebra u 0 is a compact real form of g, that is, u 0 ⊕ iu 0 = g R , and this is a Cartan decomposition of the real Lie algebra g R [4, 7.5, p. 185]. Here we may take a compact connected Lie group G corresponding to u 0 (which we may take without center) and consider the principal orbits of G by the adjoint action on the complementary subspace (iu 0 ) or (suppressing the irrelevant factor i) the adjoint action of G on u 0 . In both cases (for g S and u 0 ) using Proposition 2.1 one has to make the corresponding "simplifications". For instance Φ (g 0 , a 0 ) = Φ (g, h) and ρ, σ are the identity.

Basis for g 0
Using part (ii) of Proposition 2.1, we may define k α for each α ∈ Φ (g, h) by the identity Also recalling the effect of τ ((i) of Proposition 2.1) we may consider the action of θ on x α and x α σ , that is, Now, by the definition (4.1), we have σ ( We shall need the basis constructed by Kammeyer in [5,Sec. 4]. Let us consider the σ and τ adapted Chevalley basis for (g, h) from Proposition 2.1 and set, for α ∈ Φ(g, h), These vectors are fixed by σ, so they belong to g 0 . Now setting we see that the vectors in the first row of (4.4) belong to k 0 and those in the second one to p 0 . Now, using (4.2) and definitions, we observe that: On the other hand, the vectors R α and W α shall be considered only for α real (i.e., α σ = α) and clearly R α = P α + Q α , W α = U α + V α . However, observe that, for α real, we have the equalities: and we conclude that P α is in k 0ρ(α) . Setting ρ (α) = ρ(β) = λ, similar computations for α ∈ Φ C , β ∈ Φ R with the vectors in (4.4) show that 4.1. Basis for k 0,λ and p 0,λ , λ ∈ Φ + (g 0 , a 0 ). Consider now for λ ∈ Φ + (g 0 , a 0 ) the set ρ −1 (λ) = {α ∈ Φ + (g, h) : ρ (α) = λ} and split it separating the real roots from the complex ones. So we set For a root α in ρ −1 (λ) C we have α σ = α; then we define, as in [5], the set ρ −1 (λ) * C where we place one of the two elements in {α, α σ } for each α ∈ ρ −1 (λ) C . Now for λ, µ ∈ Φ + (g 0 , a 0 ) take the sets (4.8) By (4.7), Ξ k (λ) ⊂ k 0λ and Ξ p (µ) ⊂ p 0µ , and each set is linearly independent over R. Since the equal cardinalities of Ξ k (λ) and Ξ p (λ) coincide with the dimensions of k 0,λ and p 0,λ , we have a basis for each of these subspaces. Obviously, there is a one to one correspondence between Ξ k (λ) and Ξ p (λ). As a consequence of (4.6), for the members of the bases Ξ k (λ) and Ξ p (λ) we have: which is coherent with their one to one correspondence.

Distribution
The roots of Φ + (g 0 , a 0 ) are written in terms of ∆ (g 0 , a 0 ) as a Z linear combination with non-negative coefficients. It is usual to define the height of a root as the sum of these coefficients, and we may consider in Φ + (g 0 , a 0 ) the subsets Ω and Γ of roots of odd and even height respectively, Φ + (g 0 , a 0 ) = Ω ∪ Γ. We may consider, associated to the set Ω, a subspace D E (Ω) ⊂ T E (M ) (see (3.1)) defined by D E (Ω) = λ∈Ω p 0λ . This subspace is invariant by the action of the isotropy subgroup at E. The union of the sets Ξ p (λ) with λ ∈ Ω is a basis for D E (Ω). Since Proceeding similarly with the vectors V β and W α , we get the local fields To understand the nature of D (Ω) we compute the brackets of the fields constructed above by using the Levi-Civita connection on M which is torsion free.

Covariant derivatives.
We use the fact that M ⊂ p 0 and (p 0 , B θ ) is a Euclidean space. So we may compute the Euclidean covariant derivative in p 0 , which we denote by ∇ E , of each field in (5.2) in the direction of each vector of Ξ p (µ) for µ ∈ Ω. Since they are all similar, we compute only one of them. Let us take γ ∈ ρ −1 (λ), ϕ ∈ ρ −1 (µ) (we may have λ = µ but in that case γ = ϕ).
To compute ∇ E Uϕ U F γ , we need to know the field U F γ restricted to a curve whose tangent vector at E is U ϕ , and to obtain it let us consider first the curve on M passing through E defined on an adequate interval (−ε, ε) ⊂ R by u (t) = Ad (exp (t (F P ϕ ))) E, for t ∈ (−ε, ε). Here F is the factor needed so that F P ϕ ∈ S 0, r 2 ; it will be irrelevant at the end so we keep it as a non-zero undefined constant associated to P ϕ . By (4.9) we have Then (since E is regular and ϕ ∈ ρ −1 (µ), ϕ(E) = µ(E)) we may write Ad (exp (tF P ϕ )) E.
So U ϕ is the tangent vector (at t = 0, i.e., at E) to the curve in M defined on (−ε, ε) ⊂ R by Now we need the restriction of the field U F γ to the curve ω(t). By the definition (5.1) we see that this restriction U F γ (ω(t)) is Then we may compute Ad (exp (tF P ϕ )) U γ and get Now we may obtain ∇ Uϕ U F γ by taking the tangential component of ∇ E Uϕ U F γ . So we have:

CRISTIÁN U. SÁNCHEZ
In (5.4) we have brackets of fields on the left side and products in g 0 on the right side. We use the words brackets for fields and products for vectors in g 0 . Recall that for λ, µ ∈ Ω, we have bases Ξ p (λ) for p 0λ and Ξ p (µ) for p 0µ , respectively. To fix notation we set them as Each of these tangent vectors at E generates a corresponding field around E So we have nine possible brackets of these fields.

Some required lemmata
It is convenient to introduce the following notation. For a root α contained either in Φ (g, h) or in Φ (g 0 , a 0 ) we shall write Recall that ∆ (g 0 , a 0 ) ⊂ Φ + (g 0 , a 0 ) is a system of simple roots for Φ (g 0 , a 0 ). In this Part 1, we are assuming that Φ (g 0 , a 0 ) is the system of roots of a complex simple Lie algebra (i.e., it is reduced).
Proof. This lemma is obtained by inspection of the table of roots in [2, pp. 528-531]. The mentioned table contains the form of the positive roots for the four types of classical algebras and the five exceptional ones. In the case of the classical algebras, if we take γ ∈ Γ, it must have an even number of coefficients 1, therefore it must contain a coefficient 1 at the left of the first obligatory filling 1 (first from the left, underlined in the table in [2]). Therefore, eliminating from γ the root corresponding to the coefficient 1 at the extreme left position (corresponding to some α j ∈ ∆ (g 0 , a 0 )) we obtain a root β in Ω. Then we may write γ = β + α j and clearly |β − α j | is not a root of Φ (g 0 , a 0 ).
On the other hand, for the four exceptional algebras e C 6 , e C 7 , e C 8 and f C 4 the tables are ordered by increasing height (altitudes in [2]) so the roots of Γ, in each case, are the ones contained in the rows in even position from the top while those in Ω are in the other rows. The roots in the 2k-th row are constructed from those in the (2k − 1)-th row by adding one of the simple roots in the first row. Then we see that any γ ∈ Γ can be written as γ = η + δ, with η, δ ∈ Ω and |δ − η| is not a root. Lemma 6.2. For g 2 , given γ ∈ Γ ⊂ Φ + (g 2 , a), we can find η = δ in Ω ⊂ Φ + (g 0 , a 0 ) such that either γ = η + δ and |η − δ| is not a root of Φ + (g 0 , a 0 ) or γ = |η − δ| and η + δ is not a root of Φ + (g 0 , a 0 ).

Proof of Theorem 1.1
Here we use the results in Subsection 8.5.1 of the Appendix.
Let us observe that in order to prove Theorem 1.1, it is enough to show that, for each λ ∈ Γ, each vector of the basis Ξ p (λ) of p 0λ ⊂ T E (M ) may be computed as a sum of brackets of local fields (defined around E) that belong to the distribution D (Ω). Let us take then λ ∈ Γ and recall the basis of p 0λ given in (4.8).
This completes the proof of Theorem 1.1 when Φ (g 0 , a 0 ) is reduced.

Comment on the split cases.
Since in the split cases Φ + (g, h) = Φ + (g 0 , a 0 ) and ρ and σ are the identity, Lemma 6.1 takes care of this case. In fact all roots are real and Lemma 6.1 indicates that given γ ∈ Γ ⊂ Φ + (g 0 , a 0 ) = Φ + (g, h), we can find η and δ in Ω ⊂ Φ + (g 0 , a 0 ) such that η = δ and they satisfy γ = η + δ and |η − δ| is not a root of Φ + (g 0 , a 0 ). Considering formulae (8.4) and (8.6) (for the roots η and δ) we have (since η −δ and δ −η are not roots) that H 1 = T 2 = 0. Then we may consider formulae (8.13), for which we may write a shortened version: and therefore the vector W γ is a bracket (evaluated at E) of local fields defined around E that belong to the distribution D (Ω). This proves the theorem for the split cases.