June 12, 1998

On spherical nilpotent orbits and beyond

Dmitri I. Panyushev

Introduction

Let 𝔤 be a semisimple Lie algebra and G its adjoint group. The ground field 𝕜 is algebraically closed and of characteristic zero. We continue investigations started in [Pa94], which are primarily concerned with the complexity of nilpotent G-orbits (conjugacy classes) in 𝔤. Let N ⊂ 𝔤 be the nilpotent cone and {0} ≠ O ⊂ N an orbit. It was proved in [Pa94] that O is spherical (i.e., of complexity 0) if and only if ht (O) ≤ 3. The number ht (O), the height of O, can be defined as max{n ∈ ℕ | (ad e)^{n} ≠ 0, e ∈ O}. In this article we give yet another characterization of spherical nilpotent orbits in terms of minimal Levi subalgebras intersecting them, see (3.2). As an outcome, a kind of the canonical form for such orbits is proposed (3.4):

an orbit O ⊂ N is spherical if and only if it contains a representative which is a sum of root vectors corresponding to orthogonal simple roots.

Along the way, in sect. 2 and 3, we prove several auxiliary results about the height and the type of O. Minimal non-spherical orbits in simple Lie algebras are described in sect. 4. These are of complexity 1 for SL_{N} and of complexity 2 for all other simple groups. In section 5, we study the complexity of nilpotent orbits for Vinberg’s θ-groups. Recall that associated with a finite order automorphism θ of 𝔤, one has the periodic grading 𝔤 = ⨁_{j∈ℤ_{m}} 𝔤_{j} and the connected reductive group G_{0} acting on 𝔤_{1}. In this situation we are

interested in the complexity of G_{0}-orbits in N ∩ 𝔤_{1}. The main results are:

• a monotonicity result for the complexity of Ge and G_{0}e (e ∈ N ∩ 𝔤_{1}), see (5.1) ;

• a formula for the complexity of G_{0}e in terms of a bi-grading of 𝔤, see (5.4);

• in case θ is of order 2, an almost complete description of spherical orbits is found, see (5.7). However, the situation for θ-groups is not so simple, as it could be. It may happen that irreducible components of Ge ∩ 𝔤_{1}, which are G_{0}-orbits, have different complexity, see (5.10). Finally, section 6 is a collection of observations and questions related to spherical nilpotent orbits.

As usual, algebraic groups are denoted by capital Roman letters, and their Lie algebras by the corresponding small Gothik letters. For x ∈ 𝔤, we write Gx in place of (ad G)x. Acknowledgements. I would like to thank A.G. Elashvili and E.B. Vinberg for illuminating conversations about nilpotent orbits. Thanks are also due to D. Shmel’kin for questions reviving my interest to the subject. This research was carried out while I was visiting University of Poitiers and MPI (Bonn). I am grateful to both institutions for hospitality and support.

1 Recollections on nilpotent orbits and the complexity

Let 𝔤 be a semisimple Lie algebra with a fixed triangular decomposition 𝔤 = 𝔲_{−}⊕𝔱⊕𝔲_{+}, Δ the corresponding root system, and Π = {α_{1}, …, α_{p}} the set of simple roots. Let N ⊂ 𝔤 be the nilpotent cone. By the Morozov-Jacobson theorem, any nonzero element e ∈ N can be included in an 𝔰𝔩_{2}-triple {e, h, f} (i.e., [e, f] = h, [h, e] = 2e, [h, f] = −2f). The semisimple element h defines the ℤ-grading in 𝔤:

𝔤 = ⨁_{i∈ℤ} 𝔤(i) ,

where 𝔤(i) = { x ∈ 𝔤 | [h, x] = ix }. It is well known that all 𝔰𝔩_{2}-triples containing e are G_{e}-conjugate. Therefore the structure of this ℤ-grading does not depend on a particular choice of h.

Following E.B. Dynkin, we shall say that h is a characteristic of e. The orbit Gh contains a unique element h_{+} such that h_{+} ∈ 𝔱 and α(h_{+}) ≥ 0 for all α ∈ Π. The Dynkin diagram of 𝔤 equipped with the numerical labels α_{i}(h_{+}), α_{i} ∈ Π at the corresponding nodes is called the weighted Dynkin diagram of e. After Dynkin and Kostant, it is known (see e.g. [SpSt]) that

(a) α_{i}(h_{+}) ∈ {0, 1, 2};

(b) elements e, e′ ∈ N are G-conjugate if and only if their characteristics h and h′ are G-conjugate if and only if their weighted Dynkin diagrams coincide.

We shall need the following standard results on the structure of the stabilizer G_{e} ⊂ G and the centralizer 𝔤_{e} ⊂ 𝔤 (see [SpSt, ch. III]).

1.1 Proposition. (i) The Lie algebra 𝔤_{e} (resp. 𝔤_{f}) is positively (resp. negatively) graded; 𝔤_{e} = ⨁_{i≥0} 𝔤_{e}(i), where 𝔤_{e}(i) = 𝔤_{e} ∩ 𝔤(i), and likewise for 𝔤_{f}; (ii) Let L := G(0) be the connected subgroup in G corresponding to 𝔤(0) and K := L_{e}^{1}. Then K = G_{e} ∩ G_{f} and it is a maximal reductive subgroup in both G_{e} and G_{f}; (iii) For any i, there are K -stable decompositions:

𝔤(i) = 𝔤_{e}(i) ⊕ [f, 𝔤(i + 2)], 𝔤(i) = 𝔤_{f}(i) ⊕ [e, 𝔤(i − 2)].

In particular, ad f : 𝔤(i) → 𝔤(i − 2) is injective when i ≥ 1 and surjective when i ≤ 1; (iv) (ad f)^{i} : 𝔤(i) → 𝔤(−i) is one-to-one. □

The notation related to the ℤ-grading associated with a nilpotent orbit will be fixed throughout the paper.

1.2 Proposition. For i even (resp. odd), 𝔤(i) is an orthogonal (resp. symplectic) K -module. In particular, dim 𝔤(i) is even for i odd.

1 K may not be connected.

Proof. For i ≥ 0, consider the bilinear form Ψ_{i} on 𝔤(i) given by (x, y) ↦ ⟨(ad f)^{i}x, y⟩, where ⟨·, ·⟩ is a G-invariant inner product on 𝔤. By (1.1)(ii),(iv), Ψ_{i} is nondegenerate and K -invariant. It follows from G-invariance of ⟨·, ·⟩ that Ψ_{i} is symmetric for i even and alternate for i odd. □

Recall that e or Ge is called

• even whenever 𝔤(i) = 0 for i odd or, equivalently, if all α_{i}(h_{+}) ∈ {0, 2};

• distinguished, if 𝔤_{e} contains no semisimple elements, i.e., K is finite.

Let R be a reductive group and X an irreducible R-variety. Then X is called (R)- spherical if B_{R}, a Borel subgroup of R, has an open orbit in X. The complexity of X relative to R, which is denoted by c_{R}(X), is equal to the minimal codimension of B_{R}-orbits in X. Clearly, c_{R}(X) = c_{R^{o}}(X), where R^{o} stands for the identity component of R.

2 The height of a nilpotent orbit

Definition. The integer max{i | 𝔤(i) ≠ 0} is called the height of e or the orbit O = Ge and is denoted by ht (e) or ht (O).

Since e ∈ 𝔤(2), we have ht (e) ≥ 2 for any e ∈ N \ {0}. Let Λ ∈ Δ_{+} be the highest root, Λ = ∑_{i=1}^{p} n_{i}α_{i}. Clearly, we then have

ht (e) = Λ(h_{+}) = ∑_{i=1}^{p} α_{i}(h_{+})n_{i}. (2.1)

An immediate consequence of (1.1) is an intrinsic characterization of the height:

ht (e) = max{ n ∈ ℕ | (ad e)^{n} ≠ 0 } . (2.2)

For the classical Lie algebras 𝔰𝔩(V), 𝔰𝔭(V), and 𝔰𝔬(V), it is sometimes more convenient to describe nilpotent orbits by the sizes of blocks in the Jordan normal form, i.e., in terms of partitions (a_{1}, …, a_{t}), where a_{1} ≥ a_{2} ≥ … ≥ a_{t} and ∑_{i=1}^{t} a_{i} = dim V . As is well known, this correspondence is one-to-one in case of 𝔰𝔩(V). For 𝔰𝔬(V) and 𝔰𝔭(V), there is a correspondence between the nilpotent orbits and partitions satisfying a special condition. That is, in the symplectic (resp. orthogonal) case, one considers the partitions whose odd (resp. even) parts occur pairwise. This correspondence turns out to be a bijection, the only exception being that for 𝔰𝔬(V) with dim V ≡ 0 (mod 4), each partition whose all parts are even (“a very even partition”) arises from the two SO(V)- orbits, see [CM93, 5.1]. Since these two SO(V)-orbits form a single O(V)-orbit, these have the same height and complexity. In the sequel, we shall identify “classical” nilpotent orbits with corresponding partitions, keeping in mind this exception.

Let us give simple formulas for the height of nilpotent orbits in the classical Lie algebras.

2.3 Theorem. Let O = (a_{1}, …, a_{t}) be a nilpotent orbit in a classical Lie algebra 𝔤 (a_{1} ≥ a_{2} ≥ … ≥ a_{t}).

1. If 𝔤 = 𝔰𝔩(V) or 𝔰𝔭(V), then ht (O) = 2(a_{1} − 1);

2. If 𝔤 = 𝔰𝔬(V), then ht (O) = \{ \begin{matrix}a_{1} + a_{2} − 2, & if a_{2} ≥ a_{1} − 1 \\ 2a_{1} − 4, & if a_{2} ≤ a_{1} − 2 . \end{matrix}

In particular, either ht (O) is even or ht (O) ≡3 (mod 4).

Proof. Let 𝔞 ⊂ 𝔤 be a simple 3-dimensional subalgebra containing e ∈ O. Denote by R(n) a simple 𝔞-module of dimension n + 1. Considering 𝔤 as the 𝔞-module, say 𝔤 = ⨁_{i} R(n_{i}), one sees that ht (e) = max{n_{i}}. On the other hand, V = ⨁_{i=1}^{t} R(a_{i} − 1) as 𝔞-module. The relationship between V and the adjoint representation is well known:

𝔤 = \{ \begin{matrix}V ⊗V^{∗} ⊖ I1 & for 𝔰𝔩(V) \\ S^{2}V & for 𝔰𝔭(V) \\ ∧^{2}V & for 𝔰𝔬(V) . \end{matrix}

Combining these relations with the Clebsch–Gordan formula R(n) ⊗ R(m) = R(n + m) + R(n − 1) ⊗ R(m − 1) and with the decomposition of S^{2}R(n_{i}) and ∧^{2}R(n_{i}), one easily detects the biggest 𝔞-submodule in 𝔤. Whence the formulas for the height. In the orthogonal case, the constraint on pairity must be satisfied. That is, the equality a_{2} = a_{1} − 1 is only possible, if a_{1} is odd. □

Remark. The above relationship between the adjoint and the simplest representations of classical algebras has been used in [El85] for obtaining a quick classification of distinguished nilpotent elements.

The orbits with odd height, in all simple Lie algebras, are not numerous and my feeling is that these ought to have some interesting properties. The following is a simple observation for them:

2.4 Proposition. Suppose ht (e) is odd. Then the weighted Dynkin diagram of e contains no 2’s.

Proof. By (2.3), such elements e do not exist in 𝔰𝔩(V) and 𝔰𝔭(V). For 𝔰𝔬(V), we must then have a_{2} = a_{1} −1. Then a formula for the weighted Dynkin diagram (see [SpSt, 2.32] or [CM93, 5.3]) shows that α_{i}(h_{+}) ∈ {0, 1}. In the exceptional cases, one can consult Elashvili’s tables for the weighted Dynkin diagrams [El75] (see also [CM93, ch. 8]). □

The ℤ-gradings associated with 𝔰𝔩_{2}-triples form only a small part among all possible ℤgradings. Many interesting features of the former were described in [Ka80]. The following assertion concerns the same subject.

2.5 Proposition. Suppose that 𝔤(2n) = 0 for the ℤ-grading associated with a nilpotent element e. Then ht (e) ≤ 2n − 1.

Proof. It suffices to prove that 𝔤(2n + 1) = 0. Suppose not. Then 𝔤(2n − 1) ≠ 0 as well. To reach 𝔤(2n + 1) from 𝔤(2n − 1), we must have in 𝔤(2) a root vector e_{α_{i}} for some α_{i} ∈ Π. Thus, the weighted Dynkin diagram must contain a label ”2”. Then (2.4) says that ht (e) is even, which is not the case, if 𝔤(2n + 1) ≠ 0. □

A relationship between the complexity and the height of nilpotent orbits is given by the following theorem proved in [Pa94]:

2.6 Theorem. A nilpotent orbit Ge is spherical if and only if ht (e) ≤ 3. □

3 The type of a nilpotent orbit and sphericity

In this section we characterize spherical nilpotent orbits in terms of minimal Levi subalgebras intersecting them. For any e ∈ N, there exists a unique, up to conjugation, minimal Levi subalgebra 𝔷 intersecting Ge and, moreover, the orbit Ze ⊂ 𝔷′ := [𝔷, 𝔷] is distinguished. This fact is usually attributed to Bala and Carter [BC76]. Not everybody has observed that a much more general assertion, in the context of graded Lie algebras, has independently been proved by E.B. Vinberg (see [Vi75] and [Vi79]). Construction itself is quite simple. Let 𝔥 be a Cartan subalgebra in 𝔨 = 𝔤_{e}(0). Then the centralizer 𝔷 := 𝔷_{𝔤}(𝔥) is the desired Levi subalgebra. Put 𝔮 := 𝔷_{𝔤}(𝔥)′. Obviously, the elements e, h, and f lie in 𝔷 and hence in 𝔮. We also have 𝔷 = ⨁_{i∈ℤ} 𝔷(i) and 𝔮 = ⨁_{i∈ℤ} 𝔮(i).

3.1 Lemma. 1. 𝔮(i) = 𝔷(i) for i ≠ 0;

2. 𝔮(i) = 0 whenever i is odd.

Proof. 1. Since 𝔷 contains h, the centre of 𝔷 is contained in 𝔤(0) ∩ 𝔷 = 𝔷(0).

2. Obviously, 𝔮(0)_{e} = 𝔮_{e}(0) = 0, i.e., e is distinguished in 𝔮. By a result of Bala-Carter [BC76] and Vinberg [Vi79], any distinguished nilpotent element is even^{2}. □

The Cartan label of the semisimple subalgebra 𝔮 ⊂ 𝔤 is called the type of Ge. Indication of the type forms a part of the notation for the nilpotent orbits in the exceptional Lie algebras used in [BC76]^{3}. In case of two root lengths, if a simple component of 𝔮 involves only short roots, then one places tilde over its Cartan label. The type does not determine the orbit uniquely. For instance, if e is distinguished in 𝔤, then 𝔥 = {0} and 𝔮 = 𝔤. In order to distinguish different distinguished orbits, the Cartan label is accompanied by additional symbols, but we do not need this in our exposition. See [BC76] or [CM93, 8.4] for more details.

2Both proofs were case-by-case. An elegant a priori proof was found by Jantzen, see [Ka80, Note added in proof]

3Actually, this is a truncation of the data that were already used by E.B. Dynkin in [Dy52].

The notion of type applies to the classical Lie algebras as well and it is worth to write down explicit formulas for the type of a nilpotent orbit in this case. The partitions corresponding to the distinguished orbits has been pointed out in [Vi75], but the general formulas, though being known to experts, seem to be not in print.

Let O = (a_{1}, …, a_{t}) be a ‘classical’ nilpotent orbit. \underline{For SL(V)}: By the theory of Jordan normal form, we have 𝔮 = A_{a_{1}−1} + … + A_{a_{t}−1}. \underline{For Sp(V)}: If all the a_{i}’s are distinct and even, then e is distinguished and 𝔮 = C_{n}, where n = (dim V)/2. In general, each pair of equal parts (equal Jordan blocks) a_{i} = a_{i+1} gives rise to a summand Ã_{a_{i}−1}. After deleting all equal pairs, we obtain a partition with distinct even parts. This little partition determines the last summand in 𝔮, a smaller symplectic algebra. Because there are two root lengths, one has to distinguish between A_{1} and Ã_{1}. The answer is that A_{1} occurs if and only if one obtains at the very end the partition (2). That is, formally C_{1} = A_{1}. \underline{For SO(V)}: The procedure is similar. Each pair of equal Jordan blocks gives rise to a summand A_{a_{i}−1} in 𝔮. After deleting all equal pairs, we obtain a partition with distinct odd parts. This partition determines the last summand in 𝔮, a smaller orthogonal algebra. We have again to distinguish between A_{1} and Ã_{1}. The answer is that Ã_{1} occurs if and only if one obtains at the very end the partition (3). That is, formally B_{1} = Ã_{1}.

Modulo the description of distinguished orbits, the proof immediately amounts to the claim that the orbit(s) corresponding to the partition (n, n) is of type A_{n−1} in 𝔰𝔬_{2n} and of type Ã_{n−1} in 𝔰𝔭_{2n}.

Examples. 1. O = (4, 4, 4, 3, 3, 1, 1) ∈ 𝔰𝔭_{20}. Then 𝔮 = Ã_{3} + Ã_{2} + C_{2}.

2. O = (3, 3, 3, 2, 2, 2, 2) ∈ 𝔰𝔬_{17}. Then 𝔮 = A_{2} + 2A_{1} + Ã_{1}.

3.2 Theorem. Let e ∈ N \ {0}. The following conditions are equivalent: 1. ht (e) ≤ 3;

2. 𝔤(4) = 0;

3. Ge is of type rA_{1} + lÃ_{1};

4. There exist pairwise orthogonal simple roots β_{1}, …, β_{t} such that Ge contains an element of the form ∑_{i=1}^{t} e_{i}, where e_{i} ∈ 𝔤_{β_{i}} \ {0}. (Then t = r + l and there are r long and l short roots among the β_{i}’s.)

Proof. 1⇒2 – Obvious. 2⇒3. It follows from the definition of 𝔮 and (3.1) that 𝔮 = 𝔮(−2)⊕𝔮(0)⊕𝔮(2), e ∈ 𝔮(2), and h ∈ 𝔮(0). Since 𝔮_{e}(0) = {0}, we have dim 𝔮(0) = dim 𝔮(2). On the other hand, 𝔮(2) is a spherical Q(0)-module by [Pa94, 3.2]. Hence Q(0) is a torus. Clearly, a semisimple Lie algebra having such a grading is just a sum of several 3-dimensional simple algebras. 3⇔4. This follows at once from the definition of the type. 4⇒1. Consider the Levi subalgebra 𝔣 corresponding to the roots β_{1}, …, β_{t} and the corresponding connected subgroup F ⊂ G. We may assume that e = ∑_{i=1}^{t} e_{i} and 𝔞 = ⟨e, h, f⟩ is embedded diagonally in 𝔣′ ≃ (𝔰𝔩_{2})^{t}. Take the unique F -stable decomposition 𝔤 = 𝔣⊕𝔪 and consider an arbitrary irreducible F -submodule V ⊂ 𝔪. I claim that F has finitely many orbits in V . Indeed, as F is a Levi subgroup of G, there is a ℤ^{p−t}-grading of 𝔤 whose ‘zero’-part is 𝔣. (Recall that p = rk 𝔤.) The F -module V is contained in some homogeneous subspace of this polygrading (it will actually be equal to some homogeneous subspace, but we do not need this fact). Now, finiteness follows by famous Vinberg’s lemma, see [Vi76, Lemma in §2]. Let s be the number of simple factors of F acting non-trivially on V . Then V ≃ R(d_{1})⊗…⊗R(d_{s}), where all d_{i} ≥ 1. Since F has a dense orbit in V , we have dim((SL_{2})^{s}×𝕜^{∗}) ≥ dim V , i.e. 3s+1 ≥ (d_{1}+1) … (d_{s}+1) ≥ 2^{s}. This inequality has exactly three solutions:

s = 3, d_{1} = d_{2} = d_{3} = 1;
s = 2, d_{1} ≤ 2, d_{2} = 1;
s = 1, d_{1} ≤ 3.

In each case we have ∑_{i} d_{i} ≤ 3. Together with the Clebsch–Gordan formula, this shows that the biggest irreducible 𝔞-module that can occur in V is R(3). Since 𝔣|_{𝔞} ≃ tR(2) + (p − t)R(0), the same is true for the whole Lie algebra 𝔤. But, this means precisely that ht (e) ≤ 3. □

Equivalence of conditions (1) and (2) is a particular case of Prop. 2.5. But the last proof does not appeal to case-by-case considerations as in (2.4).

3.3 Corollary (of the proof). For any μ ∈ Δ, let {α_{i_{1}}, …, α_{i_{s}}} ⊂ Π be a set of pairwise orthogonal roots such that ⟨α_{i_{j}}, μ⟩ ≠ 0, j = 1, …, s. Then s ≤ 3 (even s ≤ 2, if an addition μ is long and one of the α_{i_{j}} ’s is short).

Proof. Replacing μ by a suitable root μ′ = μ+∑_{j} n_{j}α_{i_{j}}, one can achieve that ⟨α_{i_{j}}, μ′⟩ > 0 for all j. Then μ′ is the highest weight of an irreducible (SL_{2})^{s}-submodule V of 𝔤. Since all copies of SL_{2} act non-trivially on V , the proof of 4⇒1 shows that s ≤ 3 (even s ≤ 2, if d_{1} ≥ 2). □

Remarks. 1. In case μ ∈ Π, we obtain the well-known assertion that the number of other simple roots that are not orthogonal to μ is at most 3. That is, we have given an invariant-theoretic proof of it.

2. If we drop the assumption that the α_{i_{j}} ’s are pairwise orthogonal, then it is easy to give an example with s = 5 (at least). Combining (2.6) and (3.2), we obtain a kind of normal form for spherical nilpotent orbits:

3.4 Theorem. Suppose e ∈ N. Then the orbit Ge is spherical if and only if it contains an element of the form ∑_{i=1}^{t} e_{i}, where e_{i} ∈ 𝔤_{β_{i}} and β_{1}, …, β_{t} are pairwise orthogonal simple roots. □

Examples. 1. The spherical nilpotent orbits in 𝔰𝔭_{2p} are O_{i} = (2^{i}, 1^{2p−2i})^{4} (0 ≤ i ≤ p).

Then the type of O_{i} is \{ \begin{matrix}lÃ_{1}, & if i = 2l, \\ lÃ_{1} + A_{1}, & if i = 2l + 1. \end{matrix}

In particular for i = p = 2l + 1, the respective set of simple roots is α_{1}, α_{3}, …, α_{2l+1}. (The last of them is long.)

2. The type of spherical orbit (3, 2^{2}, 1^{l}) in 𝔤 = 𝔰𝔬(V) is \{ \begin{matrix}3A_{1}, & if l is odd , \\ A_{1} + Ã_{1}, & if l is even . \end{matrix}

4 Orbits of small complexity

Recall, with some variations, a formula for the complexity of Ge. Keep the notation of sect. 1. Let S be a stabilizer in general position (= s.g.p.) for the K -action on 𝔤(2). We shall use the notation S = s.g.p.(K, 𝔤(2)) as a shorthand. The reader is referred to [VP89, §7] for the basic facts on s.g.p. As 𝔤(2) is an orthogonal K -module, a result of D. Luna asserts that S is reductive. Set 𝔤(≥j) = ⊕_{i≥j}𝔤(i).

4.1 Theorem. 1. c_{G}(Ge) = c_{L}(𝔤(2)) + c_{S}(𝔤(≥3)); 2. c_{G}(Ge) = c_{L}(𝔤(≥2)).

Proof. 1. Since Le ≃ L/K is the dense orbit in 𝔤(2), we have c_{L}(𝔤(2)) = c_{L}(L/K). Hence (1) is nothing but the first formula in [Pa94, 2.3].

2. By definition, c_{L}(𝔤(≥ 2)) = dim 𝔤(≥ 2) − dim B_{L} + dim(B_{L})_{∗}, where B_{L} is a Borel subgroup of L and (B_{L})_{∗} := s.g.p.(B_{L}, 𝔤(2)). To compute (B_{L})_{∗}, we use the decomposition 𝔤(≥2) = 𝔤(2) ⊕ 𝔤(≥3). Namely, if B_{2} := s.g.p.(B_{L}, 𝔤(2)), then (B_{L})_{∗} = s.g.p.(B_{2}, 𝔤(≥3)). Whence

c_{L}(𝔤(≥2)) = c_{L}(𝔤(2)) + min_{v∈𝔤(≥3)} codim_{𝔤(≥3)}B_{2}v .

I claim that (B_{2})^{o} = B_{S}^{5}. Granting this, we see that the last term in the above formula is just c_{S}(𝔤(≥ 3)), as required. It remains to demonstrate the claim. By [Pa87] (see also [Pa90, sect. 1]), (B_{2})^{o} is equal to a Borel subgroup of an s.g.p.(L, 𝔤(2) ⊕ 𝔤(2)^{∗}). To compute the latter, we use the same trick as for (B_{L})_{∗}:

s.g.p.(L, 𝔤(2) ⊕ 𝔤(2)^{∗}) = s.g.p.(K, 𝔤(2)^{∗}) = s.g.p.(K, 𝔤(2)) = S .

The first equality follows because L/K is the dense orbit in 𝔤(2). □

Remark. The meaning of these formulas is that computing of the complexity of Ge is reduced to that for a representation space. In case of representations, there is an explicit algorithm for doing this [Pa87]. Actually, given a representation R → GL(V) of a reductive group R, the algorithm says how to find s.g.p.(R, V ⊕ V^{∗}) =: R_{∗}. The group R_{∗} is reductive and has some other nice properties. Then c_{R}(V) = dim V − dim B_{R} + dim B_{R_{∗}}.

4As usual in the theory of partitions, a^{j} := a, …, a (j times).

5Because s.g.p.’s are defined up to conjugacy in the ambient group, this means that one obtains the equality under an appropriate choice of all the groups involved.

4.2 Proposition. If dim 𝔤(4) ≥ 2 or ht (e) ≥ 5, then c_{G}(Ge) ≥ 2.

Proof. Actually, it will be shown that c_{S}(𝔤(≥3)) ≥ 2. Let us detect two algebraically independent B_{S} -invariant rational functions on 𝔤(≥3). It follows from (3.2) that in both cases 𝔤(4) ≠ 0. Since ad f : 𝔤(4) → 𝔤(2) is injective, there is a K -module W_{1} such that 𝔤(2) ≃ 𝔤(4) ⊕ W_{1}. Because S = s.g.p.(K, 𝔤(2)), we have K(𝔤(2)^{S}) is dense in 𝔤(2). It follows that 𝔤(4)^{S} ≠ 0. Write 𝔤(4) = 𝔤(4)^{S} ⊕ W_{2}, where W_{2} is a K -module. Since 𝔤(4) is an orthogonal K -module (1.2), the same holds for W_{2}. Hence in case dim 𝔤(4) ≥ 2 we have at least two algebraically independent functions in 𝕜[𝔤(4)]^{S}. If dim 𝔤(4) = 1, then we examine the following possible cases:

(a) Let 𝔤(6) ≠ 0. Then repeating the previous argument with (ad f)^{2} in place of ad f, we obtain 𝔤(6)^{S} ≠ 0.

(b) Let 𝔤(5) ≠ 0. Then 𝔤(5) and [f, 𝔤(5)] ⊂ 𝔤(3) are two different isomorphic S - submodules in 𝔤(≥3). Obviously, this produces a non-constant B_{S} -invariant rational function on 𝔤(≥3). □

Remark. Similar arguments prove that if ht (e) ≥ 2n+1 (n ≥ 2), then c_{G}(Ge) ≥ n. But I think there ought to exist a quadratic polynomial n ↦ ϕ(n) such that c_{G}(Ge) ≥ ϕ(n). By (2.6) and (4.2), the nilpotent orbits of complexity 1 are contained among those having height 4. The latter exist in all simple Lie algebras. However, routine computations lead to the following conclusion:

4.3 Theorem. Nilpotent orbits of complexity 1 exist only for G = SL_{n}, n ≥ 3. Moreover, such an orbit is unique. The corresponding weighted Dynkin diagram is 2–0-… -0–2 and the partition is (3, 1^{n−3}). For all other simple groups, the minimal non-spherical orbits are of complexity 2.

Proof. For the classical groups, classification of the minimal non-spherical (= m.n.s.) orbits follows from (2.3), (2.6), and an explicit description, due to Gerstenhaber and Hesselink, of the closure ordering, see [CM93, 6.2]. For the exceptional groupes, one uses (3.2)(3) and the Hasse diagrams for the closure ordering, see [Spal, IV.2].

1. For SL_{n}, there is a unique m.n.s. orbit O = (3, 1^{n−3}). It is even and of height 4. For n = 3, O is the regular nilpotent orbit and the assertion follows by counting dimensions. Let n > 3. Then L = SL_{n−2} × (𝕜^{∗})^{2} and the L-modules 𝔤(2) and 𝔤(4) are as follows: 𝔤(2) = R(φ_{1}) ⊗ ε + R(φ_{1})^{∗} ⊗ μ, 𝔤(4) = R(0) ⊗ εμ. Here R(λ) stands for the irreducible representation of the semisimple part with highest weight λ, while ε and μ are basic characters of the central torus; dim 𝔤(2) = 2n − 4 and dim 𝔤(4) = 1. Obviously, 𝔤(2) is a spherical L-module. It is not hard to compute that K ≃ SL_{n−3} × 𝕜^{∗}, S ≃ SL_{n−4} × 𝕜^{∗}, and S acts trivially on 𝔤(4). Applying then (4.1), we get c_{G}(O) = 1.

The minimal nilpotent orbit lying ‘over’ O is

O′ = \{ \begin{matrix}(4), & if n = 4 \\ (3, 2, 1^{n−5}), & if n ≥ 5 \end{matrix} . Then it already turns out that c_{G}(O′) = 2.

2. There are two m.n.s. orbits for Sp_{2n} (n ≥ 3) and F_{4}, and one orbit in the other cases. More precisely, all groups of rank ≥ 3 have a unique orbit of type A_{2} (of type Ã_{2} for Sp_{2n}), which is m.n.s. Besides, F_{4} has the m.n.s. orbit of type Ã_{2}, and Sp_{2n} (n ≥ 2) has the m.n.s. orbit of type C_{4}, which corresponds to the partition (4, 1^{2n−4}). Finally, the m.n.s. orbit for G_{2} is 10-dimensional (and distinguished). All these orbits are of complexity 2. For instance, consider the orbit of type Ã_{2} for G = F_{4}. Here L = Spin_{7} × 𝕜^{∗}, 𝔤(2) = R(φ_{3})⊗ε, and 𝔤(4) = R(φ_{1})⊗ε^{2}; dim 𝔤(2) = 8, dim 𝔤(4) = 7. This information is easily being extracted from the weighted Dynkin diagram given in the Table 1 in sect. 5. Then K ≃ G_{2} and c_{L}(L/K) = 0. Next, 𝔤(2) affords the sum of the simplest (7-dimensional) and the trivial 1-dimensional representation of G_{2}. Therefore S ≃ SL_{3}. It is not hard to compute that 𝔤(4)|_{SL_{3}} = R(φ′_{1}) + R(φ′_{2}) + R(0). Whence c_{S}(𝔤(4)) = 2. □

This result confirms a claim in [Vi86, n. 9] concerning orbits of complexity 1 in the universe that ”it appears there should be few of them”.

5 The complexity of nilpotent orbits of θ-groups

Let θ be an automorphism of 𝔤, of finite order m. Fix a primitive m-th root of unity ζ. Consider the periodic grading

𝔤 = ⨁_{j∈ℤ_{m}} 𝔤_{j} ,

where 𝔤_{j} is the θ-eigenspace of 𝔤 corresponding to ζ^{j}. Following Vinberg, we shall say that the connected reductive group G_{0} acting linearly on 𝔤_{1} is a θ-group. The main references on θ-groups are [Vi75], [Vi76], [Vi79]. One of the basic results is that \overline{G_{0}e} ∋ 0 (e ∈ 𝔤_{1}) if and only if e ∈ N. Such G_{0}-orbits are called nilpotent, too. Our aim is to study the complexity of them. Throughout this section, it is assumed that e ∈ N ∩ 𝔤_{1}. The first result is:

5.1 Theorem. c_{G}(Ge) ≥ c_{G_{0}}(G_{0}e).

To demonstrate the theorem, we need the following variation on Vinberg’s themes:

5.2 Lemma. Let G → GL(V) be a representation of a reductive group, H ⊂ G a reductive subgroup, and W ⊂ V an H -stable subspace. Choose Borel subgroups in H and G such that B_{H} ⊂ B_{G}. Suppose that

(∗)  𝔟_{H}v = 𝔟_{G}v ∩ W  for all v ∈ W.

For any G-stable locally-closed subvariety X ⊂ V and each irreducible component Y of X ∩ W, we then have

c_{G}(X) ≥ c_{H}(Y) .

Proof. By [Vi86], the complexity c_{G}(X) is equal to the modality of B_{G}-action on X, i.e., to max_{X′⊂X} trdeg k(X′)^{B_{G}}, where X′ runs through the irreducible B_{G}-stable subvarieties of X. On the other side, it follows from (∗) and [Vi76, §2] that B_{G}v ∩ W is a union of finitely many B_{H} -orbits for all v ∈ W, each B_{H} -orbit being an irreducible component of B_{G}v ∩ W. Therefore, if Y′ ⊂ Y is B_{H} -stable and irreducible, then trdeg k(Y′)^{B_{H}} = trdeg k(\overline{B_{G}·Y′})^{B_{G}}. □

Proof of (5.1). The lemma applies to V = 𝔤, H = G_{0}, and W = 𝔤_{1}. The condition (∗) follows from presence of periodic grading. As (∗) holds also for 𝔤_{0} and 𝔤 in place of 𝔟_{H} and 𝔟_{G}, Vinberg’s lemma [Vi76, §2] implies that each irreducible component of Ge ∩ 𝔤_{1} is a G_{0}-orbit. In particular, one of the components is G_{0}e. □

It follows that, given a spherical orbit Ge, each irreducible component of Ge ∩ 𝔤_{1} is a spherical G_{0}-orbit, too. But a naïve hope for the converse fails to be true. For, all simple Lie algebras have a periodic grading such that G_{0} is a torus. Then all G_{0}-orbits in 𝔤_{1} are spherical, while this is not always the case for the G-orbits in G𝔤_{1}. To develop a technique for dealing with the complexity of G_{0}-orbits and, in particular, for classifying the spherical ones, we need some preparations. By a modification of the Morozov-Jacobson theorem, one may assume that {e, h, f} is adapted to θ, i.e., h ∈ 𝔤_{0} and f ∈ 𝔤_{−1}. Then 𝔤 gains a ℤ × ℤ_{m}-grading

𝔤 = ⨁_{i∈ℤ} ⨁_{j∈ℤ_{m}} 𝔤(i)_{j} .

We have e ∈ 𝔤(2)_{1}, h ∈ 𝔤(0)_{0}, and f ∈ 𝔤(−2)_{−1}. By [Vi79, Th. 1], all adapted triples containing e are (G_{0})_{e}-conjugate. Therefore the structure of this bi-grading does not depend on choice of an adapted triple. One may say that G_{0}e determines a refinement 𝔤(i) = ⨁_{j∈ℤ_{m}} 𝔤(i)_{j} of the ℤ-grading associated with Ge. However, it is worth noting that another irreducible component of Ge ∩ 𝔤_{1} may and usually does determine another refinement of the same ℤ-grading. We defer discussing of this and related problems until (5.9). Denote by L_{0} and K_{0} the identity components of the θ-fixed subgroups in L and K respectively. Then 𝔩_{0} = 𝔤(0)_{0} and it follows from (1.1) that [𝔩_{0}, e] = 𝔤(2)_{1} and (𝔩_{0})_{e} = 𝔨_{0}. Clearly, each 𝔤(i)_{j} is a K_{0}-module and ⟨𝔤(i)_{j}, 𝔤(i′)_{j′}⟩ = 0 unless i+i′ = 0 and j +j′ = 0. An extension of (1.2) to the bi-grading is given by

5.3 Proposition. For all i, j ∈ ℤ we have

1. 𝔤(2i)_{i} and 𝔤(2i)_{j} ⊕ 𝔤(2i)_{2i−j} are orthogonal K_{0}-modules;

2. 𝔤(2i + 1)_{j} and 𝔤(2i + 1)_{2i+1−j} are dual K_{0}-modules. (The subscripts are being considered as elements of ℤ/mℤ.)

Proof. Let i ≥ 0. Since (ad f)^{2i} : 𝔤(2i)_{i} → 𝔤(−2i)_{−i} is bijective, the restriction of Ψ_{i} (see (1.2)) to 𝔤(2i)_{i} is non-degenerate, and so on … □

Denote by M an s.g.p. for the K_{0}-action on 𝔤(2)_{1}. Again, M is reductive by Luna’s result, since 𝔤(2)_{1} is orthogonal. Note that there is no relation between M and S = s.g.p.(K, 𝔤(2)).

5.4 Theorem. 1. c_{G_{0}}(G_{0}e) = c_{L_{0}}(𝔤(2)_{1}) + c_{M}(𝔤(≥3)_{1}); 2. c_{G_{0}}(G_{0}e) = c_{L_{0}}(𝔤(≥2)_{1}).

Proof. 1. As well as Theorem 4.1(1), it will be a consequence of [Pa94, 1.2]. Namely, to write a formula for the complexity of G_{0}e ≃ G_{0}/(G_{0})_{e}, we use an embedding of (G_{0})_{e} into some parabolic subgroup in G_{0}. Set 𝔭 = ⨁_{i≥0} 𝔤(i). It is a parabolic subalgebra in 𝔤 and 𝔩 = 𝔤(0) is a Levi subalgebra in it. Obviously, 𝔭_{0} = ⨁_{i≥0} 𝔤(i)_{0} is parabolic in 𝔤_{0} and 𝔩_{0} is a Levi subalgebra in it. Denote by A^{u} the unipotent radical of an algebraic group A. By (1.1), we have G_{e} ⊂ P and (G_{e})^{u} ⊂ P^{u}. Write N for (G_{e})^{u} ∩ G_{0}. Then the identity component of (G_{0})_{e} is K_{0}N and N = (K_{0}N)^{u}. Clearly, K_{0}N ⊂ P_{0} and N ⊂ (P_{0})^{u}. That is, we have the right embedding in terminology of [Pa94]. Since the component group of the stabilizer does not affect the complexity, we may apply [Pa94, 1.2] to conclude

c_{G_{0}}(G_{0}e) = c_{G_{0}}(G_{0}/K_{0}N) = c_{L_{0}}(L_{0}/K_{0}) + c_{M}((𝔭_{0})^{u}/𝔫) .

Since K_{0} is the identity component of (L_{0})_{e} and L_{0}e is dense in 𝔤(2)_{1}, we have c_{L_{0}}(L_{0}/K_{0}) = c_{L_{0}}(𝔤(2)_{1}). It follows from (1.1)(iii) that

𝔭^{u} = 𝔤(≥1) = (𝔤_{e})^{u} ⊕ ad f·𝔤(≥3) .

Whence

(𝔭_{0})^{u} = (𝔭^{u})_{0} = (𝔤_{e})_{0}^{u} ⊕ ad f·𝔤(≥3)_{1} = 𝔫 ⊕ ad f·𝔤(≥3)_{1} .

Thus, (𝔭_{0})^{u}/𝔫 is isomorphic to 𝔤(≥3)_{1} as K_{0}- and hence M -module.

2. The proof is analogous that in (4.1)(2). □

Remarks. 1. All the previous results hold without changes if 𝔤 = ⊕_{j}𝔤_{j} is a ℤ-grading, i.e., formally m = ∞.

2. The paper [Pa94] has dealt not only with the complexity, but also with the rank of nilpotent orbits. Although the notions are quite different, the formulas for the both and the proofs turned out to be the same. This also holds in case of θ-groups. For instance, making use of the above embedding K_{0}N ⊂ P_{0} and Theorem 1.2 in [loc cit.], one proves the formula for the rank of G_{0}e: r_{G_{0}}(G_{0}e) = r_{L_{0}}(𝔤(2)_{1}) + r_{M}(𝔤(≥3)_{1}). Although it is already possible to give some estimates for c_{G_{0}}(G_{0}e), these are isolated and do not allow us to achieve attractive results (yet?). For this reason, we restrict ourselves to the case where θ is involutory. That is, from now on m = 2 and we intend to describe spherical nilpotent orbits for the isotropy representation of a symmetric variety G/G_{0}. Now, an adapted 𝔰𝔩_{2}-triple gives rise to the decomposition of K_{0}-modules 𝔤(i) = 𝔤(i)_{0} ⊕ 𝔤(i)_{1} (i ∈ ℤ) and as an immediate corollary of (5.3) we have

5.5 Lemma.

(i) For i odd, 𝔤(i)_{0} and 𝔤(i)_{1} are dual K_{0}-modules; (ii) For i even, 𝔤(i)_{0} and 𝔤(i)_{1} are orthogonal K_{0}-modules. □

Our idea is to characterize spherical G_{0}-orbits in terms of the G-orbits in 𝔤 these generate. In view of (5.1), one has to realize which nonspherical G-orbits may arise in this way. To this end, our main tool is Theorem 5.4.

5.6 Theorem. Suppose θ is involutory and e ∈ 𝔤_{1} ∩ N. If ht (e) ≥ 5 or 𝔤(4)_{1} ≠ 0, then c_{G_{0}}(G_{0}e) > 0.

Proof. Actually, we shall prove that 𝕜[𝔤(≥ 3)_{1}]^{M} ≠ 𝕜, which certainly implies that c_{M}(𝔤(≥3)_{1}) > 0, cf. (4.2). The situation splits into 3 cases.

(a) Assume that 𝔤(5) ≠ 0. By (5.5), 𝔤(5)_{1} and 𝔤(5)_{0} are dual K_{0}-modules and hence dual M -modules. Then (ad f)𝔤(5)_{0} and 𝔤(5)_{1} are dual M -modules in 𝔤(≥3)_{1}.

(b) Assume that 𝔤(4)_{1} ≠ 0. By (5.5), it is an orthogonal K_{0}-module and hence M -module. Thus, 𝕜[𝔤(≥3)_{1}]^{M} ≠ 𝕜.

(c) If ht (e) ≥ 5, 𝔤(5) = 0, and 𝔤(4) ⊂ 𝔤_{0}, then 0 ≠ (ad e)𝔤(4) = 𝔤(6) = 𝔤(6)_{1}. Consider (ad f)^{2}𝔤(6)_{1} = (ad f)𝔤(4) ⊂ 𝔤(2)_{1}. It is a K_{0}-submodule. Since M = s.g.p.(K_{0}, 𝔤(2)_{1}), we have K_{0}(𝔤(2)_{1}^{M}) is dense in 𝔤(2)_{1}. Whence ((ad f)^{2}𝔤(6)_{1})^{M} ≠ 0 and, finally, 𝔤(6)_{1}^{M} ≠ 0. Thus, 𝕜[𝔤(≥3)_{1}]^{M} contains a linear function. □

From (5.1) and (5.6) it follows that there is a coherent description for all G-orbits except for those of height 4:

(5.7) If θ is involutory and O is a nilpotent orbit such that O ∩ 𝔤_{1} ≠ ∅, then for ht (O) ≤ 3 each irreducible component of O ∩ 𝔤_{1} is G_{0}-spherical, while for ht (O) ≥ 5 none of the irreducible components is G_{0}-spherical. This holds regardless of θ provided only that O ∩ 𝔤_{1} ≠ ∅.

For the G-orbits of height 4, the answer already depends on relationship between θ and O, i.e., the complexity of the irreducible components of O ∩ 𝔤_{1} may or may not be equal to zero, depending on θ. A necessary condition of sphericity is that 𝔤(4) ⊂ 𝔤_{0}, but it is not sufficient. I am going to consider this problem in a forthcoming paper. Note that, given e and θ, it is easy to realize when Ge ∩ 𝔤_{1} ≠ ∅. By a result of L. Antonyan [An82, Th. 1], we have Ge ∩ 𝔤_{1} ≠ ∅ ⟺ Gh ∩ 𝔤_{1} ≠ ∅, the latter being immediately seen from the weighted Dynkin diagram of e and the Satake diagram of θ.

5.8 Example. Using (5.4), one can give a recipe for producing triples (𝔤, θ, e) such that e ∈ 𝔤_{1}, c_{G}(Ge) > 0, and c_{G_{0}}(G_{0}e) = 0. Let e be an even nilpotent element with ht (e) = 4. Then c_{G}(Ge) > 0. Define a ℤ_{2}-grading of 𝔤 by the formulas 𝔤_{0} = 𝔤(−4) ⊕ 𝔤(0) ⊕ 𝔤(4), 𝔤_{1} = 𝔤(−2) ⊕ 𝔤(2). Then L_{0} = L, 𝔤(2) = 𝔤(2)_{1}, and 𝔤(≥3)_{1} = 0. Therefore c_{G_{0}}(G_{0}e) = c_{L}(𝔤(2)). All such orbits with spherical L-module 𝔤(2) are listed in table 1. Content-Length: 16782 X-Lines: 353

Table 1:

𝔤  characteristic type of O dim 𝔤(2)  𝔤_{0}
𝔰𝔩_{n} (n ≥ 3) 2–0–… –0–2  A_{2}  2n − 4 𝔰𝔩_{n−2} ⊕ 𝔰𝔩_{2} ⊕ 𝕜
𝔰𝔩_{6}  0–2–0–2–0  2A_{2}  8  𝔰𝔩_{4} ⊕ 𝔰𝔩_{2} ⊕ 𝕜
𝔰𝔭_{2n} (n ≥ 3) 0–2–0-… ⇐0  A_{2}  4n − 8 𝔰𝔭_{2n−4} ⊕ 𝔰𝔭_{4}
𝔣_{4}  2–0⇐0–0  Ã_{2}  8  𝔰𝔬_{9}
𝔢_{6}  2–0–00–0–2  2A_{2}  16  𝔰𝔬_{10} ⊕ 𝕜

5.9 On behaviour of irreducible components. In general, different irreducible components of O ∩ 𝔤_{1} determine non-isomorphic bi-gradings, e.g. the groups L_{0} can be different, and so on. One may address the following questions in this regard:

Is it possible that these components have different complexity relative to G_{0}?

Is it possible that spherical and non-spherical components occur together? The answer to the first question is “yes” and we present below an example of triple (𝔤, θ, e) such that the complexity of irreducible components of O ∩ 𝔤_{1} takes three values. As for the second question, such situation might only occur when ht (O) = 4. But my calculations support the negative answer. I have the following partial result based on the classification of orbits and explicit calculations: Suppose that either 𝔤 is exceptional or 𝔤 = 𝔰𝔩_{N} and θ is inner. If ht (O) = 4 and O ∩ 𝔤_{1} has a spherical irreducible component, then all the components are spherical. So, I believe the second answer ought to be “no” and it would be desirable to find a classification-free proof.

5.10 Example. Let 𝔤 = 𝔰𝔩_{N}. Denote by θ_{n} an inner involution of 𝔤 such that 𝔤_{0} ≃ 𝔰𝔩_{n} ⊕ 𝔰𝔩_{N−n} ⊕ 𝕜. To emphasize the dependence on n, we shall write G_{0}^{(n)} and 𝔤_{1}^{(n)}. The elements of 𝔤_{1}^{(n)} can be thought of as pairs of counter operators (A, B), where A ∈ Hom (V, W), B ∈ Hom (W, V), dim V = n, and dim W = N − n. The orbit classification for pairs of counter operators was first obtained in [DP65]. (From the modern point of view, this is a special case of the quiver theory.) We explain this classification using the language of ab-diagrams introduced in [KP79, sect. 4]. Given O = (a_{1}, …, a_{t}), a corresponding ab-diagram is obtained if one writes a string of consecutive symbols “a” and “b”, of length a_{i}, in place of part “a_{i}”. Two ab-diagrams are proclaimed to be equivalent if these are obtained form each other by reordering ab-strings of equal length. There is a bijection between the irreducible components of O ∩ 𝔤_{1}^{(n)} and the classes of equivalent ab-diagrams such that the total number of a’s is n.

For N ≥ 6, consider O = (3, 3, 1^{N−6}). Then O ∩ 𝔤_{1}^{(n)} ≠ ∅ if and only if 2 ≤ n ≤ N − 2. From the point of view of complexity, essentially different irreducible components correspond to the following ab-diagrams:

I : (bab, bab, …); II : (aba, aba, …); III : (aba, bab, …). The strings of length 1 are uniquely determined by the requirement that the total number of the symbol “a” is as prescribed. Note that case I (resp. II) occurs if and only if N − n ≥ 4 (resp. n ≥ 4) and case III occurs if and only if 3 ≤ n ≤ N − 3. Making use of these ab-diagrams, one can write explicitly an 𝔰𝔩_{2}-triple adapted to θ_{n}. An explicit matrix form of the latter allows us to determine the decompositions 𝔤(i)_{0} ⊕ 𝔤(i)_{1} and then to compute the complexity. The answer is that the complexity relative to G_{0}^{(n)} of the irreducible components of O ∩ 𝔤_{1}^{(n)} is equal to: n − 2 in case I; N − n − 2 in case II; 1 in case III. For instance, if N = 9 and n = 4 then there are irreducible components of complexity 1, 2, and 3. Since c_{G}(O) = N − 2, one also obtains a nice illustration to Theorem 5.1.

6 Questions, observations, remarks

6.1 Irreducibility. A direct verification shows that N^{sph}, the union of all nilpotent spherical G-orbits, is irreducible. The explicit expression for the maximal spherical orbit in the classical case is derived from [Pa94, sect. 4]; in the exceptional case, look at the tables in [Spal, IV.2]. It might be interesting to give an a priori proof and to describe the ideal of this variety. However, the analogous property does not hold for arbitrary θ-groups. For instance, (N ∩ 𝔤_{1})^{sph} has 2 irreducible components for an involution of SL_{2}. A more exotic example is an involution of F_{4} of maximal rank. Making use of the known normality results (due to Broer and Kraft & Procesi), one sees that N^{sph} is normal for G = A_{p}, B_{2p}, C_{p}, D_{p}, F_{4}; and not normal for B_{2p+1} and G_{2}. It is likely that N^{sph} is normal for E_{p} and there is a unified proof for A-D-E.

6.2 Dimension. By definition, dim O ≤ dim B_{G} for O ⊂ N^{sph}. Our aim here is a bit more exact inequality. Without loss of generality, we may assume that 𝔤 is simple. Denote by k_{0} (resp. k_{1}) the number of even (resp. odd) exponents of 𝔤. Then k_{0} + k_{1} = rk 𝔤 =: p.

Proposition. If O ⊂ N^{sph}, then dim O ≤ dim B_{G} − k_{0}. Proof. This is easily verified case-by-case, but we give a conceptual proof. Let 𝔤 = 𝔤_{0}^{max} ⊕ 𝔤_{1}^{max} be the decomposition corresponding to an involution of maximal rank θ^{max}. This means 𝔤_{1}^{max} contains a Cartan subalgebra (= C.s.a.) of 𝔤. By [An82, th. 2], we have O ∩ 𝔤_{1}^{max} ≠ ∅. By [KR71, prop. 5], the equality dim(O ∩ 𝔤_{1}) = \frac{1}{2} dim O holds for any involution (provided that O ∩ 𝔤_{1} ≠ ∅). As each irreducible component of O ∩ 𝔤_{1}^{max} is G_{0}^{max}-spherical (5.1), we have

\frac{1}{2} dim O = dim(O ∩ 𝔤_{1}^{max}) ≤ dim B_{0}^{max} ,

where B_{0}^{max} is a Borel subgroup of G_{0}^{max}. Thus, it remains to demonstrate that dim B_{0}^{max} = \frac{1}{2}(dim B_{G} − k_{0}). Since dim G_{0}^{max} = dim B_{G} − p, it amounts to proving that rk G_{0}^{max} = k_{1}, see next lemma. □

Lemma. rk G_{0}^{max} = k_{1}. Proof. Let 𝔥 be a θ^{max}-stable C.s.a. of 𝔤 such that 𝔥_{0} is a C.s.a. of 𝔤_{0}^{max}. Since the Gorbit of a characteristic of a regular nilpotent element intersects 𝔤_{0}^{max} (consider an adapted 𝔰𝔩_{2}-triple!), we see that 𝔥_{0} contains regular semisimple elements. Set σ := θ^{max} |_{𝔥}. By [Sp74, 6.5], the eigenvalues of σ are ε_{i}^{−1} (1 ≤ i ≤ p), where the ε_{i}’s are the “eigenvalues of θ^{max} on the set of basic invariants”. The latter means that homogeneous generators F_{1}, …, F_{p} of the polynomial algebra 𝕜[𝔤]^{G} can be chosen so that θ^{max}·F_{i} = ε_{i}F_{i}. (Of course, ε_{i} ∈ {1, −1}.) On the other side, there is a C.s.a. 𝔥′ ⊂ 𝔤_{1}^{max}, i.e., σ′ := θ^{max} |_{𝔥′}= −id. By [Sp74, 6.4(v)], the eigenvalues of σ′ are ε_{i}^{−1}(−1)^{m_{i}} (1 ≤ i ≤ p), where m_{1}, …, m_{p} are the exponents of 𝔤. Whence ε_{i} = 1 if m_{i} is odd and ε_{i} = −1 if m_{i} is even. Finally, rk G_{0}^{max} = dim 𝔥_{0} = #{i | ε_{i} = 1} = k_{0}. □

We have proven that dim N^{sph} ≤ dim B_{G} − k_{0}. Actually, the equality holds, but I do not know a unified proof. A promising approach is discussed in the next subsection. There is another curious coincidence related to another involution. By [An82, Th.3(2)], there is a unique, up to G-conjugation, inner involution θ^{int} of 𝔤 such that all orbits in N have non empty intersection with 𝔤_{1}^{int}. (θ^{int} ≠ θ^{max} for A_{p}, D_{2p+1}, and E_{6}.) One can prove that dim 𝔤_{0}^{int} − dim 𝔤_{1}^{int} = k_{0} − k_{1}. Since rk G_{0}^{int} = rk G, this implies dim 𝔤_{1}^{int} = dim B_{G} − k_{0}. That is, dim N^{sph} = dim 𝔤_{1}^{int}.

6.3 A relationship with the index of B_{G}. The index of an algebraic group (or its Lie algebra) is the minimal codimension of its orbits in the coadjoint representation. The index ind B_{G} of B_{G} for all simple groups was computed in [Tr83, §4]. This result can be stated as: ind B_{G} = k_{0}. It seems however that no explanation of this equality is known. Our observation is:

Proposition. If O ⊂ N^{sph}, then dim O ≤ dim B_{G} − ind B_{G}. Proof. By the very definition of sphericity, there is x ∈ O such that 𝔤_{x} + 𝔟_{G} = 𝔤. Taking the orthocomplements, we obtain

(♡) [𝔤, x] ∩ [𝔟_{G}, 𝔟_{G}] = {0} . Let \overline{x} be the image of x in 𝔤/[𝔟_{G}, 𝔟_{G}]. The latter is identified with the B_{G}-module (𝔟_{G})^{∗}. Since x ∈ [𝔤, x], we have \overline{x} ≠ 0. Because [𝔤, x] = [𝔟_{G}, x], equality (♡) implies that dim B_{G} − ind B_{G} ≥ dim B_{G}\overline{x} = dim B_{G}x = dim O. □

Given an ξ ∈ (𝔟_{G})^{∗} ≃ 𝔤/[𝔟_{G}, 𝔟_{G}], it would be interesting to realize, is there x ∈ 𝔤 over ξ such that Gx is spherical?

6.4 In search of a uniform proof. Recently, Fan and Stembridge found a relationship in the simply laced case between the spherical nilpotent orbits and the “commutative” elements in the Weyl group W. They proved (see [FS97, Th. 3.1(a)]) that, for a natural map ϕ : W → N /G, the set W_{c} of commutative elements is just the preimage of N_{4}/G, where N_{4} = {e ∈ N | ht (e) < 4}. But their hope [FS97, 3.4(d)] that better understanding of the fibres of ϕ may lead to a uniform proof of (2.6) seems to me groundless. For: (1) the proof of Theorem 3.1(a) in [loc.cit.] also uses classificational results; (2) while the concept of a spherical orbit is quite general, the above relationship breaks in the non simply laced case, even if one replaces “commutative elements” in W by “fully commutative” or “short-braid avoiding” ones. To see this, it suffices to consider the groups Sp_{4} and G_{2}. Actually, case-by-case (classificational) arguments in proving (2.6) were used only for the orbits of height 3. My opinion is that the right way consists in better understanding of the orbits of height 3, as special case of orbits of odd height.

References

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ul. Akad. Anokhina, d.30, kor.1, kv.7 Moscow 117602 Russia

e-mail: panyush@dpa.msk.ru