A CANONICAL DISTRIBUTION ON ISOPARAMETRIC SUBMANIFOLDS II

. The present paper continues our previous work [ Rev. Un. Mat. Argentina 61 (2020), no. 1, 113–130], which was devoted to showing 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. In that work this fact was established for the case when the system of restricted roots is reduced. Here we complete the proof of the main result for the case in which the system of restricted roots is ( BC ) q , i.e., non-reduced.


Introduction
We present here the second part of the paper [4] devoted to indicating some properties of compact, connected homogeneous isoparametric submanifolds of Euclidean spaces of codimension h ≥ 2. It is well known (see [5]) that 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. Here we study the case of those spaces in which the restricted roots form the non-reduced system Φ (g 0 , a 0 ) = (BC) q . This case was not included in [4], where only reduced systems of restricted roots were considered.
The compact, connected homogeneous isoparametric submanifolds of Euclidean spaces of codimension h ≥ 2 (considered here and in [4]) are the principal orbits of the tangential representations of the compact, connected, irreducible, symmetric spaces. In Table 1, in the next section, the reader can find the symmetric spaces whose tangential principal orbits are the isoparametric submanifolds considered here. We shall use the notation and basic facts from [4, and, hoping that the reader has the opportunity to take a look at that work, we shall not repeat these facts here (except for necessary notation and formulae). Recall that the theorem to be proved is: 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. 2020 Mathematics Subject Classification. 53C30, 53C42, 53C17.

CRISTIÁN U. SÁNCHEZ
A distribution D of r-planes (n > r ≥ 2) in a connected manifold M n is smooth [6, p. 41] if for any p ∈ M n there is an open set A containing p and r smooth vector fields {X 1 , . . . , X r } defined on A such that X j (q) ∈ D (q) and D (q) = span R {X j (q)}, 1 ≤ j ≤ r, for all q ∈ A. The distribution D is said to be completely non-integrable of step 2 if for every point p ∈ M n the above vector fields defined in A satisfy, for all q ∈ A, i.e., the generated real vector space coincides with the tangent space. The distribution D = D (Ω) mentioned in the theorem is defined in [4,Section 5]. In our present situation the system of restricted roots is Φ (g 0 , a 0 ) = (BC) q and the proof of Theorem 1.1 is naturally divided into three parts by the nature of the restricted roots.
This paper is organized as follows. In the next section we indicate, in Table 1, the symmetric spaces whose tangential representations contain the isoparametric submanifolds concerning us here. In Section 3 we recall the root system (BC) q (compare [2, p. 475, 3.25]) and in Section 4 we present the required four lemmata about the relation between the roots of g C 0 and their restricted counterpart Φ (g 0 , a 0 ). In Section 5 we recall required notation introduced in [4], and in Section 6 the formulae, from [4], needed in the proof of Theorem 1.1. Finally Section 7 contains the proof of Theorem 1.1 itself.
We include also an Appendix with proofs of the lemmata in Section 3, for the spaces in Table 1 but restricting the sizes of the root sistems (BC) q only to the case q = 2. This is intended to be an example of the way to obtain the required lemmata for the spaces in Table 1.

The spaces considered
In Table 1 we indicate the list of the symmetric spaces whose tangential principal orbits are the isoparametric submanifolds considered in the present paper. They are the spaces for which the corresponding systems of restricted roots are nonreduced, that is, Φ (g 0 , a 0 ) = (BC) n (for n ≥ 1). We indicate only the compact spaces; they can, of course, be replaced by their corresponding non-compact duals.

The system (BC) q
As mentioned above, we consider here the case in which Φ (g 0 , a 0 ) = (BC) q . So we start recalling the system (BC) q . We use the description in [2, p. 475, 3.25], for this non-reduced system of roots. The roots (written in terms of a set {ε 1 , . . . , ε q }) are: Table 1.
simple ones. The double roots are and the others are (3.1)
Proof. We may assume that q ≥ 2 because in (BC) 1 , Γ = {2λ 1 }. To prove the Lemma, we have to consider the roots in (3.1). Those in Γ (written in terms of the simple ones) have an even number of coefficients equal to 1, while those in Ω have an odd number of them. Then γ ∈ Γ must have at least two coefficients equal to 1. So we may suppress the simple root of lower index (call it η) and calling δ the sum of the remaining terms we have obviously γ = η + δ and |η − δ| is not a root of (BC) q . This completes the proof of Lemma 4.1.
The proofs of the following three lemmata are obtained by inspection in the pairs Φ + (g, h) , (BC) q . We include in the Appendix an example of the proof of Lemmata 4.2, 4.3, and 4.4.

Basic notation
In order to avoid repetitions on the notation and basic facts we appeal to the patience of the reader and expect that he/she has the opportunity to take a look at [4,, where the essential notation is introduced. We shall indicate the numbers of the formulae there in the corresponding references.
Remark 7.1. It is important to observe that in order to prove Theorem 1.1, it is enough to show (for each λ ∈ Γ) that the vectors 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). So this is the objective here.
Let us consider the case (i) in (7.4). We see, by the argument above, that (7.3) holds in this case and it takes the form Now, we see (for the roots (ξ, ω)) that formulae (6.2) and (6.3) yield H 1 = T 2 = 0, and then (6.6) holds. Then we may write W ϕ = W (ξ+ω) using formula (6.9) (for the roots (ξ, ω)) and therefore the vector W ϕ is a bracket (evaluated at E) of local fields defined around E that belong to the distribution D (Ω).
On the other hand, in the case (ii) of (7.4) we have that (7.3) holds for the pair of complex roots (ξ, ω), which again yields H 1 = T 2 = 0. Then by formulae (6.7) (for ϕ = ξ + ω) we have that also in this case W ϕ is a sum of brackets (evaluated at E) of local fields that belong to the distribution D (Ω). This completes the proof of Theorem 1.1 in Case (A).

Case (B).
Let us take λ ∈ Γ a double root, λ = 2µ with µ ∈ Ω, and consider again the basis (5.1) for p 0λ . As above we have to consider the two situations We shall take first ϕ ∈ ρ −1 (λ) R . Then we have to consider the vector W ϕ . By Lemma 4.3 there exist roots α, β ∈ ρ −1 (µ) such that α = β and ϕ = α + β, and we have then (for α and β) the alternative (7.4), that is, either α and β are both real or they are both complex roots.

Case (C).
Here λ ∈ Γ is a double root, λ = 2µ and µ ∈ Γ. Since µ ∈ Γ is not a double root, by Lemma 4.1 there exist two roots η = δ in Ω such that µ = η + δ and |η − δ| is not a root of Φ + (g 0 , a 0 ). Also, by Lemma 4.4, for the root In order to simplify our notation we set We have to study again the vectors of the basis (5.1) for our λ ∈ Γ. But before doing this, we are going take a look at U α12 , V α12 , U β12 , and V β12 . For the roots involved, we have the following possibilities: Let us start taking γ ∈ ρ −1 (2µ) * C ; then we are in situation (a) and considering again the basis (5.1), we have to study the vectors U γ and V γ . But first we take a look at the vectors U α12 , V α12 , U β12 , and V β12 . 7.3.1. Situation (a). By (6.1) (with (α 1 , α 2 ) instead of (δ, ϕ)) we have The extra terms H 1 and T 2 have expressions similar to (6.2) and (6.3) (with (α 1 , α 2 ) instead of (δ, ϕ)). That is, For U β12 and V β12 we have similar expressions (with β instead of α).
Then we see that in our present situation the equalities (6.1) become (7.11) Now we may replace the fields inside the brackets on the right side of (7.11) by their expressions in (7.10). By doing this for ΘU γ in (7.11) we see that the bracket U F α12 , U F β12 in the first term (using (7.10) and multiplying by ab = 0) is (7.12) By proceeding similarly with the second term V F α12 , V F β12 in the first line of (7.11) we have (7.13) Now by expanding the brackets in each of the four terms in (7.12) and (7.13) and computing the difference of the expanded expressions (dividing by ab) we finally obtain the first line of (7.11). That is,

all brackets should be evaluated at E.
By computing similarly (using again (7.10)), the second identity in (7.11) turns out to be where all the brackets should be evaluated at E. Then we see that Θ ab U γ and Then we have the proof of the theorem in the situation (a). Let us take now γ ∈ ρ −1 (2µ) R ; then we have to study situations (b) and (c).

CRISTIÁN U. SÁNCHEZ
This identity extends locally as we did to get (7.10) and recalling (7.8) we may write (7.17) where both sides are evaluated at the same point around E.

Appendix
We include here as an example a proof of the lemmata in Section 4 for the spaces in Table 2. For simplicity we consider only the root system (BC) 2 . Note that for EIII this is no restriction. We hope that this will serve as an example. Table 2.
Then we have Lemma 4.3 for AIII.
8.2. The space CII (p > 2). For p > 2 the restricted roots form (BC) 2 . Let us set r = p + 2. The algebra of Sp (r) is C r . Since p > 2 we have here r > 4.
Again the system of simple roots π = {α 1 , . . . , α r } and the roots of C r are We observe that for those roots of type (2) in (8.2) we have the particular case j = r which are the roots The restriction rule is The multiplicities are m(λ 1 ) = 4, m(λ 2 ) = 4(r − 4), m(2λ 2 ) = 3.
8.3. The space DIII (p = 5). The system of roots for SO (10) is D 5 . In the present case the restriction rule is and the multiplicities are We need to indicate the roots of D 5 in the notation from Bourbaki. If {e j } is the canonical basis of R 5 the roots are ±e j ± e k , 1 ≤ j < k ≤ p.
Then we have Lemma 4.2 for DIII. Now we consider the root in ρ −1 (2λ 2 ). There is only one: To prove Lemma 4.3 for DIII we take γ and have to find α, β ∈ ρ −1 (λ 2 ) such that α = β and γ = α + β. Then we may take Then we have Lemma 4.3 for DIII.
Then we have Lemma 4.2 for EIII.
Then we have Lemma 4.3 for EIII.