The Index of Centralizers of

Elements of Reductive Lie Algebras

Jean-Yves Charbonnel and Anne Moreau

Received: April 28 2009 Revised: March 18, 2010

Communicated by Eckhard Meinrenken

Abstract. For a finite dimensional complex Lie algebra, its index is the minimal dimension of stabilizers for the coadjoint action. A famous conjecture due to A.G. Elashvili says that the index of the centralizer of an element of a reductive Lie algebra is equal to the rank. That conjecture caught attention of several Lie theorists for years. It reduces to the case of nilpotent elements. In [Pa03a] and [Pa03b], D.I. Panyushev proved the conjecture for some classes of nilpotent elements (e.g. regular, subregular and spherical nilpotent elements). Then the conjecture has been proven for the classical Lie algebras in [Y06a] and checked with a computer programme for the exceptional ones [deG08]. In this paper we give an almost general proof of that conjecture.

2010 Mathematics Subject Classification: 22E46, 17B80, 17B20, 14L24 Keywords and Phrases: reductive Lie algebra; index; centralizer; argument shift method; Poisson-commutative family of polynomials; rigid nilpotent orbit; Slodowy slice

1. Introduction

In this note 𝕜 is an algebraically closed field of characteristic 0.

1.1. Let 𝔤 be a finite dimensional Lie algebra over 𝕜 and consider the coadjoint representation of 𝔤. By definition, the index of 𝔤 is the minimal dimension of stabilizers 𝔤^{x}, x ∈ 𝔤^{∗}, for the coadjoint representation:

ind𝔤 := min{dim𝔤^{x}; x ∈ 𝔤^{∗}}.

The definition of the index goes back to Dixmier [Di74]. It is a very important notion in representation theory and in invariant theory. By Rosenlicht’s theorem [Ro63], generic orbits of an arbitrary algebraic action of a linear algebraic group on an irreducible algebraic variety are separated by rational invariants; in particular, if 𝔤 is an algebraic Lie algebra,

ind𝔤 = deg tr 𝕜(𝔤^{∗})^{𝔤},

where 𝕜(𝔤^{∗})^{𝔤} is the field of 𝔤-invariant rational functions on 𝔤^{∗}. The index of a reductive algebra equals its rank. For an arbitrary Lie algebra, computing its index seems to be a wild problem. However, there is a large number of interesting results for several classes of nonreductive subalgebras of reductive Lie algebras. For instance, parabolic subalgebras and their relatives as nilpotent radicals, seaweeds, are considered in [Pa03a], [TY04], [J07]. The centralizers, or normalizers of centralizers, of elements form another interesting class of such subalgebras, [E85a], [Pa03a], [Mo06b]. The last topic is closely related to the theory of integrable Hamiltonian systems [Bol91]. Let us precise this link. From now on, 𝔤 is supposed to be reductive. Denote by G the adjoint group of 𝔤. The symmetric algebra S(𝔤) carries a natural Poisson structure. By the so-called argument shift method, for x in 𝔤^{∗}, we can construct a Poisson-commutative family ℱ_{x} in S(𝔤) = 𝕜[𝔤^{∗}]; see [MF78] or Remark 1.4. It is generated by the derivatives of all orders in the direction x ∈ 𝔤^{∗} of all elements of the algebra S(𝔤)^{𝔤} of 𝔤-invariants of S(𝔤). Moreover, if G.x denotes the coadjoint orbit of x ∈ 𝔤^{∗}:

Theorem 1.1 ([Bol91], Theorems 2.1 and 3.2). There is a Poisson-

commutative family of polynomial functions on 𝔤^{∗}, constructed by the argument shift method, such that its restriction to G.x contains \frac{1}{2}dim(G.x) algebraically independent functions if and only if ind𝔤^{x} = ind𝔤.

Denote by rk𝔤 the rank of 𝔤. Motivated by the preceding result of Bolsinov, A.G. Elashvili formulated a conjecture:

Conjecture 1.2 (Elashvili). Let 𝔤 be a reductive Lie algebra. Then ind𝔤^{x} = rk𝔤 for all x ∈ 𝔤^{∗}.

Elashvili’s conjecture also appears in the following problem: Is the algebra S(𝔤^{x})^{𝔤^{x}} of invariants in S(𝔤^{x}) under the adjoint action a polynomial algebra? This question was formulated by A. Premet in [PPY07, Conjecture 0.1]. After that, O. Yakimova discovered a counterexample [Y07], but the question remains very interesting. As an example, under certain hypothesis and under the condition that Elashvili’s conjecture holds, the algebra of invariants S(𝔤^{x})^{𝔤^{x}} is polynomial in rk𝔤 variables, [PPY07, Theorem 0.3]. During the last decade, Elashvili’s conjecture caught attention of many invariant theorists [Pa03a], [Ch04], [Y06a], [deG08]. To begin with, describe some easy but useful reductions. Since the 𝔤-modules 𝔤 and 𝔤^{∗} are isomorphic, it is equivalent to prove Conjecture 1.2 for centralizers of elements of 𝔤. On the other hand, by a result due to E.B. Vinberg [Pa03a], the inequality ind𝔤^{x} ≥ rk𝔤 holds for all x ∈ 𝔤. So it only remains to prove the opposite one. Given x ∈ 𝔤, let x = x_{s} + x_{n} be its Jordan decomposition. Then 𝔤^{x} = (𝔤^{x_{s}})^{x_{n}}. The subalgebra 𝔤^{x_{s}} is reductive of rank rk𝔤. Thus, the verification of Conjecture 1.2 reduces to the case of nilpotent elements. At last, one can clearly restrict oneself to the case of simple 𝔤. Review now the main results obtained so far on Elashvili’s conjecture. If x is regular, then 𝔤^{x} is a commutative Lie algebra of dimension rk𝔤. So, Conjecture 1.2 is obviously true in that case. Further, the conjecture is known for subregular nilpotent elements and nilpotent elements of height 2 and 3, [Pa03a], [Pa03b]. Remind that the height of a nilpotent element e is the maximal integer m such that (ade)^{m} ≠ 0. More recently, O. Yakimova proved the conjecture in the classical case [Y06a]. To valid the conjecture in the exceptional types, W. de Graaf used the computer programme GAP, see [deG08]. Since there are many nilpotent orbits in the Lie algebras of exceptional type, it is difficult to present the results of such computations in a concise way. In 2004, the first author published a case-free proof of Conjecture 1.2 applicable to all simple Lie algebras; see [Ch04]. Unfortunately, the argument in [Ch04] has a gap in the final part of the proof which was pointed out by L. Rybnikov. To summarize, so far, there is no conceptual proof of Conjecture 1.2. Nevertheless, according to Yakimova’s works and de Graaf’s works, we can claim:

Theorem 1.3 ([Y06a], [deG08]). Let 𝔤 be a reductive Lie algebra. Then ind𝔤^{x} = rk𝔤 for all x ∈ 𝔤^{∗}.

Because of the importance of Elashvili’s conjecture in invariant theory, it would be very appreciated to find a general proof of Theorem 1.3 applicable to all finite-dimensional simple Lie algebras. The proof we propose in this paper is fresh and almost general. More precisely, it remains 7 isolated cases; one nilpotent orbit in type E_{7} and six nilpotent orbits in type E_{8} have to be considered separately. For these 7 orbits, the use of GAP is unfortunately necessary. In order to provide a complete proof of Theorem 1.3, we include in this paper the computations using GAP we made to deal with these remaining seven cases.

1.2. Description of the paper. Let us briefly explain our approach. Denote by 𝒩(𝔤) the nilpotent cone of 𝔤. As noticed previously, it suffices to prove ind𝔤^{e} = rk𝔤 for all e in 𝒩(𝔤). If the equality holds for e, it does for all elements of G.e; we shortly say that G.e satisfies Elashvili’s conjecture. From a nilpotent orbit 𝒪_{𝔩} of a reductive factor 𝔩 of a parabolic subalgebra of 𝔤, we can construct a nilpotent orbit of 𝔤 having the same codimension in 𝔤 as 𝒪_{𝔩} in 𝔩 and having other remarkable properties. The nilpotent orbits obtained in such a way are called induced; the other ones are called rigid. We refer the reader to Subsection 2.3 for more precisions about this topic. Using Bolsinov’s criterion of Theorem 1.1, we first prove Theorem 1.3 for all induced nilpotent orbits and so the conjecture reduces to the case of rigid nilpotent orbits. To deal with rigid nilpotent orbits, we use methods developed in [Ch04] by the first author, and resumed in [Mo06a] by the second author, based on nice properties of Slodowy slices of nilpotent orbits.

In more details, the paper is organized as follows:

We state in Section 2 the necessary preliminary results. In particular, we investigate in Subsection 2.2 extensions of Bolsinov’s criterion and we establish an important result (Theorem 2.7) which will be used repeatedly in the sequel. We prove in Section 3 the conjecture for all induced nilpotent orbits (Theorem 3.3) so that Elashvili’s conjecture reduces to the case of rigid nilpotent orbits (Theorem 3.3). From Section 4, we handle the rigid nilpotent orbits: we introduce and study in Section 4 a property (P) given by Definition 4.2. Then, in Section 5, we are able to deal with almost all rigid nilpotent orbits. Still in Section 5, the remaining cases are dealt with set-apart by using a different approach.

1.3. Notations. • If E is a subset of a vector space V, we denote by span(E) the vector subspace of V generated by E. The grassmanian of all d-dimensional subspaces of V is denoted by Gr_{d}(V). By a cone of V, we mean a subset of V invariant under the natural action of 𝕜^{∗} := 𝕜\{0} and by a bicone of V × V we mean a subset of V × V invariant under the natural action of 𝕜^{∗} × 𝕜^{∗} on V × V.

• From now on, we assume that 𝔤 is semisimple of rank ℓ and we denote by ⟨., .⟩ the Killing form of 𝔤. We identify 𝔤 to 𝔤^{∗} through ⟨., .⟩. Unless otherwise specified, the notion of orthogonality refers to the bilinear form ⟨., .⟩.

• Denote by S(𝔤)^{𝔤} the algebra of 𝔤-invariant elements of S(𝔤). Let f_{1}, …, f_{ℓ} be homogeneous generators of S(𝔤)^{𝔤} of degrees d_{1}, …,d_{ℓ} respectively. We choose the polynomials f_{1}, …,f_{ℓ} so that d_{1}≤ ⋯ ≤d_{ℓ}. For i = 1, …, ℓ and (x, y) ∈ 𝔤 × 𝔤, we may consider a shift of f_{i} in direction y: f_{i}(x + ty) where t ∈ 𝕜. Expanding f_{i}(x + ty) as a polynomial in t, we obtain

(1)  f_{i}(x + ty) = ∑_{m=0}^{d_{i}} f_{i}^{(m)}(x, y)t^{m}; ∀(t, x, y) ∈ 𝕜 × 𝔤 × 𝔤

where y ↦ (m!)f_{i}^{(m)}(x, y) is the differential at x of f_{i} of the order m in the direction y. The elements f_{i}^{(m)} as defined by (1) are invariant elements of S(𝔤) ⊗_{𝕜} S(𝔤) under the diagonal action of G on 𝔤 × 𝔤. Note that f_{i}^{(0)}(x, y) = f_{i}(x) while f_{i}^{(d_{i})}(x, y) = f_{i}(y) for all (x, y) ∈ 𝔤 × 𝔤.

Remark 1.4. The family ℱ_{x} := {f_{i}^{(m)}(x, .); 1 ≤ i ≤ ℓ, 1 ≤ m ≤ d_{i}} for x ∈ 𝔤, is a Poisson-commutative family of S(𝔤) by Mishchenko-Fomenko [MF78]. One says that the family ℱ_{x} is constructed by the argument shift method.

• Let i ∈ {1, …, ℓ}. For x in 𝔤, we denote by φ_{i}(x) the element of 𝔤 satisfying (df_{i})_{x}(y) = f_{i}^{(1)}(x, y) = ⟨φ_{i}(x), y⟩, for all y in 𝔤. Thereby, φ_{i} is an invariant element of S(𝔤) ⊗_{𝕜}𝔤 under the canonical action of G. We denote by φ_{i}^{(m)}, for 0 ≤ m ≤ d_{i} − 1, the elements of S(𝔤) ⊗_{𝕜} S(𝔤) ⊗_{𝕜} 𝔤 defined by the equality:

(2)  φ_{i}(x + ty) = ∑_{m=0}^{d_{i}−1} φ_{i}^{(m)}(x, y)t^{m}, ∀(t, x, y) ∈ 𝕜 × 𝔤 × 𝔤.

• For x ∈ 𝔤, we denote by 𝔤^{x} = {y ∈ 𝔤 | [y, x] = 0} the centralizer of x in 𝔤 and by 𝔷(𝔤^{x}) the center of 𝔤^{x}. The set of regular elements of 𝔤 is

𝔤_{reg} := {x ∈ 𝔤 | dim𝔤^{x} = ℓ}

and we denote by 𝔤_{reg,ss} the set of regular semisimple elements of 𝔤. Both 𝔤_{reg} and 𝔤_{reg,ss} are G-invariant dense open subsets of 𝔤. We denote by C(x) the G-invariant cone generated by x and we denote by x_{s} and x_{n} the semisimple and nilpotent components of x respectively.

• The nilpotent cone of 𝔤 is 𝒩(𝔤). As a rule, for e ∈ 𝒩(𝔤), we choose an 𝔰𝔩_{2}-triple (e, h, f) in 𝔤 given by the Jacobson-Morozov theorem [CMa93, Theorem 3.3.1]. In particular, it satisfies the equalities:

[h, e] = 2e,  [e, f] = h,  [h, f] = −2f

The action of adh on 𝔤 induces a ℤ-grading:

𝔤 = ⨁_{i∈ℤ}𝔤(i) , 𝔤(i) = {x ∈ 𝔤 | [h, x] = ix}.

Recall that e, or G.e, is said to be even if 𝔤(i) = 0 for odd i. Note that e ∈ 𝔤(2), f ∈ 𝔤(−2) and that 𝔤^{e}, 𝔷(𝔤^{e}) and 𝔤^{f} are all adh-stable.

• All topological terms refer to the Zariski topology. If Y is a subset of a topological space X, we denote by \overline{Y} the closure of Y in X.

1.4. Acknowledgments. We would like to thank O. Yakimova for her interest and useful discussions and more particularly for bringing Bolsinov’s paper to our attention. We also thank A.G. Elashvili for suggesting Lawther-Testerman’s paper [LT08] about the centers of centralizers of nilpotent elements.

Contents

1. Introduction 387 2. Preliminary results 392 3. Proof of Theorem 1.3 for induced nilpotent orbits 398 4. The Slodowy slice and the property (P) 400 5. Proof of Theorem 1.3 for rigid nilpotent orbits 408 Appendix A. Proof of Proposition 5.10: explicit computations. 413 References 418

2. Preliminary results

We start in this section by reviewing some facts about the differentials of generators of S(𝔤)^{𝔤}. Then, the goal of Subsection 2.2 is Theorem 2.7. We collect in Subsection 2.3 basic facts about induced nilpotent orbits.

2.1. Differentials of generators of S(𝔤)^{𝔤}. According to subsec-

tion 1.3, the elements φ_{1}, …,φ_{ℓ} of S(𝔤) ⊗_{𝕜} 𝔤 are the differentials of f_{1}, …,f_{ℓ} respectively. Since f_{i}(g(x)) = f_{i}(x) for all (x, g) ∈ 𝔤 × G, the element φ_{i}(x) centralizes x for all x ∈ 𝔤. Moreover:

Lemma 2.1. (i)[Ri87, Lemma 2.1] The elements φ_{1}(x), …,φ_{ℓ}(x) belong to 𝔷(𝔤^{e}). (ii)[Ko63, Theorem 9] The elements φ_{1}(x), …,φ_{ℓ}(x) are linearly independent elements of 𝔤 if and only if x is regular. Moreover, if so, φ_{1}(x), …,φ_{ℓ}(x) is a basis of 𝔤^{x}.

We turn now to the elements φ_{i}^{(m)}, for i = 1, …, ℓ and 0 ≤ m ≤ d_{i} − 1, defined in Subsection 1.3 by (2). Recall that d_{i} is the degree of the homogeneous polynomial f_{i}, for i = 1, …, ℓ. The integers d_{1}−1, …, d_{ℓ}− 1 are thus the exponents of 𝔤. By a classical result [Bou02, Ch. V, §5, Proposition 3], we have ∑ d_{i} = b_{𝔤} where b_{𝔤} is the dimension of Borel subalgebras of 𝔤. For (x, y) in 𝔤 × 𝔤, we set:

V_{x,y} := span{φ_{i}^{(m)}(x, y) ; 1 ≤ i ≤ ℓ, 0 ≤ m ≤ d_{i} − 1}. (3)

The subspaces V_{x,y} will play a central role throughout the note.

Remark 2.2. (1) For (x, y) ∈ 𝔤 × 𝔤, the dimension of V_{x,y} is at most b_{𝔤} since ∑ d_{i} = b_{𝔤}. Moreover, for all (x, y) in a nonempty open subset of 𝔤 × 𝔤, the equality holds [Bol91]. Actually, in this note, we do not need this observation.

(2) By Lemma 2.1(ii), if x is regular, then 𝔤^{x} is contained in V_{x,y} for all y ∈ 𝔤. In particular, if so, dim[x, V_{x,y}] = dimV_{x,y} − ℓ.

The subspaces V_{x,y} were introduced and studied by Bolsinov in [Bol91], motivated by the maximality of Poisson-commutative families in S(𝔤). These subspaces have been recently exploited in [PY08] and [CMo08]. The following results are mostly due to Bosinov, [Bol91]. We refer to [PY08] for a more recent account about this topic. We present them in a slightly different way:

Lemma 2.3. Let (x, y) be in 𝔤_{reg} × 𝔤.

(i) The subspace V_{x,y} of 𝔤 is the sum of the subspaces 𝔤^{x+ty} where t runs through any nonempty open subset of 𝕜 such that x + ty is regular for all t in this subset. (ii) The subspace 𝔤^{y} +V_{x,y} is a totally isotropic subspace of 𝔤 with respect to the Kirillov form K_{y} on 𝔤 × 𝔤, (v, w) ↦ ⟨y, [v, w]⟩. Furthermore, dim(𝔤^{y} + V_{x,y})^{⊥} ≥ \frac{1}{2}dimG.y. (iii) The subspaces [x, V_{x,y}] and [y, V_{x,y}] are equal.

Proof. (i) Let O be a nonempty open subset of 𝕜 such that x + ty is regular for all t in O. Such an open subset does exist since x is regular. Denote by V_{O} the sum of all the subspaces 𝔤^{x+ty} where t runs through O. For all t in O, 𝔤^{x+ty} is generated by φ_{1}(x + ty), …,φ_{ℓ}(x + ty), cf. Lemma 2.1(ii). As a consequence, V_{O} is contained in V_{x,y}. Conversely, for i = 1, …, ℓ and for t_{1}, …,t_{d_{i}} pairwise different elements of O, φ_{i}^{(m)}(x, y) is a linear combination of φ_{i}(x + t_{1}y), …, φ_{i}(x + t_{d_{i}}y); hence φ_{i}^{(m)}(x, y) belongs to V_{O}. Thus V_{x,y} is equal to V_{O}, whence the assertion. (ii) results from [PY08, Proposition A4]. Notice that in (ii) the inequality is an easy consequence of the first statement. At last, [PY08, Lemma A2] gives us (iii). □

Let σ and σ_{i}, for i = 1, …, ℓ, be the maps

\begin{matrix}𝔤 × 𝔤 & ⟶^{σ} & 𝕜^{b_{𝔤}+ℓ} \\ (x, y) & ⟼ & (f_{i}^{(m)}(x, y))_{{1≤i≤ℓ, \atop 0≤m≤d_{i}}} \end{matrix} ,  \begin{matrix}𝔤 × 𝔤 & ⟶^{σ_{i}} & 𝕜^{d_{i}+1} \\ (x, y) & ⟼ & (f_{i}^{(m)}(x, y))_{0≤m≤d_{i}} \end{matrix}

respectively, and denote by σ′(x, y) and σ′_{i}(x, y) the tangent map at (x, y) of σ and σ_{i} respectively. Then σ′_{i}(x, y) is given by the differentials of the f_{i}^{(m)}’s at (x, y) and σ′(x, y) is given by the elements σ′_{i}(x, y).

Lemma 2.4. Let (x, y) and (v, w) be in 𝔤 × 𝔤.

(i) For i = 1, …, ℓ, σ′_{i}(x, y) maps (v, w) to

(⟨φ_{i}(x), v⟩, ⟨φ_{i}^{(1)}(x, y), v⟩ + ⟨φ_{i}^{(0)}(x, y), w⟩,

…, ⟨φ_{i}^{(d_{i}−1)}(x, y), v⟩ + ⟨φ_{i}^{(d_{i}−2)}(x, y), w⟩, ⟨φ_{i}(y), w⟩).

(ii) Suppose that σ′(x, y)(v, w) = 0. Then, for w′ in 𝔤, σ′(x, y)(v, w′) = 0 if and only if w − w′ is orthogonal to V_{x,y}. (iii) For x ∈ 𝔤_{reg}, σ′(x, y)(v, w′) = 0 for some w′ ∈ 𝔤 if and only if v ∈ [x, 𝔤].

Proof. (i) The verifications are easy and left to the reader. (ii) Since σ′(x, y)(v, w) = 0, σ′(x, y)(v, w′) = 0 if and only if σ′(x, y)(v, w − w′) = 0 whence the statement by (i). (iii) Suppose that x is regular and suppose that σ′(x, y)(v, w′) = 0 for some w′ ∈ 𝔤. Then by (i), v is orthogonal to the elements φ_{1}(x), …,φ_{ℓ}(x). So by Lemma 2.1(ii), v is orthogonal to 𝔤^{x}. Since 𝔤^{x} is the orthogonal complement of [x, 𝔤] in 𝔤, we deduce that v lies in [x, 𝔤]. Conversely, since σ(x, y) = σ(g(x), g(y)) for all g in G, the element ([u, x], [u, y]) belongs to the kernel of σ′(x, y) for all u ∈ 𝔤. So, the converse implication follows. □

2.2. On Bolsinov’s criterion. Let a be in 𝔤 and denote by π the map

𝔤 × G.a ⟶^{π} 𝔤 × 𝕜^{b_{𝔤}+ℓ}
(x, y) ⟼ (x, σ(x, y)).

Remark 2.5. Recall that the family (ℱ_{x})_{x∈𝔤} constructed by the argument shift method consists of all elements f_{i}^{(m)}(x, .) for i = 1, …, ℓ and 1 ≤ m ≤ d_{i}, see Remark 1.4. By definition of the morphism π, there is a family constructed by the argument shift method whose restriction to G.a contains \frac{1}{2}dimG.a algebraically independent functions if and only if π has a fiber of dimension \frac{1}{2}dimG.a.

In view of Theorem 1.1 and the above remark, we now concentrate on the fibers of π. For (x, y) ∈ 𝔤 × G.a, denote by F_{x,y} the fiber of π at π(x, y):

F_{x,y} := {x} × {y′ ∈ G.a | σ(x, y′) = σ(x, y)}.

Lemma 2.6. Let (x, y) be in 𝔤 × G.a.

(i) The irreducible components of F_{x,y} have dimension at least \frac{1}{2}dimG.a.

(ii) The fiber F_{x,y} has dimension \frac{1}{2}dimG.a if and only if any irreducible component of F_{x,y} contains an element (x, y′) such that (𝔤^{y′} + V_{x,y′})^{⊥} has dimension \frac{1}{2}dimG.a.

Proof. We prove (i) and (ii) all together. The tangent space T_{x,y′}(F_{x,y}) of F_{x,y} at (x, y′) in F_{x,y} identifies to the subspace of elements w of [y′, 𝔤] such that σ′(x, y′)(0, w) = 0. Hence, by Lemma 2.4(ii),

T_{x,y′}(F_{x,y}) = [y′, 𝔤] ∩ V_{x,y′}^{⊥} = (𝔤^{y′} + V_{x,y′})^{⊥},

since [y′, 𝔤] = (𝔤^{y′})^{⊥}. But by Lemma 2.3(ii), (𝔤^{y′} + V_{x,y′})^{⊥} has dimension at least \frac{1}{2}dimG.a; so does T_{x,y′}(F_{x,y}). This proves (i). Moreover, the equality holds if and only if (𝔤^{y′} + V_{x,y′})^{⊥} has dimension \frac{1}{2}dimG.a, whence the statement (ii). □

Theorem 2.7. The following conditions are equivalent:

(1) ind𝔤^{a} = ℓ;

(2) π has a fiber of dimension \frac{1}{2}dimG.a;

(3) there exists (x, y) ∈ 𝔤 × G.a such that (𝔤^{y} + V_{x,y})^{⊥} has dimension \frac{1}{2}dimG.a;

(4) there exists x in 𝔤_{reg} such that dim(𝔤^{a}+V_{x,a}) = \frac{1}{2}(dim𝔤+dim𝔤^{a});

(5) there exists x in 𝔤_{reg} such that dimV_{x,a} = \frac{1}{2}dimG.a + ℓ;

(6) σ(𝔤 × \{a\}) has dimension \frac{1}{2}dimG.a + ℓ.

Proof. By Theorem 1.1 and Remark 2.5, we have (1)⇔(2). Moreover, by Lemma 2.6(ii), we have (2)⇔(3). (3)⇔(4): If (4) holds, so does (3). Indeed, if so,

dim𝔤 − \frac{1}{2}dimG.a = \frac{1}{2}(dim𝔤 + dim𝔤^{a}) = dim(𝔤^{a} + V_{x,a}).

Conversely, suppose that (3) holds. By Lemma 2.3(ii), 𝔤^{y} + V_{x,y} has maximal dimension \frac{1}{2}(dim𝔤 + dim𝔤^{y}). So the same goes for all (x, y) in a G-invariant nonempty open subset of 𝔤 × G.a. Hence, since the map (x, y) ↦ V_{x,y} is G-equivariant, there exists x in 𝔤_{reg} such that

dim(V_{x,a} + 𝔤^{a}) = \frac{1}{2}(dim𝔤 + dim𝔤^{a}).

(4)⇔(5): Let x be in 𝔤_{reg}. By Lemma 2.3(iii), [x, V_{x,a}] = [a, V_{x,a}]. Hence 𝔤^{a} ∩ V_{x,a} has dimension ℓ by Remark 2.2(2). As a consequence,

dim(𝔤^{a} + V_{x,a}) = dim𝔤^{a} + dimV_{x,a} − ℓ,

whence the equivalence.

(2)⇔(6): Suppose that (2) holds. By Lemma 2.6, \frac{1}{2}dimG.a is the minimal dimension of the fibers of π. So, π(𝔤 × G.a) has dimension

dim𝔤 + dimG.a − \frac{1}{2}dimG.a = dim𝔤 + \frac{1}{2}dimG.a.

Denote by τ the restriction to π(𝔤 × G.a) of the projection map 𝔤 × 𝕜^{b_{𝔤}+ℓ} → 𝕜^{b_{𝔤}+ℓ}. Then τ^{∘}π is the restriction of σ to 𝔤 × G.a. Since σ is a G-invariant map, σ(𝔤×{a}) = σ(𝔤×G.a). Let (x, y) ∈ 𝔤_{reg,ss} ×G.a. The fiber of τ at z = σ(x, y) is G.x since x is a regular semisimple element of 𝔤. Hence,

dimσ(𝔤 × \{a\}) = dimπ(𝔤 × G.a) − (dim𝔤 − ℓ) = \frac{1}{2}dimG.a + ℓ

and we obtain (6). Conversely, suppose that (6) holds. Then π(𝔤 × G.a) has dimension dim𝔤 + \frac{1}{2}dimG.a by the above equality. So the minimal dimension of the fibers of π is equal to

dim𝔤 + dimG.a − (dim𝔤 + \frac{1}{2}dimG.a) = \frac{1}{2}dimG.a

and (2) holds. □

2.3. Induced and rigid nilpotent orbits. The definitions and results

of this subsection are mostly extracted from [Di74], [Di75], [LS79] and [BoK79]. We refer to [CMa93] and [TY05] for recent surveys. Let 𝔭 be a proper parabolic subalgebra of 𝔤 and let 𝔩 be a reductive factor of 𝔭. We denote by 𝔭_{u} the nilpotent radical of 𝔭. Denote by L the connected closed subgroup of G whose Lie algebra is ad𝔩 and denote by P the normalizer of 𝔭 in G.

Theorem 2.8 ([CMa93],Theorem 7.1.1). Let 𝒪_{𝔩} be a nilpotent orbit of 𝔩.

There exists a unique nilpotent orbit 𝒪_{𝔤} in 𝔤 whose intersection with 𝒪_{𝔩} + 𝔭_{u} is a dense open subset of 𝒪_{𝔩} + 𝔭_{u}. Moreover, the intersection of 𝒪_{𝔤} and 𝒪_{𝔩}+𝔭_{u} consists of a single P -orbit and codim_{𝔤}(𝒪_{𝔤}) = codim_{𝔩}(𝒪_{𝔩}).

The orbit 𝒪_{𝔤} only depends on 𝔩 and not on the choice of a parabolic subalgebra 𝔭 containing it [CMa93, Theorem 7.1.3]. By definition, the orbit 𝒪_{𝔤} is called the induced orbit from 𝒪_{𝔩}; it is denoted by Ind_{𝔩}^{𝔤}(𝒪_{𝔩}). If 𝒪_{𝔩} = 0, then we call 𝒪_{𝔤} a Richardson orbit. For example all even nilpotent orbits are Richardson [CMa93, Corollary 7.1.7]. In turn, not all nilpotent orbits are induced from another one. A nilpotent orbit which is not induced in a proper way from another one is called rigid. We shall say that e ∈ 𝒩(𝔤) is an induced (respectively rigid) nilpotent element of 𝔤 if the G-orbit of e is an induced (respectively rigid) nilpotent orbit of 𝔤. The following results are deeply linked to the properties of the sheets of 𝔤 and the deformations of its G-orbits. We refer to [BoK79] about these notions.

Theorem 2.9. (i) Let x be a non nilpotent element of 𝔤 and let 𝒪_{𝔤} be the induced nilpotent orbit from the adjoint orbit of x_{n} in 𝔤^{x_{s}}. Then 𝒪_{𝔤} is the unique nilpotent orbit contained in \overline{C(x)} whose dimension is dimG.x. Furthermore, \overline{C(x)} ∩ 𝒩(𝔤) = \overline{𝒪_{𝔤}} and \overline{C(x)} ∩ 𝒩(𝔤) is the nullvariety in \overline{C(x)} of f_{i} where i is an element of \{1, …, ℓ\} such that f_{i}(x) ≠ 0. (ii) Conversely, if 𝒪_{𝔤} is an induced nilpotent orbit, there exists a non nilpotent element x of 𝔤 such that \overline{C(x)} ∩ 𝒩(𝔤) = \overline{𝒪_{𝔤}}.

Proof. (i) Let 𝔭 be a parabolic subalgebra of 𝔤 having 𝔤^{x_{s}} as a Levi factor. Denote by 𝔭_{u} its nilpotent radical and by P the normalizer of 𝔭 in G. Let 𝒪′ be the adjoint orbit of x_{n} in 𝔤^{x_{s}}.

Claim 2.10. Let C be the P -invariant closed cone generated by x and let C_{0} be the subset of nilpotent elements of C. Then C = 𝕜x_{s} +\overline{𝒪′} +𝔭_{u}, C_{0} = \overline{𝒪′} + 𝔭_{u} and C_{0} is an irreducible subset of dimension dimP (x).

Proof. The subset x_{s} + \overline{𝒪′} + 𝔭_{u} is an irreducible closed subset of 𝔭 containing P (x). Moreover, its dimension is equal to

dim𝒪′ + dim𝔭_{u} = dim𝔤^{x_{s}} − dim𝔤^{x} + dim𝔭_{u} = dim𝔭 − dim𝔤^{x}.

Since the closure of P (x) and x_{s} +\overline{𝒪′} +𝔭_{u} are both irreducible subsets of 𝔤, they coincide. As a consequence, the set 𝕜x_{s} + \overline{𝒪′} + 𝔭_{u} is contained in C. Since the former set is clearly a closed conical subset of 𝔤 containing x, C = 𝕜x_{s} + \overline{𝒪′} + 𝔭_{u}. Then we deduce that C_{0} = \overline{𝒪′} + 𝔭_{u}. □

Denote by G ×_{P} 𝔤 the quotient of G × 𝔤 under the right action of P given by (g, z).p := (gp, p^{−1}(z)). The map (g, z) ↦ g(z) from G × 𝔤 to 𝔤 factorizes through the quotient map from G × 𝔤 to G ×_{P} 𝔤. Since G/P is a projective variety, the so obtained map from G ×_{P} 𝔤 to 𝔤 is closed. Since C and C_{0} are closed P -invariant subsets of 𝔤, G ×_{P} C and G×_{P} C_{0} are closed subsets of G×_{P} 𝔤. Hence \overline{C(x)} = G(C) and G(C_{0}) is a closed subset of 𝔤. So, by the claim, the subset of nilpotent elements of \overline{C(x)} is irreducible since C_{0} is irreducible. Since there are finitely many nilpotent orbits, the subset of nilpotent elements of \overline{C(x)} is the closure of one nilpotent orbit. Denote it by 𝒪̃ and prove 𝒪̃ = 𝒪_{𝔤}. For all k, l in {1, …, ℓ}, denote by p_{k,l} the polynomial function

p_{k,l} := f_{k}(x)^{d_{l}}f_{l}^{d_{k}} − f_{l}(x)^{d_{k}}f_{k}^{d_{l}}

Then p_{k,l} is G-invariant and homogeneous of degree d_{k}d_{l}. Moreover p_{k,l}(x) = 0. As a consequence, \overline{C(x)} is contained in the nullvariety of the functions p_{k,l}, 1 ≤ k, l ≤ ℓ. Hence the nullvariety of f_{i} in \overline{C(x)} is contained in the nilpotent cone of 𝔤 since it is the nullvariety in 𝔤 of the functions f_{1}, …,f_{ℓ}. Then dim𝒪̃ = dim\overline{C(x)} − 1 = dimG.x. Since 𝒪′ + 𝔭_{u} is contained in \overline{C(x)}, Theorem 2.8 tells us that 𝒪_{𝔤} is contained in \overline{C(x)}. Moreover by Theorem 2.8, 𝒪_{𝔤} has dimension dimG.x, whence 𝒪̃ = 𝒪_{𝔤}. All statements of (i) are now clear. (ii) By hypothesis, 𝒪_{𝔤} = Ind_{𝔩}^{𝔤}(𝒪_{𝔩}), where 𝔩 is a proper Levi subalgebra of 𝔤 and 𝒪_{𝔩} a nilpotent orbit in 𝔩. Let x_{s} be an element of the center of 𝔩 such that 𝔤^{x_{s}} = 𝔩, let x_{n} be an element of 𝒪_{𝔩} and set x = x_{s} + x_{n}. Since 𝔩 is a proper subalgebra, the element x is not nilpotent. So by (i), the subset of nilpotent elements of \overline{C(x)} is the closure of 𝒪_{𝔤}. □

3. Proof of Theorem 1.3 for induced nilpotent orbits

Let e be an induced nilpotent element. Let x be a non nilpotent element of 𝔤 such that \overline{C(x)} ∩ 𝒩(𝔤) = \overline{G.e}. Such an element does exist by Theorem 2.9(ii). As an abbreviation, we set:

\begin{aligned}𝕜\underline{d} & := & 𝕜^{d_{1}+1} × ⋯ × 𝕜^{d_{ℓ}+1} ≃ 𝕜^{b_{𝔤}+ℓ}, \\ 𝕜^{\underline{d}×} & := & (𝕜^{d_{1}+1} \ \{0\}) × ⋯ × (𝕜^{d_{ℓ}+1} \ \{0\}), \\ ℙ\underline{d} & := & ℙ(𝕜^{d_{1}+1}) × ⋯ × ℙ(𝕜^{d_{ℓ}+1}) = ℙ^{d_{1}} × ⋯ × ℙ^{d_{ℓ}}. \end{aligned}

For j = 1, …, ℓ, recall that σ_{j} is the map:

𝔤 × 𝔤 ⟶^{σ_{j}} 𝕜^{d_{j}+1}, (x, y) ↦ (f_{j}^{(m)}(x, y))_{0≤m≤d_{j}}.

Let ℬ_{j} be the nullvariety of σ_{j} in 𝔤 × 𝔤 and let ℬ be the union of ℬ_{1}, …,ℬ_{ℓ}; it is a bicone of 𝔤 × 𝔤. Denote by ρ and τ the canonical maps:

\begin{matrix}(𝔤 × 𝔤) \ \{0\} & ⟶^{ρ} & ℙ(𝔤 × 𝔤) \\ 𝕜^{d×} & ⟶^{τ} & ℙ\underline{d}. \end{matrix}

Let σ^{∗} be the restriction to (𝔤 × 𝔤) \ ℬ of σ; it has values in 𝕜^{\underline{d}×}. Since σ_{j}(sx, sy) = s^{d_{j}}σ_{j}(x, y) for all (x, y) ∈ 𝔤 × 𝔤 and j = 1, …, ℓ, the map τ^{∘}σ^{∗} factors through ρ. Denote by \overline{σ^{∗}} the map from ρ(𝔤 × 𝔤 \ ℬ) to ℙ\underline{d} making the following diagram commutative:

\xymatrix{𝔤 × 𝔤 \ ℬ \ar[r]^{σ^{∗}=σ|_{𝔤×𝔤\ℬ}} \ar[d]_{ρ} & 𝕜^{\underline{d}×} \ar[d]_{τ} \\ ρ(𝔤 × 𝔤 \ ℬ) \ar@{-->}[r]^{\overline{σ^{∗}}} & ℙ\underline{d}}

and let Γ be the graph of the restriction to ρ(𝔤 × \overline{C(x)} \ ℬ) of \overline{σ^{∗}}.

Documenta Mathematica 15 (2010) 387–421

Lemma 3.1. The set Γ is a closed subset of ℙ(𝔤 × 𝔤) × ℙ\underline{d}.

Proof. Let Γ̃ be the inverse image of Γ by the map ρ × τ. Then Γ̃ is the intersection of the graph of σ and (𝔤 × \overline{C(x)} \ ℬ) × 𝕜^{\underline{d}×}. Since σ^{−1}(𝕜\underline{d} \ 𝕜^{\underline{d}×}) is contained in ℬ, Γ̃ is the intersection of the graph of σ and (𝔤 × \overline{C(x)} \ \{0\}) × 𝕜^{\underline{d}×}; so Γ̃ is closed in (𝔤 × \overline{C(x)}) \ \{0\} × 𝕜^{\underline{d}×}. As a consequence, Γ is closed in ℙ(𝔤 × 𝔤) × ℙ\underline{d}, since ℙ(𝔤 × 𝔤) × ℙ\underline{d} is endowed with the quotient topology. □

Denote by Z the closure of σ(𝔤 × \overline{C(x)}) in 𝕜\underline{d}.

Lemma 3.2. There exists an open subset U of Z, contained in σ(𝔤×\overline{C(x)}), such that U ∩ σ(𝔤 × G.e) is not empty.

Proof. Let Γ_{2} be the projection of Γ ⊂ ℙ(𝔤×𝔤)×ℙ\underline{d} to ℙ\underline{d}. By Lemma 3.1, Γ_{2} is a closed subset of ℙ\underline{d} since ℙ(𝔤 × 𝔤) is complete. So τ^{−1}(Γ_{2}) is a closed subset of 𝕜^{\underline{d}×}. Moreover,

τ^{−1}(Γ_{2}) = σ(𝔤 × \overline{C(x)} \ ℬ)

since σ(𝔤×\overline{C(x)}) is stable under the action of 𝕜^{∗} ×⋯×𝕜^{∗} on 𝕜\underline{d}. Hence the open subset Z ∩ 𝕜^{\underline{d}×} of Z is contained in σ(𝔤 × \overline{C(x)}). But for all y in 𝔤 such that f_{j}(y) ≠ 0 for any j, σ(y, e) belongs to 𝕜^{\underline{d}×}. Thus, the open subset U = Z ∩ 𝕜^{\underline{d}×} of Z is convenient and the lemma follows. □

We are now ready to prove the main result of this section:

Theorem 3.3. Assume that ind𝔞^{x} = rk𝔞 for all reductive subalgebras 𝔞 strictly contained in 𝔤 and for all x in 𝔞. Then for all induced nilpotent orbits 𝒪_{𝔤} in 𝔤 and for all e in 𝒪_{𝔤}, ind𝔤^{e} = ℓ.

Proof. Let 𝒪_{𝔤} be an induced nilpotent orbit and let e be in 𝒪_{𝔤}. Using Theorem 2.9(ii), we let x be a non nilpotent element of 𝔤 such that \overline{C(x)} ∩ 𝒩(𝔤) = \overline{𝒪_{𝔤}}. Since x is not nilpotent, 𝔤^{x} is the centralizer in the reductive Lie algebra 𝔤^{x_{s}} of the nilpotent element x_{n} of 𝔤^{x_{s}}. Since 𝔤^{x_{s}} is strictly contained in 𝔤 and has rank ℓ, the index of 𝔤^{x} is equal to ℓ by hypothesis. Besides, by Theorem 2.7, (1)⇒(6), applied to x,

dimσ(𝔤 × \{x\}) = \frac{1}{2}dimG.x + ℓ.

Since σ is G-invariant, σ(𝔤 × {x}) = σ(𝔤 × G.x). Hence for all z in a dense subset of σ(𝔤 × G.x), the fiber of the restriction of σ to 𝔤 × G.x at z has minimal dimension

dim𝔤 + dimG.x − (\frac{1}{2}dimG.x + ℓ) = dim𝔤 + \frac{1}{2}dimG.x − ℓ.

Denote by Z the closure of σ(𝔤×\overline{C(x)}) in 𝕜\underline{d}. We deduce from the above equality that Z has dimension

dim𝔤 + dimC(x) − (dim𝔤 + \frac{1}{2}dimG.x − ℓ)

= dimC(x) − \frac{1}{2}dimG.x + ℓ
= \frac{1}{2}dimG.e + ℓ + 1,

since dimC(x) = dimG.x + 1 = dimG.e + 1. By Lemma 3.2, there exists an open subset U of Z contained in σ(𝔤 × \overline{C(x)}) having a nonempty intersection with σ(𝔤 × G.e). Let i be in \{1, …, ℓ\} such that f_{i}(x) ≠ 0. For z ∈ 𝕜\underline{d}, we write z = (z_{i,j})_{{1≤i≤ℓ \atop 0≤j≤d_{i}}} its

coordinates. Let 𝒱_{i} be the nullvariety in U of the coordinate z_{i,d_{i}}. Then 𝒱_{i} is not empty by the choice of U. Since U is irreducible and since z_{i,d_{i}} is not identically zero on U, 𝒱_{i} is equidimensional of dimension \frac{1}{2}dimG.e + ℓ. By Theorem 2.9(i), the nullvariety of f_{i} in \overline{C(x)} is equal to \overline{G.e}. Hence σ^{−1}(𝒱_{i}) ∩ (𝔤 × \overline{C(x)}) = σ^{−1}(U) ∩ (𝔤 × \overline{G.e}) is an open subset of 𝔤 × \overline{G.e}. So σ(𝔤 × G.e) has dimension \frac{1}{2}dimG.e + ℓ. Then by Theorem 2.7, (6)⇒(1), the index of 𝔤^{e} is equal to ℓ. □

From that point, our goal is to prove Theorem 1.3 for rigid nilpotent elements; Theorem 3.3 tells us that this is enough to complete the proof.

4. The Slodowy slice and the property (P)

In this section, we introduce a property (P) in Definition 4.2 and we prove that e ∈ 𝒩(𝔤) has Property (P) if and only if ind𝔤^{e} = ℓ (Theorem 4.13). Then, we will show in the next section that all rigid nilpotent orbits of 𝔤 but seven orbits (one in the type E_{7} and six in the type E_{8}) do have Property (P).

4.1. Blowing up of 𝒮. Let e be a nilpotent element of 𝔤 and consider an 𝔰𝔩_{2}-triple (e, h, f) containing e as in Subsection 1.3. The Slodowy slice is the affine subspace 𝒮 := e + 𝔤^{f} of 𝔤 which is a transverse variety to the adjoint orbit G.e. Denote by B_{e}(𝒮) the blowing up of 𝒮 centered at e and let p : B_{e}(𝒮) → 𝒮 be the canonical morphism. The variety 𝒮 is smooth and p^{−1}(e) is a smooth irreducible hypersurface of B_{e}(𝒮). The use of the blowing-up B_{e}(𝒮) for the computation of the index was initiated by the first author in [Ch04] and resumed by the second author in [Mo06a]. Here, we use again this technique to study the index of 𝔤^{e}. Describe first the main tools extracted from [Ch04] we need.

For Y an open subset of B_{e}(𝒮), we denote by 𝕜[Y] the algebra of regular functions on Y. By [Ch04, Théorème 3.3], we have:

Theorem 4.1. The following two assertions are equivalent: (A) the equality ind𝔤^{e} = ℓ holds, (B) there exists an affine open subset Y ⊂ B_{e}(𝒮) such that Y ∩p^{−1}(e) ≠ ∅ and satisfying the following property:

for any regular map φ ∈ 𝕜[Y] ⊗_{𝕜} 𝔤 such that φ(x) ∈

[𝔤, p(x)] for all x ∈ Y, there exists ψ ∈ 𝕜[Y] ⊗_{𝕜} 𝔤 such
that φ(x) = [ψ(x), p(x)] for all x ∈ Y.

An open subset Ω ⊂ B_{e}(𝒮) is called a big open subset if B_{e}(𝒮) \ Ω has codimension at least 2 in B_{e}(𝒮). As explained in [Ch04, Section 2], there exists a big open subset Ω of B_{e}(𝒮) and a regular map

α : Ω → Gr_{ℓ}(𝔤)

such that α(x) = 𝔤^{p(x)} if p(x) is regular. Furthermore, the map α is uniquely defined by this condition. In fact, this result is a consequence of [Sh94, Ch. VI, Theorem 1]. From now on, α stands for the so-defined map. Since p^{−1}(e) is an hypersurface and since Ω is a big open subset of B_{e}(𝒮), note that Ω ∩ p^{−1}(e) is a nonempty set. In addition, α(x) ⊂ 𝔤^{p(x)} for all x ∈ Ω.

Definition 4.2. We say that e has Property (P) if 𝔷(𝔤^{e}) ⊂ α(x) for all x in Ω ∩ p^{−1}(e).

Remark 4.3. Suppose that e is regular. Then 𝔤^{e} is a commutative algebra, i.e. 𝔷(𝔤^{e}) = 𝔤^{e}. If x ∈ Ω ∩ p^{−1}(e), then α(x) = 𝔤^{e} since p(x) = e is regular in this case. On the other hand, ind𝔤^{e} = dim𝔤^{e} = ℓ since e is regular. So e has Property (P) and ind𝔤^{e} = ℓ.

4.2. On the property (P). This subsection aims to show: Property (P) holds for e if and only if ind𝔤^{e} = ℓ. As a consequence of Remark 4.3, we can (and will) assume that e is a nonregular nilpotent element of 𝔤. As a first step, we will state in Corollary 4.12 that, if (P) holds, then so does the assertion (B) of Theorem 4.1. Let L_{𝔤} be the S(𝔤)-submodule of φ ∈ S(𝔤) ⊗_{𝕜} 𝔤 satisfying [φ(x), x] = 0 for all x in 𝔤. It is known that L_{𝔤} is a free module of basis φ_{1}, …, φ_{ℓ}, cf. [Di79]. We investigate an analogous property for the Slodowy slice 𝒮 = e + 𝔤^{f}. We denote by 𝒮_{reg} the intersection of 𝒮 and 𝔤_{reg}. As e is nonregular, the set (𝒮 \ 𝒮_{reg}) contains e.

Lemma 4.4. The set 𝒮 \ 𝒮_{reg} has codimension 3 in 𝒮 and each irreducible component of 𝒮 \ 𝒮_{reg} contains e.

Proof. Let us consider the morphism

G × 𝒮 ⟶ 𝔤
(g, x) ⟼ g(x)

By a Slodowy’s result [Sl80], this morphism is a smooth morphism. So its fibers are equidimensional of dimension dim𝔤^{f}. In addition, by [V72], 𝔤\𝔤_{reg} is a G-invariant equidimensional closed subset of 𝔤 of codimension 3. Hence 𝒮\𝒮_{reg} is an equidimensional closed subset of 𝒮 of codimension 3. Denoting by t ↦ g(t) the one parameter subgroup of G generated by adh, 𝒮 and 𝒮 \ 𝒮_{reg} are stable under the action of t^{−2}g(t) for all t in 𝕜^{∗}. Furthermore, for all x in 𝒮, t^{−2}g(t)(x) goes to e when t goes to ∞, whence the lemma. □

Denote by 𝕜[𝒮] the algebra of regular functions on 𝒮 and denote by L_{𝒮} the 𝕜[𝒮]-submodule of φ ∈ 𝕜[𝒮] ⊗_{𝕜} 𝔤 satisfying [φ(x), x] = 0 for all x in 𝒮.

Lemma 4.5. The module L_{𝒮} is a free module of basis φ_{1}|_{𝒮}, …, φ_{ℓ}|_{𝒮} where φ_{i}|_{𝒮} is the restriction to 𝒮 of φ_{i} for i = 1, …, ℓ.

Proof. Let φ be in L_{𝒮}. There are regular functions a_{1}, …, a_{ℓ} on 𝒮_{reg} satisfying

φ(x) = a_{1}(x)φ_{1}|_{𝒮}(x) + ⋯ + a_{ℓ}(x)φ_{ℓ}|_{𝒮}(x)

for all x ∈ 𝒮_{reg}, by Lemma 2.1(ii). By Lemma 4.4, 𝒮 \ 𝒮_{reg} has codimension 3 in 𝒮. Hence a_{1}, …, a_{ℓ} have polynomial extensions to 𝒮 since 𝒮 is normal. So the maps φ_{1}|_{𝒮}, …, φ_{ℓ}|_{𝒮} generate L_{𝒮}. Moreover, by Lemma 2.1(ii) for all x ∈ 𝒮_{reg}, φ_{1}(x), …, φ_{ℓ}(x) are linearly independent, whence the statement. □

The following proposition accounts for an important step to interpret Assertion (B) of Theorem 4.1:

Proposition 4.6. Let φ be in 𝕜[𝒮] ⊗_{𝕜} 𝔤 such that φ(x) ∈ [𝔤, x] for all x in a nonempty open subset of 𝔤. Then there exists a polynomial map ψ ∈ 𝕜[𝒮] ⊗_{𝕜} 𝔤 such that φ(x) = [ψ(x), x] for all x ∈ 𝒮.

Proof. Since 𝔤^{x} is the orthogonal complement of [x, 𝔤] in 𝔤, our hypothesis says that φ(x) is orthogonal to 𝔤^{x} for all x in a nonempty open subset 𝒮′ of 𝒮. The intersection 𝒮′ ∩ 𝒮_{reg} is not empty; so by Lemma 2.1(ii), ⟨φ(x), φ_{i}|_{𝒮}(x)⟩ = 0 for all i = 1, …, ℓ and for all x ∈ 𝒮′ ∩𝒮_{reg}. Therefore, by continuity, ⟨φ(x), φ_{i}|_{𝒮}(x)⟩ = 0 for all i = 1, …, ℓ and all x ∈ 𝒮. Hence φ(x) ∈ [x, 𝔤] for all x ∈ 𝒮_{reg} by Lemma 2.1(ii) again. Consequently by Lemma 4.4, Lemma 4.5 and the proof of the main theorem of [Di79], there exists an element ψ ∈ 𝕜[𝒮] ⊗_{𝕜}𝔤 which satisfies the condition of the proposition. □

Let u_{1}, …,u_{m} be a basis of 𝔤^{f} and let u_{1}^{∗}, …, u_{m}^{∗} be the corresponding coordinate system of 𝒮 = e + 𝔤^{f}. There is an affine open subset Y ⊂ B_{e}(𝒮) with Y ∩p^{−1}(e) ≠ ∅ such that 𝕜[Y] is the set of linear combinations of monomials in (u_{1}^{∗})^{−1}, u_{1}^{∗}, …, u_{m}^{∗} whose total degree is nonnegative. In particular, we have a global coordinates system u_{1}^{∗}, v_{2}^{∗}, …, v_{m}^{∗} on Y satisfying the relations:

(4)  u_{2}^{∗} = u_{1}^{∗}v_{2}^{∗} , …, u_{m}^{∗} = u_{1}^{∗}v_{m}^{∗}.

Note that, for x ∈ Y, we so have: p(x) = e + u_{1}^{∗}(x)(u_{1} + v_{2}^{∗}(x)u_{2} + ⋯ + v_{m}^{∗}(x)u_{m}). So, the image of Y by p is the union of {e} and the complementary in 𝒮 of the nullvariety of u_{1}^{∗}. Let Y′ be an affine open subset of Y contained in Ω and having a nonempty intersection with p^{−1}(e). Denote by L_{Y′} the set of regular maps φ from Y′ to 𝔤 satisfying [φ(x), p(x)] = 0 for all x ∈ Y′.

Lemma 4.7. Suppose that e has Property (P). For each z ∈ 𝔷(𝔤^{e}), there exists ψ_{z} ∈ 𝕜[Y′] ⊗_{𝕜} 𝔤 such that z − u_{1}^{∗}ψ_{z} belongs to L_{Y′}.

Proof. Let z be in 𝔷(𝔤^{e}). Since Y′ ⊂ Ω, for each y ∈ Y′, there exists an affine open subset U_{y} of Y′ containing y and regular maps ν_{1}, …,ν_{ℓ} from U_{y} to 𝔤 such that ν_{1}(x), …,ν_{ℓ}(x) is a basis of α(x) for all x ∈ U_{y}. Let y be in Y′. We consider two cases:

(1) Suppose p(y) = e. Since e has Property (P), there exist regular functions a_{1}, …,a_{ℓ} on U_{y} satisfying

z = a_{1}(x)ν_{1}(x) + ⋯ + a_{ℓ}(x)ν_{ℓ}(x),

for all x ∈ U_{y} ∩ p^{−1}(e). The intersection U_{y} ∩ p^{−1}(e) is the set of zeroes of u_{1}^{∗} in U_{y}. So there exists a regular map ψ from U_{y} to 𝔤 which satisfies the equality:

z − u_{1}^{∗}ψ = a_{1}ν_{1} + ⋯ + a_{ℓ}ν_{ℓ}.

Hence [z − u_{1}^{∗}(x)ψ(x), p(x)] = 0 for all x ∈ U_{y} since α(x) is contained in 𝔤^{p(x)} for all x ∈ Ω.

(2) Suppose p(y) ≠ e. Then we can assume that U_{y} ∩ p^{−1}(e) = ∅ and the map ψ = (u_{1}^{∗})^{−1}z satisfies the condition: [z − u_{1}^{∗}(x)ψ(x), p(x)] = 0 for all x ∈ U_{y}. In both cases (1) or (2), we have found a regular map ψ_{y} from U_{y} to 𝔤 satisfying: [z − (u_{1}^{∗}ψ_{y})(x), p(x)] = 0 for all x ∈ U_{y}. Let y_{1}, …,y_{k} be in Y′ such that the open subsets U_{y_{1}}, …,U_{y_{k}} cover Y′. For i = 1, …, k, we denote by ψ_{i} a regular map from U_{y_{i}} to 𝔤 such that z − u_{1}^{∗}ψ_{i} is in Γ(U_{y_{i}}, ℒ) where ℒ is the localization of L_{Y′} on Y′.

Then for i, j = 1, …, k, ψ_{i} − ψ_{j} is in Γ(U_{y_{i}} ∩ U_{y_{j}}, ℒ). Since Y′ is affine, H^{1}(Y′, ℒ) = 0. So, for i = 1, …, l, there exists ψ̃_{i} in Γ(U_{y_{i}}, ℒ) such that ψ̃_{i} − ψ̃_{j} is equal to ψ_{i} − ψ_{j} on U_{y_{i}} ∩ U_{y_{j}} for all i, j. Then there exists a well-defined map ψ_{z} from Y′ to 𝔤 whose restriction to U_{y_{i}} is equal to ψ_{i} − ψ̃_{i} for all i, and such that z − u_{1}^{∗}ψ_{z} belongs to L_{Y′}. Finally, the map ψ_{z} verifies the required property. □

Let z be in 𝔷(𝔤^{e}). We denote by φ_{z} the regular map from Y to 𝔤 defined by:

φ_{z}(x) = [z, u_{1}] + v_{2}^{∗}(x)[z, u_{2}] + ⋯ + v_{m}^{∗}(x)[z, u_{m}], for all x ∈ Y. (5)

Corollary 4.8. Suppose that e has Property (P) and let z be in 𝔷(𝔤^{e}). There exists ψ_{z} in 𝕜[Y′] ⊗_{𝕜} 𝔤 such that φ_{z}(x) = [ψ_{z}(x), p(x)] for all x ∈ Y′.

Proof. By Lemma 4.7, there exists ψ_{z} in 𝕜[Y′] ⊗_{𝕜} 𝔤 such that z − u_{1}^{∗}ψ_{z} is in L_{Y′}. Then

u_{1}^{∗}φ_{z}(x) = [z, p(x)] = [z − u_{1}^{∗}ψ_{z}(x), p(x)] + u_{1}^{∗}[ψ_{z}(x), p(x)],

for all x ∈ Y′. So the map ψ_{z} is convenient, since u_{1}^{∗} is not identically zero on Y′. □

The following lemma is easy but helpful for Proposition 4.10:

Lemma 4.9. Let v be in 𝔤^{e}. Then, v belongs to 𝔷(𝔤^{e}) if and only if [v, 𝔤^{f}] ⊂ [e, 𝔤].

Proof. Since [x, 𝔤] is the orthogonal complement of 𝔤^{x} in 𝔤 for all x ∈ 𝔤, we have:

[v, 𝔤^{f}] ⊂ [e, 𝔤] ⟺ ⟨[v, 𝔤^{f}], 𝔤^{e}⟩ = 0
⟺ ⟨[v, 𝔤^{e}], 𝔤^{f}⟩ = 0 ⟺ [v, 𝔤^{e}] ⊂ [f, 𝔤].

But 𝔤 is the direct sum of 𝔤^{e} and [f, 𝔤] and [v, 𝔤^{e}] is contained in 𝔤^{e} since v ∈ 𝔤^{e}. Hence [v, 𝔤^{f}] is contained in [e, 𝔤] if and only if v is in 𝔷(𝔤^{e}). □

Proposition 4.10. Suppose that e has Property (P) and let φ be in 𝕜[Y] ⊗_{𝕜} 𝔤 such that φ(x) ∈ [𝔤, p(x)] for all x ∈ Y. Then there exists ψ in 𝕜[Y′] ⊗_{𝕜} 𝔤 such that φ(x) = [ψ(x), p(x)] for all x ∈ Y′.

Proof. Since φ is a regular map from Y to 𝔤, there is a nonnegative integer d and φ̃ ∈ 𝕜[𝒮] ⊗_{𝕜} 𝔤 such that

(u_{1}^{∗})^{d}(x)φ(x) = (φ̃^{∘}p)(x), ∀x ∈ Y (6)

and φ̃ is a linear combination of monomials in u_{1}^{∗}, …, u_{m}^{∗} whose total degree is at least d. By hypothesis on φ, we deduce that for all x ∈ 𝒮 such that u_{1}^{∗}(x) ≠ 0, φ̃(x) is in [𝔤, x]. Hence by Proposition 4.6, there exists ψ̃ in 𝕜[𝒮] ⊗_{k} 𝔤 such that φ̃(x) = [ψ̃(x), x] for all x ∈ 𝒮. Denote by ψ̃′ the sum of monomials of degree at least d in ψ̃ and denote by ψ′ the element of 𝕜[Y] ⊗_{𝕜} 𝔤 satisfying

(u_{1}^{∗})^{d}(x)ψ′(x) = (ψ̃′^{∘}p)(x), ∀x ∈ Y. (7)

Then we set, for x ∈ Y, φ′(x) := φ(x) − [ψ′(x), p(x)]. We have to prove the existence of an element ψ′′ in 𝕜[Y′] ⊗_{𝕜} 𝔤 such that φ′(x) = [ψ′′(x), p(x)] for all x ∈ Y′.

• If d = 0, then φ = φ̃^{∘}p, ψ′ = ψ and φ′ = 0; so ψ′ is convenient in that case.

• If d = 1, we can write

u_{1}^{∗}(x)φ(x) = φ̃(p(x))
= [ψ̃(p(x)), e + u_{1}^{∗}(x)(u_{1} + v_{2}^{∗}(x)u_{2} + ⋯ + v_{m}^{∗}(x)u_{m})],

for all x ∈ Y, whence we deduce

u_{1}^{∗}(x)(φ(x) − [ψ′(x), p(x)])
= [ψ̃(e), e + u_{1}^{∗}(x)(u_{1} + v_{2}^{∗}(x)u_{2} + ⋯ + v_{m}^{∗}(x)u_{m})]

for all x ∈ Y. Hence ψ̃(e) belongs to 𝔤^{e} and [ψ̃(e), u_{i}] ∈ [e, 𝔤] for all i = 1, …, m, since φ(x) ∈ [e, 𝔤] for all x ∈ Y ∩ p^{−1}(e). Then ψ̃(e) is in 𝔷(𝔤^{e}) by Lemma 4.9. So by Corollary 4.8, φ′ has the desired property.

• Suppose d > 1. For \underline{i} = (i_{1}, …, i_{m}) ∈ ℕ^{m}, we set |\underline{i}| := i_{1}+⋯+i_{m} and we denote by ψ\underline{i} the coefficient of (u_{1}^{∗})^{i_{1}} ⋯ (u_{m}^{∗})^{i_{m}} in ψ̃. By Corollary 4.8, it suffices to prove:

\{ \begin{matrix}ψ\underline{i} = 0 & if |\underline{i}| < d − 1; \\ ψ\underline{i} ∈ 𝔷(𝔤^{e}) & if |\underline{i}| = d − 1 \end{matrix} .

For \underline{i} ∈ ℕ^{m} and j ∈ \{1, …, m\}, we define the element \underline{i}(j) of ℕ^{m} by:

i(j) := (i_{1}, …, i_{j−1}, i_{j} + 1, i_{j+1}, …, i_{m}).

It suffices to prove:

Claim 4.11. For |i| ≤ d − 1, ψ\underline{i} is an element of 𝔤^{e} such that [ψ\underline{i}, u_{j}] + [ψ_{\underline{i}(j)}, e] = 0 for j = 1, …, m.

Indeed, by Lemma 4.9, if

[ψ\underline{i}, u_{j}] + [ψ_{\underline{i}(j)}, e] = 0 and ψ\underline{i} ∈ 𝔤^{e}

for all j = 1, …, m, then ψ\underline{i} ∈ 𝔷(𝔤^{e}). Furthermore, if

[ψ\underline{i}, u_{j}] + [ψ_{\underline{i}(j)}, e] = 0 and ψ\underline{i} ∈ 𝔤^{e} and ψ_{\underline{i}(j)} ∈ 𝔤^{e}

for all j = 1, …, m, then ψ\underline{i} = 0 since 𝔷(𝔤^{e}) ∩ 𝔤^{f} = 0. So only remains to prove Claim 4.11. We prove the claim by induction on |\underline{i}|. Arguing as in the case d = 1, we prove the claim for |\underline{i}| = 0. We suppose the claim true for all |\underline{i}| ≤ l − 1 for some 0 < l ≤ d − 2. We have to prove the statement for all |i| ≤ l. By what foregoes and by induction hypothesis, ψ\underline{i} = 0 for |\underline{i}| ≤ l − 2. For k = l + 1, l + 2, we consider the ring 𝕜[τ_{k}] where τ_{k}^{k} = 0. Since (u_{1}^{∗})^{d} vanishes on the set of 𝕜[τ_{l+1}]-points x = x_{0} + x_{1}τ_{l+1} + ⋯ + x_{l}τ_{l+1}^{l} of Y whose source x_{0} is a zero of u_{1}^{∗},

0 = [ψ̃(e + τ_{l+1}v), e + τ_{l+1}v] = ∑_{|i|=l} τ_{l+1}^{l}[ψ\underline{i}, e](u_{1}^{∗})^{i_{1}} ⋯ (u_{m}^{∗})^{i_{m}}(v),

for all v ∈ 𝔤^{f}. So ψ\underline{i} ∈ 𝔤^{e} for |\underline{i}| = l. For |\underline{i}| equal to l, the term in

τ_{l+2}^{l+1}(u_{1}^{∗})^{i_{1}} ⋯ (u_{i_{j−1}}^{∗})^{i_{j−1}}(u_{i_{j}+1}^{∗})^{i_{j}+1}(u_{i_{j+1}}^{∗})^{i_{j+1}} ⋯ (u_{m}^{∗})^{i_{m}}(v)

of [ψ̃(e + τ_{l+2}v), e + τ_{l+2}v] is equal to [ψ_{\underline{i}(j)}, e] + [ψ\underline{i}, u_{j}]. Since (u_{1}^{∗})^{d} vanishes on the set of 𝕜[τ_{l+2}]-points of Y whose source is a zero of u_{1}^{∗}, this term is equal to 0, whence the claim. □

Recall that Y′ is an affine open subset of Y contained in Ω and having a nonempty intersection with p^{−1}(e).

Corollary 4.12. Suppose that e has Property (P). Let φ be in 𝕜[Y′]⊗_{𝕜}𝔤 such that φ(x) ∈ [𝔤, p(x)] for all x ∈ Y′. Then there exists ψ in 𝕜[Y′]⊗_{𝕜}𝔤 such that φ(x) = [ψ(x), p(x)] for all x ∈ Y′.

Proof. For a ∈ 𝕜[Y], denote by D(a) the principal open subset defined by a. Let D(a_{1}), …, D(a_{m}) be an open covering of Y′ by principal open subsets of Y, with a_{1}, …, a_{k} in 𝕜[Y]. Since φ is a regular map from Y′ to 𝔤, there is m_{i} ≥ 0 such that a_{i}^{m_{i}}φ is the restriction to Y′ of some regular map φ_{i} from Y to 𝔤. For m_{i} big enough, φ_{i} vanishes on Y \D(a_{i}); hence φ_{i}(x) ∈ [𝔤, p(x)] for all x ∈ Y. So, by Proposition 4.6, there is a regular map ψ_{i} from Y′ to 𝔤 such that φ_{i}(x) = [ψ_{i}(x), p(x)] for all x ∈ Y′. Then for all x ∈ D(a_{i}), we have φ(x) = [a_{i}(x)^{−m_{i}}ψ_{i}(x), p(x)]. Since Y′ is an affine open subset of Y, there exists a regular map ψ from Y′ to 𝔤 which satisfies the condition of the corollary. □

We are now in position to prove the main result of this section:

Theorem 4.13. The equality ind𝔤^{e} = ℓ holds if and only if e has Property (P).

Proof. By Corollary 4.12, if e has Property (P), then Assertion (B) of Theorem 4.1 is satisfied. Conversely, suppose that ind𝔤^{e} = ℓ and show that e has Property (P). By Theorem 4.1, (A)⇒(B), Assertion (B) is satisfied. We choose an affine open subset Y′ of Y, contained in Ω, such that Y′ ∩ p^{−1}(e) ≠ and verifying the condition of the assertion (B). Let z ∈ 𝔷(𝔤^{e}). Recall that the map φ_{z} is defined by (5). Let x be in Y′. If u_{1}^{∗}(x) ≠ 0, then φ_{z}(x) belongs to [𝔤, p(x)] by (5). If u_{1}^{∗}(x) = 0 , then by Lemma 4.9, φ_{z}(x) belongs to [e, 𝔤]. So there exists a regular map ψ from Y′ to 𝔤 such that φ_{z}(x) = [ψ(x), p(x)] for all x ∈ Y′ by Assertion (B). Hence we have

[z − u_{1}^{∗}ψ(x), p(x)] = 0,

for all x ∈ Y′ since (u_{1}^{∗}φ_{z})(x) = [z, p(x)] for all x ∈ Y. So α(x) contains z for all x in Ω ∩ Y′ ∩ p^{−1}(e). Since p^{−1}(e) is irreducible, we deduce that e has Property (P). □

4.3. A new formulation of the property (P). Recall that Property

(P) is introduced in Definition 4.2. As has been noticed in the proof of Lemma 4.4, the morphism G × 𝒮 → 𝔤, (g, x) ↦ g(x) is smooth. As a consequence, the set 𝒮_{reg} of v ∈ 𝒮 such that v is regular is a nonempty open subset of 𝒮. For x in 𝒮_{reg}, 𝔤^{e+t(x−e)} has dimension ℓ for all t in a nonempty open subset of 𝕜 since x = e+(x−e) is regular. Furthermore, since 𝕜 has dimension 1, [Sh94, Ch. VI, Theorem 1] asserts that there is a unique regular map

β_{x} : 𝕜 → Gr_{ℓ}(𝔤)

satisfying β_{x}(t) = 𝔤^{e+t(x−e)} for all t in a nonempty open subset of 𝕜. Recall that Y is an affine open subset of B_{e}(𝒮) with Y ∩ p^{−1}(e) ≠ ∅ and that u_{1}^{∗}, v_{2}^{∗}, …, v_{m}^{∗} is a global coordinates system of Y, cf. (4). Let 𝒮′_{reg} be the subset of x in 𝒮_{reg} such that u_{1}^{∗}(x) ≠ 0. For x in 𝒮′_{reg}, we denote by x̃ the element of Y whose coordinates are 0, v_{2}^{∗}(x), …, v_{m}^{∗}(x).

Lemma 4.14. Let x be in 𝒮′_{reg}.

(i) The subspace β_{x}(0) is contained in 𝔤^{e}. (ii) If x̃ ∈ Ω, then α(x̃) = β_{x}(0).

Proof. (i) The map β_{x} is a regular map and [β_{x}(t), e + t(x − e)] = 0 for all t in a nonempty open subset of 𝕜. So, β_{x}(0) is contained in 𝔤^{e}. (ii) Since 𝒮′_{reg} has an empty intersection with the nullvariety of u_{1}^{∗} in 𝒮, the restriction of p to p^{−1}(𝒮′_{reg}) is an isomorphism from p^{−1}(𝒮′_{reg}) to 𝒮′_{reg}. Furthermore, β_{x}(t) = α(p^{−1}(e + tx − te)) for any t in 𝕜 such that e + t(x − e) belongs to 𝒮′_{reg} and p^{−1}(e + tx − te) goes to x̃ when t goes to 0. Hence β_{x}(0) is equal to α(x̃) since α and β are regular maps. □

Corollary 4.15. The element e has Property (P) if and only if 𝔷(𝔤^{e}) ⊂ β_{x}(0) for all x in a nonempty open subset of 𝒮_{reg}.

Proof. The map x ↦ x̃ from 𝒮′_{reg} to Y is well-defined and its image is an open subset of Y ∩ p^{−1}(e). Let 𝒮′′_{reg} be the set of x ∈ 𝒮′_{reg} such that x̃ ∈ Ω and let Y′′ be the image of 𝒮′′_{reg} by the map x ↦ x̃. Then 𝒮′′_{reg} is open in 𝒮_{reg} and Y′′ is dense in Ω ∩ p^{−1}(e) since p^{−1}(e) is irreducible. Furthermore, the image of a dense open subset of 𝒮′′_{reg} by the map x ↦ x̃ is dense in Y′′. Since α is regular, e has property (P) if and only if α(x) contains 𝔷(𝔤^{e}) for all x in a dense subset of Y′′. By Lemma 4.14(ii), the latter property is equivalent to the fact that β_{x}(0) contains 𝔷(𝔤^{e}) for all x in a dense open subset of 𝒮′′_{reg}. □

Corollary 4.16. (i) If 𝔷(𝔤^{e}) is generated by φ_{1}(e), …, φ_{ℓ}(e), then e has Property (P). (ii) If 𝔷(𝔤^{e}) has dimension 1, then e has Property (P).

Proof. Recall that φ_{i}(e) belongs to 𝔷(𝔤^{e}), for all i = 1, …, ℓ, by Lemma 2.1(i). Moreover, for all x in 𝒮_{reg} and all i = 1, …, ℓ, φ_{i}(e + t(x − e)) belongs to 𝔤^{e+t(x−e)} for any t in 𝕜. So by continuity, φ_{i}(e) belongs to β_{x}(0). As a consequence, whenever 𝔷(𝔤^{e}) is generated by φ_{1}(e), …,φ_{ℓ}(e), e has Property (P) by Corollary 4.15. (ii) is an immediate consequence of (i) since φ_{1}(e) = e by our choice of d_{1}. □

5. Proof of Theorem 1.3 for rigid nilpotent orbits

We intend to prove in this section the following theorem:

Theorem 5.1. Suppose that 𝔤 is reductive and let e be a rigid nilpotent element of 𝔤. Then the index of 𝔤^{e} is equal to ℓ.

Theorem 5.1 will complete the proof of Theorem 1.3 by Theorem 3.3. As explained in introduction, we can assume that 𝔤 is simple. We consider two cases, according to 𝔤 has classical type or exceptional type.

5.1. The classical case. Assume that 𝔤 is simple of classical type. More precisely, assume that 𝔤 is one of the Lie algebras 𝔰𝔩_{ℓ+1}(𝕜), 𝔰𝔬_{2ℓ+1}(𝕜), 𝔰𝔭_{2ℓ}(𝕜), 𝔰𝔬_{2ℓ}(𝕜).

Lemma 5.2. Let m be a positive integer such that x^{m} − trx^{m} belongs to 𝔤 for all x in 𝔤. Then e^{m} belongs to the subspace generated by φ_{1}(e), …,φ_{ℓ}(e).

Proof. Recall that L_{𝔤} is the submodule of elements φ of S(𝔤) ⊗_{𝕜} 𝔤 such that [x, φ(x)] = 0 for all x in 𝔤. According to [Di79], L_{𝔤} is a free module generated by the φ′_{i}s. For all x in 𝔤, [x, x^{m}] = 0. Hence there exist polynomial functions a_{1}, …,a_{ℓ} on 𝔤 such that

x^{m} − trx^{m} = a_{1}(x)φ_{1}(x) + ⋯ + a_{ℓ}(x)φ_{ℓ}(x)

for all x in 𝔤, whence the lemma. □

Theorem 5.3. Let e be a rigid nilpotent element. Then 𝔷(𝔤^{e}) is generated by powers of e. In particular, the index of 𝔤^{e} is equal to ℓ.

Proof. Let us prove the first assertion. If 𝔤 has type A or C, then 𝔷(𝔤^{e}) is generated by powers of e by [Mo06c, Théorème 1.1.8] or [Y06b]. So we can assume that 𝔤 has type B or D. Set n := 2ℓ + 1 if 𝔤 has type B_{ℓ} and n := 2ℓ if 𝔤 has type D_{ℓ}. Denote by (n_{1}, …,n_{k}), with n_{1}≥ ⋯ ≥n_{k}, the partition of n corresponding to the nilpotent element e. By [Mo06c, Théorème 1.1.8] or [Y06b], 𝔷(𝔤^{e}) is not generated by powers of e if and only if n_{1} and n_{2} are both odd integers and n_{3} < n_{2}. On the other hand, since e is rigid, n_{k} is equal to 1, n_{i} ≤ n_{i+1} ≤ n_{i} + 1 and all odd integers of the partition (n_{1}, …,n_{k}) have a multiplicity different from 2 [Ke83, Sp82, ch. II] or [CMa93, Corollary 7.3.5]. Hence, the preceding criterion is not satisfied for e. Then, the second assertion results from Lemma 5.2, Corollary 4.16(i) and Theorem 4.13. □

Remark 5.4. Yakimova’s proof of Elashvili’s conjecture in the classical case is shorter and more elementary [Y06a]. The results of Section 4 will serve the exceptional case in a more relevant way.

5.2. The exceptional case. We let in this subsection 𝔤 be simple of exceptional type and we assume that e is a nonzero rigid nilpotent element of 𝔤. The dimension of the center of centralizers of nilpotent elements has been recently described in [LT08, Theorem 4]. On the other hand, we have explicit computations for the rigid nilpotent orbits in the exceptional types due to A.G. Elashvili. These computations are collected in [Sp82, Appendix of Chap. II] and a complete version was published later in [E85b]. From all this, we observe that the center of 𝔤^{e} has dimension 1 in most cases. In more details, we have:

Proposition 5.5. Let e be nonzero rigid nilpotent element of 𝔤.

(i) Suppose that 𝔤 has type G_{2}, F_{4} or E_{6}. Then dim𝔷(𝔤^{e}) = 1. (ii) Suppose that 𝔤 has type E_{7}. If 𝔤^{e} has dimension 41, then dim𝔷(𝔤^{e}) = 2; otherwise dim𝔷(𝔤^{e}) = 1. (iii) Suppose that 𝔤 has type E_{8}. If 𝔤^{e} has dimension 112, 84, 76, or 46, then dim𝔷(𝔤^{e}) = 2, if 𝔤^{e} has dimension 72, then dim𝔷(𝔤^{e}) = 3; otherwise dim𝔷(𝔤^{e}) = 1.

By Corollary 4.16(ii), ind𝔤^{e} = ℓ whenever dim𝔷(𝔤^{e}) = 1. So, as an immediate consequence of Proposition 5.5, we obtain:

Corollary 5.6. Suppose that either 𝔤 has type G_{2}, F_{4}, E_{6}, or 𝔤 has type E_{7} and dim𝔤^{e} ≠ 41, or 𝔤 has type E_{8} and dim𝔤^{e} ∉ {112, 84, 76, 72, 46}. Then dim𝔷(𝔤^{e}) = 1 and the index of 𝔤^{e} is equal to ℓ.

According to Corollary 5.6, it remains 7 cases; there are indeed two rigid nilpotent orbits of codimension 46 in E_{8}. We handle now these remaining cases. We process here in a different way; we study technical conditions on 𝔤^{e} under which ind𝔤^{e} = ℓ. For the moment, we state general results about the index. Let 𝔞 be an algebraic Lie algebra. Recall that the stabilizer of ξ ∈ 𝔞^{∗} for the coadjoint representation is denoted by 𝔞^{ξ} and that ξ is regular if dim𝔞^{ξ} = ind𝔞. Choose a commutative subalgebra 𝔱 of 𝔞 consisted of semisimple elements of 𝔞 and denote by 𝔷_{𝔞}(𝔱) the centralizer of 𝔱 in 𝔞. Then 𝔞 = 𝔷_{𝔞}(𝔱)⊕[𝔱, 𝔞]. The dual 𝔷_{𝔞}(𝔱)^{∗} of 𝔷_{𝔞}(𝔱) identifies to the orthogonal complement of [𝔱, 𝔞] in 𝔞^{∗}. Thus, ξ ∈ 𝔷_{𝔞}(𝔱)^{∗} if and only if 𝔱 is contained in 𝔞^{ξ}.

Lemma 5.7. Suppose that there exists ξ in 𝔷_{𝔞}(𝔱)^{∗} such that dim(𝔞^{ξ} ∩ [𝔱, 𝔞]) ≤ 2. Then

ind𝔞 ≤ ind𝔷_{𝔞}(𝔱) + 1.

Proof. Let T be the closure in 𝔷_{𝔞}(𝔱)^{∗}×Gr_{3}([𝔱, 𝔞]) of the subset of elements (η, E) such that η is a regular element of 𝔷_{𝔞}(𝔱)^{∗} and E is contained in 𝔞^{η}. The image T_{1} of T by the projection from 𝔷_{𝔞}(𝔱)^{∗} × Gr_{3}([𝔱, 𝔞]) to 𝔷_{𝔞}(𝔱)^{∗} is closed in 𝔷_{𝔞}(𝔱)^{∗}. By hypothesis, T_{1} is not equal to 𝔷_{𝔞}(𝔱)^{∗} since for all η in T_{1}, dim(𝔞^{η} ∩ [𝔱, 𝔞]) ≥ 3. Hence there exists a regular element ξ_{0} in 𝔷_{𝔞}(𝔱)^{∗} such that dim(𝔞^{ξ_{0}} ∩ [𝔱, 𝔞]) ≤ 2. Since 𝔱 is contained in 𝔞^{ξ_{0}},

𝔞^{ξ_{0}} = 𝔷_{𝔞}(𝔱)^{ξ_{0}} ⊕ 𝔞^{ξ_{0}} ∩ [𝔱, 𝔞].

If [𝔱, 𝔞] ∩ 𝔞^{ξ_{0}} = {0} then ind𝔞 is at most ind𝔷_{𝔞}(𝔱). Otherwise, 𝔞^{ξ_{0}} is not a commutative subalgebra since 𝔱 is contained in 𝔞^{ξ_{0}}. Hence ξ_{0} is not a regular element of 𝔞^{∗}, so ind𝔞 < dim𝔞^{ξ_{0}}. Since dim𝔞^{ξ_{0}} ≤ ind𝔷_{𝔞}(𝔱) + 2, the lemma follows. □

From now on, we assume that 𝔞 = 𝔤^{e}. As a rigid nilpotent element of 𝔤, e is a nondistinguished nilpotent element. So we can choose a nonzero commutative subalgebra 𝔱 of 𝔤^{e} consisted of semisimple elements. Denote by 𝔩 the centralizer of 𝔱 in 𝔤. As a Levi subalgebra of 𝔤, 𝔩 is a reductive Lie algebra whose rank is ℓ. Moreover its dimension is strictly smaller than dim𝔤. In the preceding notations, we have 𝔷_{𝔤^{e}}(𝔱) = 𝔷_{𝔤}(𝔱)^{e} = 𝔩^{e}. Let 𝔱_{1} be a commutative subalgebra of 𝔩^{e} containing 𝔱 and consisting of semisimple elements of 𝔩. Then [𝔱, 𝔤^{e}] is stable under the adjoint action of 𝔱_{1}. For λ in 𝔱_{1}^{∗}, denote by 𝔤_{λ}^{e} the λ-weight space of the adjoint action of 𝔱_{1} in 𝔤^{e}.

Lemma 5.8. Let λ ∈ 𝔱_{1}^{∗} be a nonzero weight of the adjoint action of 𝔱_{1} in 𝔤^{e}. Then −λ is also a weight for this action and λ and −λ have the same multiplicity. Moreover, 𝔤_{λ}^{e} is contained in [𝔱, 𝔤^{e}] if and only if the restriction of λ to 𝔱 is not identically zero.

Proof. By definition, 𝔤_{λ}^{e} ∩ 𝔩^{e} = {0} if and only if the restriction of λ to 𝔱 is not identically zero. So 𝔤_{λ}^{e} is contained in [𝔱, 𝔤^{e}] if and only if the restriction of λ to 𝔱 is not equal to 0 since

𝔤_{λ}^{e} = (𝔤_{λ}^{e} ∩ 𝔩^{e}) ⊕ (𝔤_{λ}^{e} ∩ [𝔱, 𝔤^{e}]).

The subalgebra 𝔱_{1} is contained in a reductive factor of 𝔤^{e}. So we can choose h and f such that 𝔱_{1} is contained in 𝔤^{e}∩𝔤^{f}. As a consequence, any weight of the adjoint action of 𝔱_{1} in 𝔤^{f} is a weight of the adjoint action of 𝔱_{1} in 𝔤^{e} with the same multiplicity. Furthermore, the 𝔱_{1}-module 𝔤^{f} for the ajoint action is isomorphic to the 𝔱_{1}-module (𝔤^{e})^{∗} for the coadjoint action. So −λ is a weight of the adjoint action of 𝔱_{1} in 𝔤^{f} with the same multiplicity as λ. Hence −λ is a weight of the adjoint action of 𝔱_{1} in 𝔤^{e} with the same multiplicity as λ, whence the lemma. □

Choose pairwise different elements λ_{1}, …,λ_{r} of 𝔱_{1}^{∗} so that the weights of the adjoint action of 𝔱_{1} in 𝔤^{e} which are not identically zero on 𝔱 are precisely the elements ±λ_{i}. For i = 1, …, r, let v_{i,1}, …,v_{i,m_{i}} and w_{i,1}, …,w_{i,m_{i}} be basis of 𝔤_{λ_{i}}^{e} and 𝔤_{−λ_{i}}^{e} respectively. Then we set:

q_{i} := det ([v_{i,k}, w_{i,l}])_{1≤k,l≤m_{i}} ∈ S(𝔩^{e}).

Proposition 5.9. Suppose that ind𝔩^{e} = ℓ and suppose that one of the following two conditions is satisfied:

(1) for i = 1, …, r, q_{i} ≠ 0,

(2) there exists j in {1, …, r} such that q_{i} ≠ 0 for all i ≠ j and such that the basis v_{j,1}, …,v_{j,m_{j}} and w_{j,1}, …,w_{j,m_{j}} can be chosen so that

det ([v_{j,k}, w_{j,l}])_{1≤k,l≤m_{j}−1} ≠ 0.

Then, ind𝔤^{e} = ℓ.

Proof. First, observe that ind𝔤^{e} − ind𝔤 is an even integer. Indeed, we have:

ind𝔤^{e} − ind𝔤 = (ind𝔤^{e} − dim𝔤^{e}) + (dim𝔤^{e} − dim𝔤) + (dim𝔤 − ind𝔤).

But the integers ind𝔤^{e} − dim𝔤^{e}, dim𝔤^{e} − dim𝔤 and dim𝔤 − ind𝔤 are all even integers. Thereby, if ind𝔤^{e} ≤ ind𝔤 + 1, then ind𝔤^{e} ≤ ind𝔤. In turn, by Vinberg’s inequality (cf. Introduction), we have ind𝔤^{e} ≥ ind𝔤. Hence, it suffices to prove ind𝔤^{e} ≤ ind𝔩^{e} + 1 since our hypothesis says that ind𝔩^{e} = ℓ = ind𝔤. Now, by Lemma 5.7, if there exists ξ in (𝔩^{e})^{∗} such that (𝔤^{e})^{ξ} ∩ [𝔱, 𝔤^{e}] has dimension at most 2, then we are done. Denote by 𝔩_{1} the centralizer of 𝔱_{1} in 𝔤. Then 𝔩_{1} is contained in 𝔩 and 𝔩^{e} = 𝔩_{1}^{e} ⊕ [𝔱_{1}, 𝔩^{e}] and (𝔩_{1}^{e})^{∗} identifies to the orthogonal of [𝔱_{1}, 𝔩^{e}] in the dual of 𝔩^{e}. Moreover, for i = 1, …, r, q_{i} belongs to S(𝔩_{1}^{e}). For ξ in (𝔩_{1}^{e})^{∗}, denote by B_{ξ} the bilinear form

\begin{matrix}[𝔱, 𝔤^{e}] × [𝔱, 𝔤^{e}] & ⟶ & 𝕜 \\ (v, w) & ⟼ & ξ([v, w]) \end{matrix}

and denote by ker B_{ξ} its kernel. For i = 1, …, r, −q_{i}(ξ)^{2} is the determinant of the restriction of B_{ξ} to the subspace

(𝔤_{λ_{i}}^{e} ⊕ 𝔤_{−λ_{i}}^{e}) × (𝔤_{λ_{i}}^{e} ⊕ 𝔤_{−λ_{i}}^{e})

in the basis v_{i,1}, …,v_{i,m_{i}}, w_{i,1}, …,w_{i,m_{i}}. If (1) holds, we can find ξ in (𝔩_{1}^{e})^{∗} such that ker B_{ξ} is zero. If (2) holds, we can find ξ in (𝔩_{1}^{e})^{∗} such that ker B_{ξ} has dimension 2 since B_{ξ} is invariant under the adjoint action of 𝔱_{1}. But ker B_{ξ} is equal to (𝔤^{e})^{ξ} ∩[𝔱, 𝔤^{e}]. Hence such a ξ satisfies the required inequality and the proposition follows. □

The proof of the following proposition is given in Appendix A since it relies on explicit computations:

Proposition 5.10. (i) Suppose that either 𝔤 has type E_{7} and dim𝔤^{e} = 41 or, 𝔤 has type E_{8} and dim𝔤^{e} ∈ {112, 72}. Then, for suitable choices of 𝔱 and 𝔱_{1}, Condition (1) of Proposition 5.9 is satisfied. (ii) Suppose that 𝔤 has type E_{8} and that 𝔤^{e} has dimension 84, 76, or 46. Then, for suitable choices of 𝔱 and 𝔱_{1}, Condition (2) of Proposition 5.5 is satisfied.

5.3. Proof of Theorem 1.3. We are now in position to complete the proof of Theorem 1.3:

Proof of Theorem 1.3. We argue by induction on the dimension of 𝔤. If 𝔤 has dimension 3, the statement is known. Assume now that ind𝔩^{e′} = rk𝔩 for any reductive Lie algebras 𝔩 of dimension at most dim𝔤 − 1 and any e′ ∈ 𝒩(𝔩). Let e ∈ 𝒩(𝔤) be a nilpotent element of 𝔤. By Theorem 3.3 and Theorem 5.3, we can assume that e is rigid and that 𝔤 is simple of exceptional type. Furthermore by Corollary 5.6, we can assume that dim𝔷(𝔤^{e}) > 1. Then we consider the different cases given by Proposition 5.10. If, either 𝔤 has type E_{7} and dim𝔤^{e} = 41, or 𝔤 has type E_{8} and dim𝔤^{e} equals 112, 72, or 46, then Condition (1) of Proposition 5.9 applies for suitable choices of 𝔱 and 𝔱_{1} by Proposition 5.10. Moreover, if 𝔩 = 𝔷_{𝔤}(𝔱), then 𝔩 is a reductive Lie algebra of rank ℓ and strictly contained in 𝔤. So, from our induction hypothesis, we deduce that ind𝔤^{e} = ℓ by Proposition 5.9. If 𝔤 has type E_{8} and dim𝔤^{e} equals 84, 76, or 46, then Condition (2) of Proposition 5.9 applies for suitable choices of 𝔱 and 𝔱_{1} by Proposition 5.10. Arguing as above, we deduce that ind𝔤^{e} = ℓ. □

Appendix A. Proof of Proposition 5.10: explicit computations.

This appendix aims to prove Proposition 5.10. We prove Proposition 5.10 for each case by using explicit computations made with the help of GAP; our programmes are presented below (two cases are detailed; the other ones are similar). Explain the general approach. In our programmes, x[1], … are root vectors generating the nilradical of the Borel subalgebra 𝔟 of 𝔤 and the representative e (denoted by e in our programmes) of the rigid orbit is chosen so that e and h belong to 𝔟 and 𝔥 respectively. The element e is given by the tables of [GQT80]. In fact, in [GQT80], they use the programme Lie which induces minor changes in the numbering. Then, we exhibit suitable tori 𝔱 and 𝔱_{1} of 𝔤 contained in 𝔤^{e} which satisfies conditions (1) or (2) of Proposition 5.9. In each case, our torus 𝔱 is one dimensional; we define it by a generator, called t in our programmes. Its centralizer in 𝔤^{e} is denoted by le. The torus 𝔱_{1} has dimension at most 4. It is defined by a basis denoted by Bt1. The weights of 𝔱_{1} for the adjoint action of 𝔱_{1} on 𝔤^{e} are given by their values on the basis Bt1 of 𝔱_{1}. We list in a matrix W almost all weights which have a positive value at Bt1. The other weights have multiplicity 1. In our programmes, by the term S we check that no weight is forgotten; this term has to be zero. Then, the matrices corresponding to the weights given by W are given by a function A. Their determinants correspond to the q_{i}’s in the notations of Proposition 5.9. If there is only one other weight, the corresponding matrix is denoted by a. At last, we verify that these matrices have the desired property depending on the situations (i) or (ii) of Proposition 5.10. As examples, we detail below two cases:

(1) the case of E_{7} with dim𝔤^{e} = 41 where we intend to check that Condition (1) of Proposition 5.9 is satisfied;

(2) the case of E_{8} with dim𝔤^{e} = 84 where we intend to check that Condition (2) of Proposition 5.9 is satisfied.

(1) E_{7}, dim𝔤^{e} = 41: In this case, with our choices, dim𝔱 = 1, dim𝔩^{e} = 23 and dim𝔱_{1} = 3. The order of matrices to be considered is at most 2.

L := SimpleLieAlgebra("E",7,Rationals);;
R := RootSystem(L);;
x := PositiveRootVectors(R);; y := NegativeRootVectors(R);;
e := x[14]+x[26]+x[28]+x[49];;
c := LieCentralizer(L,Subspace(L,[e]));
Bc := BasisVectors(Basis(c));;
> <Lie algebra of dimension 41 over Rationals>
z := LieCentre(c);; Bz := BasisVectors(Basis(z));;
t := Bc[Dimension(c)];;
le := LieCentralizer(L,Subspace(L,[t,e]));
> <Lie algebra of dimension 23 over Rationals>
n := function(k)
     if k=2 then return 1;;
     elif k=-2 then return 1;;
     elif k=1 then return 8;;
     elif k=-1 then return 8;; fi;; end;;
#The function n assigns to each weight of t
#the dimension of the corresponding
#weight subspace.
M := function(k)  local m;;
   m := function(j,k)
      if j=1 then
          return Position(List([1..Dimension(c)],
              i->t*Bc[i]-k*Bc[i]),0*x[1]);;
          else
             return m(j-1,k)
                 +Position(List([m(j-1,k)+1..Dimension(c)],
          i->t*Bc[i]-k*Bc[i]),0*x[1]);;
          fi;;
     end;;
     return List([1..n(k)],i->m(i,k));;
end;;
Bt1 := [Bc[41],Bc[40],Bc[39]];;
N := function(k,p) local n;;
     n := function(j,k,p)
          if j=1 then
              return Position(List([1..8],
          i->Bt1[2]*Bc[M(k)[i]]-p*Bc[M(k)[i]]),0*x[1]);;
          else

              return n(j-1,k,p)
                   +Position(List([n(j-1,k,p)+1..8],
          i->Bt1[2]*Bc[M(k)[i]]-p*Bc[M(k)[i]]),0*x[1]);;
          fi;;
     end;;
     return List([1..4],i->M(k)[n(i,k,p)]);;
end;;
r := function(t)
     if t=1 then return 1;
     elif t=-1 then return 1;;
     elif t=0 then return 2;;
     fi;;
end;;
Q := function(k,s,t) local q;;
  q := function(j,k,s,t)
     if j=1 then
          return Position(List([1..4],
     i->Bt1[3]*Bc[N(k,s)[i]]-t*Bc[N(k,s)[i]]),0*x[1]);;
     else
          return q(j-1,k,s,t)
                  +Position(List([q(j-1,k,s,t)+1..4],
     i->Bt1[3]*Bc[N(k,s)[i]]-t*Bc[N(k,s)[i]]),0*x[1]);;
          fi;;
     end;;
     return List([1..r(t)],i->N(k,s)[q(i,k,s,t)]);;
end;;
W := [[1,1,1],[1,-1,1],[1,1,-1],[1,-1,-1],
                               [1,1,0],[1,-1,0]];;
S := 2*(1+Sum(List([1..Length(W)],
                i->Length(Q(W[i][1],W[i][2],W[i][3])))))
                    +Dimension(le)-Dimension(c);
> 0
A := function(i) return
          List([1..r(W[i][3])],t->List([1..r(W[i][3])],
     s->Bc[Q(W[i][1],W[i][2],W[i][3])[s]]*
                  Bc[Q(-W[i][1],-W[i][2],-W[i][3])[t]]));;
end;;
A(1);A(2);A(3);A(4);A(5);A(6);
> [ [ (-1)*v.63 ] ]
> [ [ v.63 ] ]
> [ [ v.63 ] ]

> [ [ (-1)*v.63 ] ]
> [ [ (-1)*v.57+(-1)*v.60, (-1)*v.63 ],
                           [ (-1)*v.63, 0*v.1 ] ]
> [ [ (-1)*v.57+(-1)*v.60, (-1)*v.63 ],
                              [ (-1)*v.63, 0*v.1 ] ]
a := Bc[M(2)[1]]*Bc[M(-2)[1]];
> v.133

In conclusion, Condition (1) of Proposition 5.9 is satisfied for 𝔱 := 𝕜t and 𝔱_{1} :=span(Bt1).

(2) E_{8}, dim𝔤^{e} = 84: In this case, with our choices, dim𝔱 = 1, dim𝔩^{e} = 48 and dim𝔱_{1} = 3. The matrix A(7) has order 5 and it is singular of rank 4. The order of the other matrices is at most 2.

L := SimpleLieAlgebra("E",8,Rationals);;
R := RootSystem(L);;
x := PositiveRootVectors(R);; y := NegativeRootVectors(R);;
e := x[54]+x[61]+x[77]+x[97];;
c := LieCentralizer(L,Subspace(L,[e]));
Bc := BasisVectors(Basis(c));;
> <Lie algebra of dimension 84 over Rationals>
z := LieCentre(c);; Bz := BasisVectors(Basis(z));;
t := Bc[Dimension(c)];;
le := LieCentralizer(L,Subspace(L,[t,e]));
> <Lie algebra of dimension 48 over Rationals>
n := function(k)
     if k=2 then return 1;;
     elif k=-2 then return 1;;
     elif k=1 then return 17;;
     elif k=-1 then return 17;;
     fi;;
end;;
M := function(k)  local m;;
   m := function(j,k)
      if j=1 then
          return Position(List([1..Dimension(c)],
        i->Bc[84]*Bc[i]-k*Bc[i]),0*x[1]);;
      else
          return m(j-1,k)
            +Position(List([m(j-1,k)+1..Dimension(c)],
          i->Bc[84]*Bc[i]-k*Bc[i]), 0*x[1]);;
          fi;;

end;;
return List([1..n(k)],i->m(i,k));;
end;;
r := function(k,t)
if k=1 and t=1  then return 4;;
elif k=-1 and t=-1  then return 4;;
elif k=1 and t=-1  then return 4;;
elif k=-1 and t=1  then return 4;;
elif k=1 and t=0  then return 9;;
elif k=-1 and t=0  then return 9;;
fi;;
end;;
Bt1 := [Bc[84],Bc[83],Bc[82]];;
N := function(k,t) local p;;
p := function(j,k,t)
 if j=1 then
    return Position(List([1..n(k)],
  i->Bt1[2]*Bc[M(k)[i]]-t*Bc[M(k)[i]]),0*x[1]);;
 else
    return p(j-1,k,t)
         +Position(List([p(j-1,k,t)+1..n(k)],
  i->Bt1[2]*Bc[M(k)[i]]-t*Bc[M(k)[i]]),0*x[1]);;
     fi;;
end;;
return List([1..r(k,t)],i->M(k)[p(i,k,t)]);;
end;;
m := function(k,s,t)
if k=1 and s=1 and t=-1 then return 2;;
elif k=-1 and s=-1 and t=1 then  return 2;;
elif k=1 and s=1 and t=0 then return 2;;
elif k=-1 and s=-1 and t=0 then return 2;;
elif k=1 and s=-1 and t=1 then  return 2;;
elif k=-1 and s=1 and t=-1 then return 2;;
elif k=1 and s=-1 and t=0 then return 2;;
elif k=-1 and s=1 and t=0 then  return 2;;
elif k=1 and s=0 and t=1 then return 2;;
elif k=-1 and s=0 and t=-1 then return 2;;
elif k=1 and s=0 and t=-1 then return 2;;
elif k=-1 and s=0 and t=1 then return 2;;
elif k=1 and s=0 and t=0 then return 5;;
elif k=-1 and s=0 and t=0 then return 5;;

     fi;;
end;;
Q := function(k,s,t) local q;;
  q := function(j,k,s,t)
    if j=1 then
        return Position(List([1..r(k,s)],
     i->Bt1[3]*Bc[N(k,s)[i]]-t*Bc[N(k,s)[i]]),0*x[1]);;
    else
        return q(j-1,k,s,t)
              +Position(List([q(j-1,k,s,t)+1..r(k,s)],
     i->Bt1[3]*Bc[N(k,s)[i]]-t*Bc[N(k,s)[i]]),0*x[1]);;
          fi;;
     end;;
     return List([1..m(k,s,t)],i->N(k,s)[q(i,k,s,t)]);;
end;;
W := [[1,1,-1],[1,1,0],[1,-1,1],[1,-1,0],
                        [1,0,1],[1,0,-1],[1,0,0]];;
S := 2 + 2*Sum(List([1..Length(W)],
               i->Length(Q(W[i][1],W[i][2],W[i][3]))))
                  + Dimension(le)-Dimension(c);;
A := function(i) return
     List([1..m(W[i][1],W[i][2],W[i][3])],
        t->List([1..m(W[i][1],W[i][2],W[i][3])],
        s->Bc[Q(W[i][1],W[i][2],W[i][3])[s]]*
               Bc[Q(-W[i][1],-W[i][2],-W[i][3])[t]]));;
end;;
# A(1), A(2), A(3), A(5), A(6) are nonsingular.
# A(7) is singular of order 5 of rank 4; its minor
List([1..4],s->List([1..4],
t->Bc[Q(W[7][1],W[7][2],W[7][3])[s]]*
              Bc[Q(-W[7][1],-W[7][2],-W[7][3])[t]]));;
# is different from 0.
a := Bc[M(2)[1]]*Bc[M(-2)[1]];;

In conclusion, Condition (2) of Proposition 5.9 is satisfied for 𝔱 := 𝕜t and 𝔱_{1} :=span(Bt1).

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Jean-Yves Charbonnel Université Paris 7 - CNRS Institut de Mathématiques de Jussieu Théorie des groupes Case 7012 Bâtiment Chevaleret 75205 Paris Cedex 13 France jyc@math.jussieu.fr Anne Moreau LMA Téléport 2 - BP 30179 Boulevard Marie et Pierre Curie 86962 Futuroscope Chasseneuil Cedex France anne.moreau@math.univpoitiers.fr 422

Documenta Mathematica 15 (2010)