Computation of Weyl groups of G-varieties

Let G be a connected reductive group. To any irreducible G-variety one associates a certain linear group generated by reflections called the Weyl group. Weyl groups play an important role in the study of embeddings of homogeneous spaces. We establish algorithms for computing Weyl groups for homogeneous spaces and affine homogeneous vector bundles. For some special classes of G-varieties (affine homogeneous vector bundles of maximal rank, affine homogeneous spaces, homogeneous spaces of maximal rank with discrete group of central automorphisms) we compute Weyl groups more or less explicitly.


Introduction
In the whole paper the base field is the field C of complex numbers. In this section G denotes a connected reductive group. We fix a Borel subgroup B ⊂ G and a maximal torus T ⊂ B. X denotes an irreducible G-variety.
1.1. Definition of the Weyl group of a G-variety. The main object considered in the present paper is the Weyl group of an irreducible G-variety. Before giving the definition we would like to make some historical remarks.
The Weyl group of G was essentially defined in Herman Weyl's paper [Wey]. Little bit later, E. Cartan, [Ca], generalized the notion of the Weyl group to symmetric spaces (so called, little Weyl groups). From the algebraic viewpoint a symmetric space is a homogeneous space G/H, where (G σ To move forward we need the notion of complexity. Definition 1.1.1. The complexity of X is the codimension of a general B-orbit in X, or, equivalently, tr. deg C(X) B . We denote the complexity of X by c G (X). A normal irreducible G-variety of complexity 0 is said to be spherical.
In particular, every symmetric space is a spherical G-variety, [Vu1]. In [Bri2] Brion constructed the Weyl group for a spherical homogeneous space G/H with #N G (H)/H < ∞. Brion's Weyl group generalizes that of symmetric spaces. In view of results of [BP], the restriction #N G (H)/H < ∞ is not essential. Knop,[Kn1], found another way to define the Weyl group for an arbitrary irreducible G-variety also generalizing the Weyl group of a symmetric space. In [Kn3] he extended Brion's definition to arbitrary in [Kn4] Knop gave a third definition of the Weyl group and proved the equivalence of all three definitions. Now we are going to introduce the definition of the Weyl group following [Kn3]. Consider the sublattice X G,X ⊂ X(T ) consisting of the weights of B-semiinvariant functions from C(X). It is called the weight lattice of X.
Definition 1.1.2. Put a G,X = X G,X ⊗ Z C. We call the subspace a G,X ⊂ t * the Cartan space of X. The dimension of a G,X is called the rank of X and is denoted by rk G (X).
Fix a W (g)-invariant scalar product on t(R). This induces the scalar product on a G,X (R) := X G,X ⊗ Z R and on a G,X (R) * . The Weyl group of X will act on a G,X preserving the weight lattice and the scalar product. To define the action we will describe its Weyl chamber. To this end we need the notion of a central G-valuation.
Definition 1.1.3. By a G-valuation of X we mean a discrete R-valued G-invariant valuation of C(X). A G-valuation is called central if it vanishes on C(X) B .
In particular, if X is spherical, then any G-invariant valuation is central. A central valuation v determines an element ϕ v ∈ a G,X (R) * by ϕ v , λ = v(f λ ), where λ ∈ X G,X , f λ ∈ C(X) (B) λ \ {0}. The element ϕ v is well-defined because v is central. (1) The map v → ϕ v is injective. Its image is a finitely generated rational convex cone in a G,X (R) * called the central valuation cone of X and denoted by V G,X .
(2) The central valuation cone is simplicial (that is, there are linearly independent vectors α 1 , . . . , α s ∈ a G,X (R) such that the cone coincides with {x| α i , x 0, i = 1, s}). Moreover, the reflections corresponding to its facets generate a finite group. This group is called the Weyl group of X and is denoted by W G,X .
(3) The lattice X G,X ⊂ a G,X (R) is W G,X -stable.
The proof of the first part of the theorem is relatively easy. It is obtained (in a greater generality) in [Kn3], Korollare 3.6, 4.2, 5.2, 6.5. The second assertion is much more complicated. It was proved by Brion in [Bri2] in the spherical case. Knop,[Kn3], used Brion's result to prove the assertion in general case. Later, he gave an alternative proof in [Kn4]. The third assertion of Theorem 1.1.4 follows easily from the construction of the Weyl group in [Kn4].
Note that the Weyl group does not depend on the scalar product used in its definition. Indeed, the set of W -invariant scalar products on t(R) * is convex. The Weyl group fixes X G,X so does not change under small variations of the scalar product.
1.2. Main problem. Our general problem is to find an algorithm computing a G,X , W G,X for an irreducible G-variety X. However, for such an algorithm to exist, the variety X should have some good form. It is reasonable to restrict ourselves to the following two classes of G-varieties: (1) Homogeneous spaces G/H, where H is an algebraic subgroup of G.
(2) Homogeneous vector bundles over affine homogeneous spaces (=affine homogeneous vector bundles) G * H V . Here H is a reductive subgroup of G and V is an H-module. There are several reasons to make these restrictions. First of all, these G-varieties have "group theoretic" and "representation theoretic" structure, so one may hope to find algorithms with "group-" or "representation theoretic" steps. Secondly, the computation of the Cartan space and the Weyl group of an arbitrary G-variety can be reduced to the computation for homogeneous spaces. Namely, for an irreducible G-variety X and a point x ∈ X in general position the equalities a G,X = a G,Gx , W G,X = W G,Gx hold (see Proposition 3.2.1). Moreover, if X is affine and smooth and x ∈ X is a point with closed G-orbit, then a G,X = a G,G * H V , W G,X = W G,G * H V , where H = G x and V is the slice module at x, that is V = T x X/g * x (Corollary 3.2.3).
So the main results of the paper are algorithms computing the Cartan spaces and the Weyl groups for G-varieties of types (1) and (2). Moreover, we compute the Weyl groups of affine homogeneous spaces more or less explicitly.
Our algorithms are quite complicated so we do not give them here. They will be presented (in a brief form) in Section 7. Roughly speaking, all our steps consist of computing some "structure characteristics" for pairs (an algebraic Lie algebra, a subalgebra), (a reductive Lie algebra, a module over this algebra). The computation of the normalizer or the unipotent radical is an example of an operation from the first group. An operation from the second group is, for instance, the decomposition of the restriction of an irreducible representation to a Levi subalgebra together with the determination of all highest vectors of the restriction.
1.3. Motivations and known results. Our main motivation comes from the theory of embeddings of homogeneous spaces.
One may say that the theory of algebraic transformation groups studies the category of varieties acted on by some algebraic group. Because of technical reasons, one usually considers actions of connected reductive groups on normal irreducible G-varieties. The first problem in the study of a category is the classification of its objects up to an isomorphism. In our case, the problem may be divided into two parts, birational and regular. The birational part is the classification of G-varieties up to a birational equivalence, or, in the algebraic setting, the classification of all finitely generated fields equipped with an action of G by automorphisms. An important special case here is the birational classification of quasihomogeneous G-varieties, i.e. those possessing an open G-orbit. Of course, an equivalent problem is the classification of algebraic subgroups of G up to conjugacy.
The regular part is the classification of G-varieties in a given class of birational equivalence. In the quasi-homogeneous case this is equivalent to the classification of all open embeddings of a given homogeneous space into normal varieties. The program to perform this classification was proposed by Luna and Vust ([LV]). Note that in that paper only the quasi-homogeneous case was considered. However, the Luna-Vust theory can be generalized to the general case, see [T1]. Using the Luna-Vust theory one obtains a combinatorial (in a certain sense) description of G-varieties of complexity not exceeding 1. The spherical case was considered already in [LV]. More self-contained and plain exposition is given, for example, in [Kn2]. The case of complexity 1 is due to Timashev, [T1]. The classification for complexity greater than 1 seems to be wild. Now we sketch the classification theory of spherical variety. Clearly, a spherical G-variety has an open orbit. So the birational part of the classification is just the classification of all spherical homogeneous spaces. To describe all embeddings of a given spherical homogeneous space one needs to know the following data: (1) The rational vector space a G,X (Q) := X G,X ⊗ Z Q.
(3) Certain colored vectors in a G,X (Q) * that are in one-to-one correspondence with Bdivisors of the spherical homogeneous space. Namely, given a prime B-divisor D we define the colored vector ϕ D by ϕ D , λ = ord D (f λ ), where λ ∈ X G,X , f λ ∈ C(X) It turns out that normal embeddings of a spherical homogeneous space X are in one-toone correspondence with certain admissible sets of cones in a G,X (R) * . Every cone from an admissible set is generated by colored vectors and elements of V G,X and satisfies certain combinatorial requirements. An admissible set is one that satisfies some additional requirements of combinatorial nature. So the solution of the regular part of the classificational problem consists in the determination of a G,X (Q), V G,X and colored vectors.
Note that a G,X (Q) = a G,X ∩ t(Q) * . The computation of a G,X is not very difficult. After a G,X is computed one can proceed to the computation of the valuation cone V G,X . Despite of the fact that a group generated by reflections has several Weyl chambers, the Weyl group determines the cone V G,X uniquely. Namely, V G,X is a unique Weyl chamber of W G,X containing the image of the negative Weyl chamber of t under the projection t(R) → a G,X (R) * . Now we discuss results concerning the computation of Weyl groups and Cartan spaces of G-varieties. D.I. Panyushev in [Pa1], see also [Pa4], reduced the computation of weight lattices for G-varieties of two types mentioned above to that for affine homogeneous spaces (in fact, together with some auxiliary datum). In [Lo1] Cartan spaces for affine homogeneous spaces were computed.
Proceed to results on the computation of Weyl groups. They are formulated in three possible ways: (1) In terms of the Weyl group itself.
(2) In terms of the central valuation cone.
(3) In terms of primitive linearly independent elements β 1 , . . . , β r ∈ X G,X such that the central valuation cone is given by the inequalities β i 0. We denote the set {β 1 , . . . , β r } by Π G,X . For spherical X such elements β i are called spherical roots of X. The Weyl group is unit iff X is horospherical, i.e., the stabilizer of any point contains a maximal unipotent subgroup. In full generality it was proved by Knop,[Kn1].
The other results (at least known to the author) on the computation of Weyl group relate to the spherical case.
In [Bri2], Brion proposed a technique allowing to extract the Weyl group W G,G/H , where G/H is a spherical homogeneous space, from the algebra ghg −1 for some special g ∈ G.
It is an open problem to describe the set of all suitable g.
As we have already mentioned above, the Weyl group of a symmetric space coincides with its little Weyl group, see [Kn1], [Bri2], [Vu2]. The Weyl groups of spherical G-modules were computed by Knop in [Kn6]. Note that in that paper the notation W V is used for the Weyl group of V * .
There are also computations of Weyl groups for some other special classes of spherical homogeneous spaces. Spherical roots for wonderful varieties of rank 2 were computed in [Wa]. The computation is based on some structure theorems on wonderful varieties. The computation for other interesting class of homogeneous spaces can be found in [Sm]. It uses the method of formal curves, established in [LV].
Of all results mentioned above we use only Wasserman's (see, however, Remark 6.1.1). Finally, let us make a remark on the classification of spherical varieties. The first step of the classification is describing all spherical homogeneous spaces. Up to now there is only one approach to this problem due to Luna, who applied it to classify spherical subgroups in groups of type A (a connected reductive group is said to be of type A if all simple ideals of its Lie algebra are of type A). Using Luna's approach, the full classification for groups of type A − D ( [Bra]) and a partial one for type A − C ( [Pe]) were obtained. The basic idea of the Luna classification is to establish a one-to-one correspondence between spherical homogeneous spaces and certain combinatorial data that are almost equivalent to those listed above. We note, however, that Luna's approach does not allow to obtain the full classification even for groups of type A − C. Besides, the computation of combinatorial data (in particular, the Weyl group) for certain homogeneous spaces plays an important role in this approach.
1.4. The structure of the paper. Every section is divided into subsections. Theorems, lemmas, definitions etc. are numbered within each subsection, while formulae and tables within each section. The first subsection of Sections 3-6 describes their content in detail.
Acknowledgements. I wish to thank Prof. E.B. Vinberg, who awoke my interest to the subject. Also I would like to thank D.A. Timashev for stimulating discussions. Some parts of this paper were written during my visit to Fourier University of Grenoble in June 2006. I express my gratitude to this institution and especially to Professor M. Brion for hospitality.

Notation and conventions
For an algebraic group denoted by a capital Latin letter we denote its Lie algebra by the corresponding small German letter. For example, the Lie algebra of L 0 is denoted by l 0 .
By a unipotent Lie algebra we mean the Lie algebra of a unipotent algebraic group. H-morphisms, H-subvarieties, etc. Let H be an algebraic group. We say that a variety X is an H-variety if an action of H on X is given. By an H-subset (resp., subvariety) in a given H-variety we mean an H-stable subset (resp., subvariety). A morphism of H-varieties is said to be an H-morphism if it is H-equivariant. The term "H-bundle" means a principal bundle with the structure group H.
Borel subgroups and maximal tori. While considering a reductive group G, we always fix its Borel subgroup B and a maximal torus T ⊂ B. In accordance with this choice, we fix the root system ∆(g) and the system of simple roots Π(g) of g. The Borel subgroup of G containing T and opposite to B is denoted by B − .
If G 1 , G 2 are reductive groups with the fixed Borel subgroups B i ⊂ G i and maximal tori T i ⊂ B i , then we take B 1 × B 2 , T 1 × T 2 for the fixed Borel subgroup and maximal torus in Suppose G 1 is a reductive algebraic group. Fix an embedding g 1 ֒→ g such that t ⊂ n g (g 1 ). Then t ∩ g 1 is a Cartan subalgebra and b ∩ g 1 is a Borel subalgebra of g 1 . For fixed Borel subgroup and maximal torus in G 1 we take those with the Lie algebras b 1 , t 1 .
Homomorphisms and representations. All homomorphisms of reductive algebraic Lie algebras (for instance, representations) are assumed to be differentials of homomorphisms of the corresponding algebraic groups.
Identification g ∼ = g * . Let G be a reductive algebraic group. There is a G-invariant symmetric bilinear form (·, ·) on G such that its restriction to t(R) is positively definite. For instance, if V is a locally effective G-module, then (ξ, η) = tr V (ξη) has the required properties. Note that if H is a reductive subgroup of G, then the restriction of (·, ·) to h is nondegenerate, so one may identify h with h * .
Parabolic subgroups and Levi subgroups. A parabolic subgroup of G is called standard (resp., antistandard) if it contains B (resp., B − ). It is known that any parabolic subgroup is G-conjugate to a unique standard (and antistandard) parabolic. Standard (as well as antistandard) parabolics are in one-to-one correspondence with subsets of Π(g). Namely, one assigns to Σ ⊂ Π(g) the standard (resp., antistandard) parabolic subgroup, whose Lie algebra is generated by b and g −α for α ∈ Σ (resp., by b − and g α , α ∈ Σ).
By a standard Levi subgroup in G we mean the Levi subgroup containing T of a standard (or an antistandard) parabolic subgroup.
Simple Lie algebras, their roots and weights. Simple roots of a simple Lie algebra g are denoted by α i . The numeration is described below. By π i we denote the fundamental weight corresponding to α i .
Classical algebras. In all cases for b (resp. t) we take the algebra of all upper triangular (resp., diagonal) matrices in g. g = sl n . Let e 1 , . . . , e n denote the standard basis in C n and e 1 , . . . , e n the dual basis in C n * . Choose the generators ε i , i = 1, n, given by . Let e 1 , . . . , e 2n+1 be the standard basis in C 2n+1 . We suppose g annihilates the form (x, y) = 2n+1 i=1 x i y 2n+2−i . Define ε i ∈ t * , i = 1, n, by ε i , diag(x 1 , . . . , x n , 0, −x n , . . . , −x 1 ) = x i . Put α i = ε i − ε i+1 , i = 1, n − 1, α n = ε n . g = sp 2n . Let e 1 , . . . , e 2n be the standard basis in C 2n . We suppose that g annihilates the form ( g = so 2n . Let e 1 , . . . , e 2n be the standard basis in C 2n . We suppose that g annihilates the form (x, y) = 2n i=1 x i y 2n+1−i . Define ε i ∈ t * , i = 1, n, in the same way as for g = sp 2n . Put Exceptional algebras. For roots and weights of exceptional Lie algebras we use the notation from [OV]. The numeration of simple roots is also taken from [OV].
Subalgebras in semisimple Lie algebra. For semisimple subalgebras of exceptional Lie algebras we use the notation from [D]. Below we explain the notation for classical algebras.
Suppose g = sl n . By sl k , so k , sp k we denote the subalgebras of sl n annihilating a subspace U ⊂ C n of dimension n−k, leaving its complement V invariant, and (for so k , sp k ) annihilating a nondegenerate orthogonal or symplectic form on V .
The subalgebras so k ⊂ so n , sp k ⊂ sp n are defined analogously. The subalgebra gl diag k is embedded into so n , sp n via the direct sum of τ, τ * and a trivial representation (here τ denotes the tautological representation of gl k ). The subalgebras sl diag k , so diag k , sp diag k ⊂ so n , sp n are defined analogously. The subalgebra G 2 (resp., spin 7 ) in so n is the image of G 2 (resp., so 7 ) under the direct sum of the 7-dimensional irreducible (resp., spinor) and the trivial representations.
Finally, let h 1 , h 2 be subalgebras of g = sl n , so n , sp n described above. While writing h 1 ⊕h 2 , we always mean that ( The description above determines a subalgebra uniquely up to conjugacy in Aut(g). Now we list some notation used in the text.
3.1. Introduction. In this section we quote known results and constructions related to Cartan spaces, Weyl groups and weight lattices. We also prove some more or less easy results for which it is difficult to give a reference. In Subsection 3.2 we present results on the computation of Cartan spaces. In the beginning of the subsection we establish an important notion of a tame inclusion of a subgroup of G into a parabolic. Then we present a reduction of the computation from the general case to the case of affine homogeneous spaces. This reduction belongs to Panyushev. Then we partially present results of [Lo1] on the computation of the Cartan spaces for affine homogeneous spaces. Finally, at the end of the section we study the behavior of Cartan spaces, Weyl groups, etc. under the twisting of the action G : X by an automorphism.
The most important part of this section is Subsection 3.3, where we review some definitions and results related to Weyl groups of G-varieties. We start with results of F. Knop,[Kn1], [Kn3], [Kn4], [Kn8]. Then we quote results of [Lo2] that provide certain reductions for computing Weyl groups. These results allow to reduce the computation of the groups W G,X to the case when G is simple and rk G (X) = rk G.
Till the end of the subsection we deal with that special case. Here we have two types of restrictions on the Weyl group. Restrictions of the first type are valid for smooth affine varieties. They are derived from results of [Lo3]. The computation in Section 5 is based on these restrictions. Their main feature is that they describe the class of conjugacy of W G,X and do not answer the question whether a given reflection lies in W G,X .
On the other hand, we have some restrictions on the form of a reflection lying in W G,X . They are used in Section 6. These restrictions are derived from the observation that the Weyl group of an arbitrary X coincides with a Weyl group of a certain wonderful variety. This observation follows mainly from results of Knop,[Kn3].
3.2. Computation of Cartan spaces. In this subsection G is a connected reductive group. The definitions of the Cartan spaces and the Weyl group of X given in Subsection 1.1 are compatible with those given in [Kn1], [Kn4], see [Kn4], Theorem 7.4 and Corollary 7.5. In particular, W G,X is a subquotient of W (g) (i.e., there exist subgroups Γ 1 , Γ 2 ⊂ W (g) such that a G,X is Γ 1 -stable, Γ 2 is the inefficiency kernel of the action Γ 1 : a G,X , and W G,X = Γ 1 /Γ 2 ). The following proposition describes functorial properties of Cartan spaces and Weyl groups.
(1) Suppose ϕ is dominant. Then a G,X 2 ⊂ a G,X 1 and W G,X 2 is a subquotient of W G,X 1 .
(2) Suppose ϕ is generically finite. Then a G,X 1 ⊂ a G,X 2 and W G,X 1 is a subquotient of W G,X 2 . (3) If ϕ is dominant and generically finite (e.g. etale), then a G,X 1 = a G,X 2 and W G,X 1 = W G,X 2 . (4) Let X be an irreducible G-variety. There is an open G-subvariety X 0 ⊂ X such that a G,Gx = a G,X , W G,Gx = W G,X for any x ∈ X 0 .
In the sequel we write a(g, h) instead of a G,G/H . Corollary 3.2.3. Let X be smooth and affine, x a point of X with closed G-orbit. Put Proof. This is a direct consequence of the Luna slice theorem for smooth points ( [PV], Subsection 6.5) and assertion 3 of Proposition 3.2.1.
Corollary 3.2.4. Let H be a reductive subgroup of G and V an H-module.
Proof. Assertion 1 follows from assertions 1,3 of Proposition 3.2.1. To prove assertion 2 note that there is an isomorphism of G-varieties G * H V ∼ = G * H (V /V H ) × V H (G is assumed to act trivially on V H ). It remains to apply assertion 4 of Proposition 3.2.1.
In the sequel we write a(g, h, Now we reduce the computation of Cartan spaces for homogeneous spaces to that for affine homogeneous vector bundles. To this end we need one fact about subgroups in G due to Weisfeller, [Wei].
Proposition 3.2.5. Let H be an algebraic subgroup of G. Then there exists a parabolic Definition 3.2.6. Under the assumptions of the previous proposition, we say that the inclusion H ⊂ Q is tame. Algorithm 7.1.1 allows one to construct a tame inclusion.
Conjugating H by an element of Q, one may assume that S ⊂ M. Besides, conjugating Q, M, H by an element of G, one may assume that Q is an antistandard parabolic and M is its standard Levi subgroup.
The following lemma and remark seem to be standard.
Lemma 3.2.7. Let Q, M, H, S be such as in the previous discussion and V an H-module, Proof. Consider the map ι : The morphism ι becomes Qequivariant. One easily checks that ι is injective. Since dim X 0 = dim X, the morphism ι is dominant. Taking into account that X is smooth, we see that ι is an open embedding. So C(X 0 ), C(X) are Q − -equivariantly isomorphic whence the claim of the lemma.
The following proposition stems from Lemma 3.2.7 and Remark 3.2.8. It is also a direct generalization of a part of Theorem 1.2 from [Pa1] (see also [Pa4], Theorem 2.5.20). Next, we reduce the case of affine homogeneous vector bundles to that of affine homogeneous spaces. To state the main result (Proposition 3.2.12) we need the notion of the distinguished component.
First of all, set (3.1) L G,X := Z G (a G,X ).
(1) Let L 1 be a normal subgroup of L 0 G,X . Then there exists a unique irreducible component X ⊂ X L 1 such that UX = X.
(2) Set P := L G,X B. Let S be a locally closed L G,X -stable subvariety of X such that (L G,X , L G,X ) acts trivially on S and the map P * L G,X S → X, [p, s] → ps, is an embedding (such S always exists, see [Kn4], Section 2, Lemma 3.1). Then X := F S, where F := R u (P ) L 1 .
Definition 3.2.11. The component X ⊂ X L 1 satisfying the assumptions of Proposition 3.2.10 is said to be distinguished.
The distinguished component for L 1 = L 0 G,X was considered by Panyushev in [Pa2].
Proposition 3.2.12. Let H be a reductive subgroup in G, V an H-module and π the natural Proof. Let x 1 be a canonical point in general position in the sense of [Pa4], Definition 5 of Subsection 2.1. This means that B x 1 = B ∩ L 1 . It follows from Theorem 2.5.20 from [Pa4] x) . Note that the L 1module π −1 (x) does not depend on the choice of a point x from the distinguished component. Now it remains to note that the distinguished component of (G/H) L 1 contains a canonical point in general position. Let S be an L G,G/H -subvariety in G/H mentioned in assertion 2 of Proposition 3.2.10 (for X = G/H). Such S consists of canonical points in general position. Algorithm 7.1.1 computes the subgroups L 0 G,V for a G-module V , see [Pa4]. Thus the computation of a G,X is reduced to the computation of the following data: (1) The spaces a(g, h), where H is a reductive subgroup of G.
(2) A point from the distinguished component of (G/H) L 1 , where L 1 = L • 0 G,G/H , for a reductive subgroup H ⊂ G. Now we are going to present results of the paper [Lo1] concerning the computation of a(g, h). Until a further notice H denotes a reductive subgroup in G.
To state our main results we need some definitions. We begin with a standard one.
Definition 3.2.13. A subalgebra h ⊂ g is said to be indecomposable, if for (h∩g 1 )⊕(h∩g 2 ) h any pair of ideals g 1 , g 2 ⊂ g with g = g 1 ⊕ g 2 .
Since a(g 1 ⊕ g 2 , h 1 ⊕ h 2 ) = a(g 1 , h 1 ) ⊕ a(g 2 , h 2 ), the computation of a(g, h) can be easily reduced to the case when h ⊂ g is indecomposable.
In virtue of Corollary 3.2.2, a(g, h) ⊂ a(g, h 1 ) for any ideal h 1 ⊂ h. This observation motivates the following definition.
Definition 3.2.14. A reductive subalgebra h ⊂ g is called a-essential if a(g, h) a(g, h 1 ) for any ideal h 1 h.
To make the presentation of our results more convenient, we introduce one more class of subalgebras.
Definition 3.2.15. An a-essential subalgebra h ⊂ g is said to be saturated if a(g, h) = a(g, h), where h := h + z(n g (h)).
Finally, we note that Lemmas 3.2.18,3.2.19 below allow to perform the computation of a(g, h) just for one subalgebra h in a given class of Aut(g)-conjugacy.
The next proposition is a part of Theorem 1.3 from [Lo1].
(1) There is a unique ideal h ess ⊂ h such that h ess is an a-essential subalgebra of g and a(g, h) = a(g, h ess ). The ideal h ess is maximal (with respect to inclusion) among all ideals of h that are a-essential subalgebras of g.
(2) All semisimple indecomposable a-essential subalgebras in g up to Aut(g)-conjugacy are listed in Table 3.1.
(3) All nonsemisimple saturated indecomposable a-essential subalgebras in g up to Aut(g)conjugacy are listed in In Theorem 1.3 from [Lo1] all essential subalgebras are classified and a way to compute the Cartan spaces for them is given. Note that, as soon as this is done, assertion 1 of Proposition 3.2.16 provides an effective method of the determination of h ess .
At the end of this subsection we consider the behavior of Cartan spaces, Weyl groups etc. under the twisting of the action G : X by an automorphism.
Let τ ∈ Aut (G). By τ X we denote the G-variety coinciding with X as a variety, the action of G being defined by (g, x) → τ −1 (g)x. The identity map is an isomorphism of σ ( τ X) and στ X for σ, τ ∈ Aut (G). If τ is an inner automorphism, τ (g) = hgh −1 , then x → hx : τ X → X is a G-isomorphism. Hence the G-variety τ X is determined up to isomorphism by the image of τ in Aut(G)/ Int (G). In particular, considering G-varieties of the form τ X, one may assume that τ (B) = B, τ (T ) = T .
Further, if L 1 is a normal subgroup in L 0 G,X , then the distinguished components in X L 1 and τ X τ (L 1 ) coincide.
Proof. Let f ∈ C(X) be a B-semiinvariant function of weight χ. Then f considered as an element of C( τ X) is B-semiinvariant of weight τ (χ). Assertions on a G,• , X G,• , L G,• , L 0 G,• follow immediately from this observation. U-orbits of the actions G : X, G : τ X coincide whence the equalities for the distinguished components. Now let v be a central valuation of λ , whence the equality for the Weyl groups. Lemma 3.2.19. Let τ ∈ Aut (G).
(2) Let H be a reductive subgroup of G and V be an H-module.

3.3.
Results about Weyl groups. Proposition 3.3.1. Let h be an algebraic subalgebra of g and h 0 a subalgebra of g lying in the closure of Gh in Gr(g, dim h). Then h 0 is an algebraic subalgebra of g, a(g, h) = a(g, h 0 ) and W (g, h 0 ) ⊂ W (g, h).
Let T 0 be a torus and π : X → X be a principal locally trivial T 0 -bundle, where T 0 is a torus. In particular, T 0 acts freely on X and X is a quotient for this action. Now let H be an algebraic group acting on X. The bundle π : X → X is said to be H-equivariant if X is equipped with an action H : X commuting with that of T 0 and such that π is an H-morphism. We can consider X, X as H × T 0 -varieties (T 0 acts trivially on X) and π as an H × T 0 -morphism. Now we want to establish a relation between the linear part of the cone V G,X , which coincides with a * G,X (R) W G,X , and a certain subgroup of Aut G (X).
by the multiplication by a constant for any λ ∈ X G,X . Central automorphisms of X form the subgroup of Aut G (X) denoted by A G (X). It turns out that A G (X) is not a birational invariant of X. However, by Corollary 5.4 from [Kn5], there is an open G-subvariety X 0 such that A G (X 0 ) = A G (X 1 ) for any open G-subvariety X 1 ⊂ X 0 . We denote A G (X 0 ) by A G,X .
Put A G,X := Hom(X G,X , C × ). The group A G,X is embedded into A G,X as follows. We assign a ϕ,λ to ϕ ∈ A G,X , λ ∈ X G,X by the formula ϕf λ = a ϕ,λ f λ , f ∈ C(X) (B) λ . The map ι G,X : A G,X → A G,X is given by λ(ι G,X (ϕ)) = a ϕ,λ . Clearly, ι G,X is a well-defined homomorphism.
Lemma 3.3.5 ([Kn5], Theorem 5.5). ι G,X is injective and its image is closed. In particular, A • G,X is a torus.
In the sequel we identify A G,X with im ι G,X . The following proposition is a corollary of Satz 8.1 and (in fact, the proof of) Satz 8.2 from [Kn3].
Until a further notice we assume that X is normal. Let us study a relation between the central valuation cones of X and of certain G-divisors of X. Our goal is to prove Corollary 3.3.9, which will be used in the end of the subsection to obtain restrictions on possible Weyl groups of homogeneous spaces (Proposition 3.3.23).
Let v 0 be a nonzero R-valued discrete geometric valuation of C(X) B (by definition, v 0 is geometric if it is a nonnegative multiple of the valuation induced by a divisor on some model of C(X) B ). We denote by V v 0 the set of all geometric G-valuations v of C(X) such that v| C(X) B = kv 0 for some k 0. Now we construct a map from V v 0 to a finite dimensional vector space similar to that from Subsection 1.1. Namely, we have an exact sequence of abelian groups This sequent splits because X G,X is free. Fix a splitting λ → f λ . We assign an element There is a statement similar to Theorem 1.1.4. ( Such v is defined uniquely up to rescaling and the shift by an element of V G,X ∩−V G,X .
The first part of this proposition stems from [Kn3], Korolläre 3.6,4.2. The second one is a reformulation of the second part of Satz 9.2 from [Kn3].
The following result is a special case of Satz 7.5 from [Kn3].
Corollary 3.3.9. Let X be an irreducible G-variety. Then there exists a spherical G-variety Proof. Replacing X with some birationally equivalent G-variety, we may assume that there is a divisor D ⊂ X satisfying the assumptions of Proposition 3.3.8. Since the valuation corresponding to D is noncentral, we have c G (D) = c G (X) − 1 ([Kn3],Satz 7.3). Now it remains to use the induction on complexity.
Now we quote some results from [Lo2] providing some reduction procedures for computing Weyl groups.
Until a further notice, X is a smooth quasiaffine G-variety.
11. Note that X 0 is a smooth quasiaffine variety. Its smoothness is a standard fact, see, for example, [Lo2], Lemma 8.6.
It is a reductive group acting on X. Note that its tangent algebra g can be naturally identified with Further, note that t ⊂ n g (g 1 ). Thus there are the distinguished maximal torus T and the Borel subgroup B in G, see It is easy to prove, see Section 4, that if X is a homogeneous space (resp., affine homogeneous vector bundle), then X is a homogeneous space (resp., affine homogeneous vector bundle) with respect to the action of G • . Proposition 3.3.11. Let H be an algebraic subgroup of G, H = S ⋌ R u (H) its Levi decomposition, Q an antistandard parabolic in G and M its standard Levi subgroup. Suppose that Proof. This follows from [Lo2], Proposition 8.13, and Lemma 3.2.7.
The following proposition may be considered as a weakened version of Proposition 3.3.6 The required claims follow now from Proposition 3.3.12 applied to the action G : G/H 1 .
The next well-known statement describes the behavior of the Weyl group under a so called parabolic induction. The only case we need is that of homogeneous spaces of rank equal to rk G. We give the proof only to illustrate our techniques.
Corollary 3.3.14. Let Q be an antistandard parabolic subgroup of G and M the standard Levi subgroup of Q. Further, let h be a subalgebra of g such that R u (q) ⊂ h, a(g, h) = t. (1) Let H be an algebraic subgroup of G such that a(g, h) = t and G/H is quasiaffine. Then Proof. To prove the first assertion we note that the stabilizer of any point of G/H in G i is conjugate to G i ∩ H and use assertion 4 of Proposition 3.2.1 and [Lo2], Proposition 8.8.
Proceed to assertion 2. All G i -orbits in G/H are of the same dimension whence closed. Choose x ∈ G/H with (G i Till the end of the section G is simple, X is a quasiaffine irreducible smooth G-variety such that rk G (X) = rk G. In this case W G,X ⊂ W (g).
At first, we consider the case when X is affine. Here we present some results on W G,X obtained in [Lo3]. Those results can be applied here because W G,X coincides up to conjugacy with the Weyl group of the Hamiltonian G-variety T * X (see [Kn4]).
For a subgroup Γ ⊂ W (g) we denote by ∆ Γ the subset of ∆(g) consisting of all α with s α ∈ Γ. ( Table 3.3. Table 3.3: Subsets ∆ Γ for large subgroups Γ ⊂ W (g) when g is classical Note that some subsets ∆ Γ appear in Table 3.3 more than once. Now we obtain a certain restriction on W G,X in terms of the action G : T * X. To do this we introduce the notions of a g-stratum and a completely perpendicular subset of ∆(g).
Definition 3.3.19. Let Y be a smooth affine variety and y ∈ Y a point with closed G-orbit.
The pair (g y , T y Y /g * y) is called the g-stratum of y. We say that Remark 3.3.20. Let us justify the terminology. Pairs (h, V ) do define some stratification of Y //G by varieties with quotient singularities. Besides, analogous objects were called "strata" in [Sch2], where the term is borrowed from.
Definition 3.3.21. A subset A ⊂ ∆(g) is called completely perpendicular if the following two conditions take place: For example, any one-element subset of ∆(g) is completely perpendicular. Let A be a nonempty completely perpendicular subset of ∆(g). By S (A) we denote the Till the end of the subsection G is simple, X is a homogeneous G-space of rank rk(G) (not necessarily quasiaffine). In this case any element of Π G,X is a positive multiple of a unique positive root from ∆(g). The set of all positive roots arising in this way is denoted by Π G,X . Note that Π G,G/H depends only on (g, h). Therefore in the sequel we write Π(g, h) instead of Π G,G/H . Proposition 3.3.23. Let G, X be such as above and G = G 2 . Then the pair (Supp(α), α), where α ∈ Π(g, h) is considered as an element of the root system associated with Supp(α), are given in the following list: (1) (A 1 , α 1 ).
Proof of Proposition 3.3.23. For an arbitrary G-variety X let P G,X denote the intersection of the stabilizers of all B-stable prime divisors of X. Thanks to Corollary 3.3.9, we may assume that X is spherical. There is a subgroup H ⊂ G containing H such that Π G,G/H = Π G,G/ e H and X G,G/ e H = Span Z (Π G,G/H ). Furthermore, the homogeneous space G/ H possesses a so called wonderful embedding G/ H ֒→ X (these two facts follow from results of [Kn5], Sections 6,7; for definitions and results concerning wonderful varieties see [Lu2] or [T2], Section 30). Since a G,G/H = t, we see that P G,X = B. So a unique closed G-orbit on X is isomorphic to G/B. For α ∈ Π G,X let X α denote the wonderful subvariety of X of rank 1 corresponding to α. Since the closed G-orbit in X α is isomorphic to G/B, we see that P G,Xα = B. Now the proposition stems from the classification of wonderful varieties of rank 1 and the computation of their spherical roots. These results are gathered in [Wa], Table 1.

Determination of distinguished components
4.1. Introduction. In this section we find an algorithm to determine the distinguished component of X L • 0 G,X in the case when X is a homogeneous space or an affine homogeneous vector bundle. This will complete the algorithm computing a G,X for the indicated classes of varieties, see Subsection 3.2. Besides, this makes possible to use Theorem 3.3.10.
By X we denote the distinguished component of X L 0 . Our first task is to describe the structure of distinguished components.
Proposition 4.1.1. Here L 0 denotes one of the groups L 0,G,X , L • 0,G,X and X is the distinguished component X L 0 .
(1) Let X = G/H be a quasiaffine homogeneous space. Then the action  To state the result concerning affine homogeneous space we need some notation. Let h be an a-essential subalgebra of g (see Definition 3.2.14), h := h + z g (n g (h)). Put h sat = h ess . By properties of the mapping h → h ess , see [Lo1], Corollary 2.8, h ⊂ h sat . It follows directly from the construction that h sat is a saturated subalgebra of g (Definition 3.2.15).
The next proposition allows one to find a point from a distinguished component of an affine homogeneous space. (1) Let H ess denote the connected subgroup of G corresponding to h ess , π the projection G/H ess → G/H, gH ess → gH, and X ′ the distinguished component of (G/H ess Suppose h is one of the subalgebras listed in Tables 3.1,3.2, and H is connected. Then X ⊂ X is stable under the action of N G (L 0 ). (5) Let h, H be such as indicated in Subsection 4.4. Then eH ∈ X.
The first three assertions reduce the problem of finding a point from (G/H) L 0 to the case when H is connected and h is a-essential, saturated and indecomposable. All such subalgebras h are listed in Tables 3.1,3.2, see Proposition 3.2.16. Assertion 5 solves the problem in this case. Assertion 4 is auxiliary.
Let us give a brief description of the section. In Subsection 4.2 we prove Propositions 4.
Proceed to the proof of assertion 2. Clearly, the projection Here π denotes the natural projection X → G/H. It follows that, as an N G (L 0 , X)-variety, X is a homogeneous vector bundle over some component of (G/H) L 0 . Note also that N G (L 0 , X) = N G (L 0 , π(X)). The last equality yields the claim on the structure of N G (L 0 , X).
It remains to verify that the distinguished component Y of (

4.
3. An auxiliary result. Here we study components of the variety X L 0 in the case when X is smooth and affine.
By the stable subalgebra in general position (shortly, s.s.g.p.) we mean the Lie algebra of the s.g.p.
Proof. Proposition 3.2.10 implies (in the notation of that proposition By [Kn1], Korollar 8.2, L 0 G,X is the s.g.p. for the action G : T * X. Let us show that the action G : T * X is stable, that is, an orbit in general position is closed. Choose a point x ∈ X with closed G-orbit.
Using the Luna slice theorem, we see that the action G : T * X is stable iff so is the representation is an orthogonal H-module, we are done by results of [Lu1].
Thanks to results of [LR], N G (L 0 ) permutes transitively irreducible (=connected) components of (T * X) L 0 whose image in ( )/2 and that if the equality holds, then Z contains a point z with closed G-orbit and This completes the proof.
4.4. Embeddings of subalgebras. In this subsection we construct embeddings h ֒→ g for the pairs (g, h) from Tables 3.1,3.2. In the next subsection we will see that the corresponding points eH ∈ G/H lie in the distinguished components of (G/H) L 0 , L 0 = L • 0 G,G/H . If eH lies in this distinguished component, then, obviously, l 0 ⊂ h and, by Proposition 4.3.2, Proposition 4.4.1. Suppose that one of the following two assumptions hold (1) (g, h) is one of the pairs from Table 3.1.
( Table 3.2. If [h, h] ֒→ g is embedded into g as described below in this subsection, then l 0 ⊂ h and (4.1) holds.
We check this assertion case by case. The case g = sl n .
The case g = sp 2n .
Remark 4.4.2. Let us explain why we choose this strange (at the first glance) embedding. The pair (g, h) = (sp 2n , sp 2k × sp 2(n−k) ) is symmetric. The corresponding involution σ acts on the annihilator of sp 2(n−k) ⊂ h (resp. sp 2k ⊂ h) in C 2n identically (resp., by −1). Under the chosen embedding, σ acts on a(g, h) by −1, in other words, a(g, h) is the Cartan space of (g, h) in the sense of the theory of symmetric spaces.
The case g = so n , n 7.
3) The subalgebra spin 7 is embedded into g = so 9 (row 11 of Table 3.1) as the annihilator of the sum of a highest vector and a lowest vector in the so 9 -module V (π 4 ). Under this choice of the embedding, dim n g (l 0 ) = 12, dim n h (l 0 ) = 9. 4) For the embedding spin 7 ֒→ so 10 (row 12 of Table 3.1) we take the composition of the embeddings spin 7 ֒→ so 9 , so 9 ֒→ so 10 defined above. In this case n g (l 0 ) = 21, n h (l 0 ) = 9. 5) G 2 is embedded into so 7 (row 13 of Table 3.1) as the annihilator of the sum of a highest vector and a lowest vector of the so 7 -module V (π 3 ). The equalities dim n g (l 0 ) = 9, dim n h (l 0 ) = 8 hold.
The case g = G 2 .
The subalgebra A 2 (row 15 of Table 3.1) is embedded into G 2 as g (∆max) , where ∆ max denotes the subsystem of all long roots. In this case dim n g (l 0 ) = 6, dim n h (l 0 ) = 4.
The case g = E 6 .
The case g = E 7 .
The case g = h × h. Fix a Borel subgroup B 0 ⊂ H and a maximal torus T 0 ⊂ B 0 and construct from them the Borel subgroup and the maximal torus of G (Section 2). The embedding h ֒→ h × h is given by ξ → (ξ, w 0 ξ). Here w 0 is an element N H (T 0 ) mapping into the element of maximal length of the Weyl group. We have dim n g (l 0 ) = 2 rk h, dim n h (l 0 ) = rk h.
In the following table we list the pairs n g (l 0 )/l 0 , n h (l 0 )/l 0 for the embeddings h ֒→ g of subalgebras from Table 3.1 constructed above. In the first column the number of a pair in Table 3.1 is given. The fourth column contains elements of a(g, h) constituting a system of simple roots in n g (l 0 )/l 0 . The table will be used in Subsection 5.5 to compute the Weyl groups of affine homogeneous spaces.
Assertion 2. Put L = L G,G/H , P = LB, L 0 = L • 0 G,G/H sat . Since L G,G/H sat = L, there is an L-stable subvariety S ⊂ G/H sat such that (L, L) acts trivially on S and the natural morphism P * L S → G/H sat is an embedding, see Proposition 3.2.10. Thence there is a P -embedding P * L π −1 (S) ֒→ X. It follows that X G,X = X L,π −1 (S) . Hence the action (L, L) : π −1 (S) is trivial. By Proposition 3.2.10, X ′ = R u (P ) e L 0 S, X = R u (P ) L 0 π −1 (S). Therefore R u (P ) e L 0 π −1 (S) is a dense subset of π −1 (X ′ ). This proves assertion 2. Assertion 3. This is obvious. Assertion 4. Put L = L G,G/H . By Proposition 4.1.1, the assertion will follow if we prove . Therefore in the proof we may always consider any group G with Lie algebra g.
The proof of (4.2) will be carried out in two steps. Put These are the subgroups in the corresponding normalizers consisting of all elements acting on l 0 by inner automorphisms. On the first step we show that . The claim of the first step will follow if we check that Z G (l 0 )/Z H (l 0 ) is connected.
Step 1. Proof of Lemma 4.5.1. The claim is well-known in the case when m 0 is commutative (see [OV], ch.3, §3). Using induction on dim m 0 , we may assume that m 0 is simple. Choose a Cartan subalgebra t ⊂ m. Any connected component of ). An element of the last group acts on t as a composition of reflections corresponding to elements of ∆(m) ⊥ ∩ ∆(g). Since β is a long root, we see that α + β ∈ ∆(g) for any α ∈ ∆(m) ⊥ , β ∈ ∆(m). Therefore ∆(m) ⊥ ∩ ∆(g) ⊂ ∆(z g (m 0 )). In particular, It remains to consider only pairs (g, h) such that g contains a simple ideal isomorphic to B l , C l , F 4 , G 2 and ∆(l) contains a short root. Such a pair (g, h) is on of the following pairs from Table 3.1: NN4 (2k − n > 1),5,6 (n > 5),8 (n is odd),16,26 (n 3),27 (n 3). In cases 4,6,27 the subalgebra [l 0 , l 0 ] coincides with sp 2l , while in case 26 it coincides with the direct sum of the subalgebras sp 2l 1 , sp 2l 2 embedded into different simple ideals of g. The centralizer of sp 2l ⊂ sp 2n in Sp(2n) is connected for all l, n. It remains to consider cases 5,8,16.
Put g = sp 2m , h = sp 2k × sp 2(m−k) , k n/2. As in the previous paragraph, it is enough to consider the case n = 2k. Let us describe the embedding L 0 ֒→ Sp(2n). The space C 4k is decomposed into the direct sum of two-dimensional spaces V 1 , . . . , V 2k . We may assume that H = Sp (V 1 . The subgroup L 0 ⊂ Sp 2n is isomorphic to the direct product of k copies of SL 2 . The i-th copy of SL 2 (we denote it by L i 0 ) acts diagonally on V 2i−1 ⊕ V 2i and trivially on V j , j = 2i − 1, 2i. It can be seen directly that In case 8 both groups Z G (L 0 ), Z H (L 0 ) contain two connected components. The components of unit in the both cases consist of all elements acting trivially on C 2n+1 /(C 2n+1 ) l 0 . This proves the required claim.
In case 16 we have Step 2. It remains to check that N G (L 0 ) = N G (L 0 ) 0 N H (L 0 ), equivalently, the images of N G (l 0 ), N H (l 0 ) in GL(l 0 ) coincide. For the pair N1 in Table 3.1 the group N G (l 0 ) is connected. If (g, h) is one of the pairs NN 4,8 (n is odd), 12,14,15,16,20, then the group of outer automorphisms of l 0 is trivial. For the pairs NN11,13,17,18,19,22,24 from Table 3.1 the algebra l 0 is simple and N H (l 0 ) contains all automorphisms of l 0 (in cases 22,24 one proves the last claim using the embeddings D 4 ⊂ F 4 ⊂ E 6 ⊂ E 7 ⊂ E 8 ). In cases 3,9,22, 5(n = 2k) the algebra l 0 is isomorphic to the direct product of some copies of sl 2 . The group of outer automorphisms of l 0 is the symmetric group on the set of simple ideals of l 0 . The image of N H (l 0 ) in Aut(l 0 )/ Int(l 0 ) coincides with the whole symmetric group. One considers case 5, n > 2k, analogously. Here Aut(l 0 )/ Int(l 0 ) is isomorphic to the symmetric group on the set of simple ideals of l 0 that are isomorphic to sl 2 and have Dynkin index 2. In case 8 (with even n) the images of both N H (l 0 ), N G (l 0 ) in Aut(l 0 ) contain an outer automorphism of l 0 induced by an element from O 2k−n \ SO 2k−n and do not contain elements from the other nontrivial connected components of Aut(l 0 ) (which exist only when 2k − n = 8).

If a pair (g, [h, h]) is contained in
) is connected too. In cases 6,7,26,27 the algebra z(l 0 ) is one-dimensional and the groups in the both sides of (4.3) act on z(l 0 ) as Z 2 . This observation yields (4.3) in these cases. It remains to check (4.3) for (g, [h, h]) = (h × h, h), (E 6 , A 5 ), (sl n , sl n−k × sl k ). The first case is obvious.
The case (g, h) = (E 6 , A 5 ). We suppose that the invariant symmetric form on g is chosen in such a way that the length of a root equals 2.
We have l 0 = α 1 − α 5 , α 2 − α 4 . The action of N H (l 0 ) on l 0 coincides with the action of the symmetric group S 3 on its unique 2-dimensional module. To see this note that l 0 is embedded into sl 6 as {diag(x, y, −x − y, −x − y, y, x)}.
Any element of length 4 lying in the intersection of l 0 and the root lattice of g is one of the following elements: ±(α 1 − α 5 ), ±(α 2 − α 4 ), ±(α 1 + α 2 − α 4 − α 5 ). Assume that the images of N G (l 0 ) and N H (l 0 ) in GL(l 0 ) differ. The image N of N G (l 0 ) ∩ N G (t) in GL(l 0 ) permutes the six elements of length 4 listed above and preserves the scalar product on l 0 ∩ t(R). It follows that −id ∈ N. Let us check that g ∈ N G (t) ∩ N G (l 0 ) cannot act on l 0 by −1.
Replacing g with gs 2ε , we may assume that g ∈ Z G ( ε ). The last group is the product of A 5 ⊂ E 6 and a one-dimensional torus and does not contain an element acting on l 0 by −1.
In Subsection 5.5 we will need to know the image of N G (l 0 ) ∩ N G (t) in GL(a(g, h)) for pairs (g, h) from Table 3.1. This information is extracted mostly from the previous proof and Let us explain the notation used in the table. In rows 2,3,5,18 a set of generators of the group is given. The symbols A, W mean the automorphism group (resp., the Weyl group) of the root system n g (l 0 )/l 0 (we assume that this group acts trivially on z(n g (l 0 )/l 0 )). The symbol Z 2 denotes the group acting by ±1 on the center of n g (l 0 )/l 0 and trivially on the semisimple part.

Computation of Weyl groups for affine homogeneous vector bundles
5.1. Introduction. The main goal of this section is to compute the group W G,X , where X = G * H V is a homogeneous vector bundle over an affine homogeneous space G/H. We recall that W G,X depends only on the triple (g, h, V ) (see Corollary 3.2.4), so we write Let X be the distinguished component of X L 0 , where L 0 = L • 0 G,X , and G = N G (L 0 , X)/L 0 . Theorem 3.3.10 allows to recover W G,X from W G • ,X . Besides, the results of Section 4 show that the G • -variety X is an affine homogeneous vector bundle and allow to determine it. So to compute Weyl groups one may restrict to the case rk G (X) = rk G. Further, Proposition 3.3.15 reduces the computation of W G,X to the case of simple G. Finally, we may assume that G is simply connected. Below in this subsection we always assume that these conditions are satisfied.
By an admissible triple we mean a triple (g, h, V ), where g is a simple Lie algebra, h its reductive subalgebra, and V is a module over the connected subgroup H ⊂ G with Lie algebra h such that a(g, h, V ) = t.
Let us state our main result. There is a minimal ideal h 0 ⊂ h such that W (g, h, V ) = W (g, h 0 , V /V h 0 ). We say that the corresponding triple (g, h 0 , V /V h 0 ) is a W -essential part of the triple (g, h, V ). The problem of computing of W (g, h, V ) may be divided into three parts: a) To find all admissible triples Clearly, a triple is Wessential iff it coincides with its W -essential part. b) To compute the groups W (g, h, V ) for all triples (g, h, V ) found on the previous step. c) To show how one can determine a W -essential part of a given admissible triple.
Theorem 5.1.2. Let (g, h, V ) be an admissible triple. If there exists an ideal h 1 ⊂ h such that the triple (g, h 1 , V /V h 1 ) is isomorphic to a triple from Table 5.1, then (g, h 1 , V /V h 1 ) is a W -essential part of (g, h, V ) and the Weyl group is presented in the fifth column of Table  5.1. Otherwise, (g, 0, 0) is a W -essential part of (g, h, V ).
Remark 5.1.3. Inspecting Table 5.1, we deduce from Theorem 5.1.2 that a W -essential part of (g, h, V ) is uniquely determined.
Remark 5.1.4. Let (g, h, V ) be a triple listed in Table 5.1. The isomorphism class of (g, h, V ) consists of more than one equivalence class precisely for the following triples: N1 (l = m), NN2-6, 26,27. In these cases an isomorphism class consists of two different equivalence classes. Now we describe the content of this section. In Subsection 5.2 we classify W -quasiessential triples.
Here H, H 1 are connected subgroups of G corresponding to h, h 1 .
Below we will see that W -quasiessential triples are precisely those listed in Table 5.1. In Subsection 5.3 we will compute the Weyl groups for all triples listed in Table 5.1. Subsection 5.4 completes the proof of Theorem 5.1.2. Finally, in Subsection 5.5 we compute the Weyl groups of affine homogeneous spaces (without restrictions on the rank) more or less explicitly.

5.2.
Classification of W -quasiessential triples. In this subsection g is a simple Lie algebra.
(1) An admissible triple (g, h, V ) is W -quasiessential iff it is listed in Table 5.1.
(2) Let (g, h, V ) be a W -quasiessential triple. If h is simple and S (α) g T * (G * H V ), then α is a long root. If h is not simple, then S (α) g T * (G * H V ) for all roots α. To prove Proposition 5.2.1 we need some technical results. Let us introduce some notation. Let H be a reductive algebraic group, s a subalgebra of h isomorphic to sl 2 . We denote by S s the h-stratum consisting of s and the direct sum of two copies of the two-dimensional irreducible s-module.
Remark 5.2.2. Let H be a reductive subgroup of G, U an H-module, (s, V ) a g-stratum. Then (s, V ) g G * H U iff there exists g ∈ G such that Ad(g)s ⊂ h and (Ad(g)s, V ) h U (the algebra Ad(g)s is represented in V via the isomorphism Ad(g −1 ) : Ad(g)s → s). Conversely, if (s, V ) h U, then (s, V ) g G * H U. Now we recall the definition of the Dynkin index ( [D]). Let h be a simple subalgebra of g. We fix an invariant non-degenerate symmetric bilinear form K g on g such that K g (α ∨ , α ∨ ) = 2 for a root α ∈ ∆(g) of the maximal length. Analogously define a form K h on h. The Dynkin index of the embedding ι : h ֒→ g is, by definition, K g (ι(x), ι(x))/K h (x, x) (the last fraction does not depend on the choice of x ∈ h such that K h (x, x) = 0). For brevity, we denote the Dynkin index of ι by i(h, g). It turns out that i(h, g) is a positive integer (see [D]).
The following lemma seems to be standard.
Lemma 5.2.3. Let h be a simple Lie algebra and s a subalgebra of h isomorphic to sl 2 . Then the following conditions are equivalent: (1) ι(s, h) = 1.
Proof. Clearly, (2) ⇒ (1). Let us check (1) ⇒ (2). Choose the standard basis e, h, f in s. We may assume that h lies in the fixed Cartan subalgebra of h. It follows from the representation theory of sl 2 that π, h is an integer for any weight π of h. Thus h ∈ Q ∨ , where Q ∨ denotes the dual root lattice. Since i(s, h) = i(h (α) , h) for a long root α ∈ ∆(h), the lengths of h, α ∨ ∈ Q ∨ coincide. It can be seen directly from the constructions of the root systems, that all elements h ∈ Q ∨ with (h, h) = (α ∨ , α ∨ ) are short dual roots, see [Bou]. Thus we may assume that h = α ∨ . It follows from the standard theorems on the conjugacy of sl 2 -triples, see, for example, [McG], that s and h (α) are conjugate.
In Table 5.2 we list all simple subalgebras of index 1 in simple classical Lie algebras. g h sl n , n 2 sl k , k n sl n , n 4 sp 2k , 2 k n/2 so n , n 7 so k , k n, k = 4 so n , n 7 sl diag k , k n/2 so n , n 8 sp diag 2k , 2 k n/4 so n , n 7 G 2 so n , n 9 spin 7 sp 2n , n 2 sp 2k , k n Lemma 5.2.4. Let α ∈ ∆(g) and h be a reductive subalgebra of g containing g (α) . Suppose that there is no proper ideal of h containing g (α) . Then (1) If α is a long root, then h is simple and ι(g (α) , h) = ι(h, g) = 1.
Proof. Since g (α) ⊂ [h, h], we see that h is semisimple. Let h 1 , h 3 be simple Lie algebras and h 2 a semisimple Lie algebra, h 1 ⊂ h 2 ⊂ h 3 . Let h 2 = h 1 2 ⊕ . . . ⊕ h k 2 be the decomposition of h 2 into the direct sum of simple ideals and ι i , i = 1, . . . , k, the composition of the embedding h 1 ֒→ h 2 and the projection h 2 → h i 2 . It is shown in [D] that This implies assertion 1.
Proceed to assertion 2b. Note that the representation of g (α) in the tautological g-module V is the sum of the trivial 2l −2-dimensional and the 3-dimensional irreducible representations. Since g (α) is not contained in a proper ideal of h, we see that the Note that the representation of g (α) in V i is nontrivial because the projection of g (α) to h i is nontrivial. From the equality dim (V 1 (α) = 3 and the Clebsh-Gordan formula it follows Proceed to assertion 2c. It follows from assertion 2a that i(h 1 , g) = i(h 2 , g) = 1. Then h 1 = sp 2m , h 2 = sp 2k , see Table 5.2.
Let us recall the notion of the index of a module over a simple Lie algebra, see [AEV]. Let h be a simple Lie algebra, U an h-module. We define a symmetric invariant bilinear form (·, ·) U on h by (x, y) U = tr U (xy). The form (·, ·) U is nondegenerate whenever U is nontrivial. By the index of U we mean the fraction (x,y) U (x,y) h . Since h is simple, the last fraction does not depend on the choice of x, y ∈ h with (x, y) h = 0. We denote the index of U by l h (U).
The numbers k h for all simple Lie algebras are given in Table 5.3.  8 48 72 120 36 16 Let H be a reductive group, V an H-module. There exists the s.g.p. for the action H : V , see [PV], Theorem 7.2. Recall that the action H : V is called stable if an orbit in general position is closed. In this case the s.g.p. is reductive.
Lemma 5.2.6. Let h be a semisimple Lie algebra, s a subalgebra of h 1 isomorphic to sl 2 , V an h-module, V 1 ⊂ V an H-submodule such that the action H : V 1 is stable. Let H 1 denote the s.g.p. for the action H : Let v 1 ∈ V 1 be such that Hv 1 is closed and H v 1 = H 1 . The slice module at v 1 is the direct sum of V /V 1 and a trivial H v 1 -module. The claim of the lemma follows from the Luna slice theorem and Remark 5.2.2.
Proof of Proposition 5.2.1. The proof is in three steps.
Step 1. Here we suppose that h is simple. α) for some α ∈ ∆(g). Suppose S s h V ⊕ V * ⊕ g/h for some subalgebra s ⊂ h isomorphic to sl 2 . From assertion 2 of Lemma 5.2.5 it follows that Thence l h (g/h) < 1. In Section 3 of [Lo1] it was shown that i(h, g) = 1. Equivalently, i(s, h) = i(s, g). All simple subalgebras h ⊂ g with l h (g/h) < 1 are listed in [Lo1], Table 5. The list (up to Aut(g)-conjugacy) is presented in Table 5.4. In column 4 the nontrivial part of the representation of h in g/h is given. By τ we denote the tautological representation of a classical Lie algebra.
Step 2. It remains to consider the situation when h is not simple. On this step we assume that h possesses the following property (2) s ∼ G g (α) , where α is a short root of g.
(3) i(s i , h i ) = 1, where s i denotes the projection of s to h i , i = 1, 2. It follows from Lemma 5.2.4 that a subalgebra s ⊂ h satisfying (1)-(3) is defined uniquely up to Int(h)-conjugacy.
It follows from the choice of s that i(s, h) = 2 whence (h, h) g = 2k g . So (5.5) and (5.3) are equivalent.
From (5.6) it follows that m 1 + m 2 = n, l h i (V ) 1 2m i +2 . Therefore the h-module g/h is the tensor product of the tautological sp 2m 1 -and sp 2m 2 -modules and, see the table from [AEV] is the direct sum of tautological Sp(2m i )-modules. Thus if m 1 < m 2 , then H 1 acts trivially on V and m 2 = m 1 + 1. This shows that a W -quasiessential triple (g, h, V ) satisfying (*) is one of NN 32,33 of Table 5.1.
Let us show that the triples NN32,33 satisfy (*). Put V 0 = g/h. This is an orthogonal whence stable, see, for instance, [Lu1], H-module. The s.s.g.p. h 0 for the H-module V 0 is the direct sum of m 1 copies of sl 2 embedded diagonally into h 1 ⊕ h 2 and m 2 − m 1 copies of sl 2 embedded into h 2 (see [E2]). We are done by Lemma 5.2.6 applied to Finally, we see that S (h (α) 2 ) then for h 2 we take the ideal of h acting on V trivially), where α is a long root of h 2 .
Step 3. It remains to show that any W -quasiessential triple (g, h, V ) satisfies (*) provided h is not simple. Assume the converse. Let us show that (**) S s h g/h ⊕ V ⊕ V * for some subalgebra s ⊂ h, s ∼ = sl 2 , not lying in any simple ideal of h. Indeed, let h 1 be a simple ideal of h and s 1 a subalgebra of h 1 isomorphic to sl 2 such that S s 1 h g/h ⊕ V ⊕ V * , or equivalently, S s 1 h 1 g/h ⊕ V ⊕ V * . Then (g, h i , V /V h i ) is one of the triples NN 1-31 from Table 5.1 and s 1 ∼ G g (α) , where α is a long root in ∆(g). If (**) does not hold, then, by step 1, S (α) g T * (G * H V ) implies that α is a long root. Since h is not simple, we see that (g, h, V ) is not W -quasiessential. So (**) is checked.
It follows from assertion 2a of Lemma 5.2.5 that there is a subalgebra s ⊂ h and simple ideals h 1 , h 2 of h such that (g, h 1 ⊕ h 2 , V /V h 1 ⊕h 2 ) satisfies condition (*) of step 2. But in this case n g (h 1 ⊕ h 2 ) = h 1 ⊕ h 2 whence h = h 1 ⊕ h 2 . Contradiction.

Computation of Weyl groups.
In this subsection we compute the groups W (g, h, V ) for triples (g, h, V ) listed in Table 5.1.
At first, we reduce the problem to the case when C[V ] H = C. Let (g, h, V ) be one of the triples from Table 5.1. Let H 0 denote the unit component of the principal isotropy subgroup for the action H : V .
In Table 5.6 we present all triples that are reduced from nonreduced triples from Table 5.1. We use the same notation as in Table 5.1. In column 2 we give the number of (g, h, V ) in Table 5.1 and, in some cases, restrictions on the algebra g or the h-module V . It turns out that (g, h 0 , V 0 ) is again contained in Table 5.1.
Lemma 5.3.1. All triples (g, h, V ) from Table 5.1 such that C[V ] H = C are listed in the second column of Table 5.6. The reduced triple for (g, h, V ) coincides with (g, h 0 , V 0 ).
Proof. The triples NN 32,33 are reduced. So we may assume that h is simple. The list of all simple linear groups with a dense orbit is well-known, see, for example, [Vi1]. It follows from the classification of the paper [E1] that the s.s.g.p's for the H-module g/h ⊕ V ⊕ V * are simple for all triples (g, h, V ), except NN1,2,4,20. By Popov's criterion, see [Po], the action H : V are stable whenever the s.s.g.p is reductive. For the remaining four triples the reduced triples are easily found case by case.
Below in this subsection we consider only reduced triples (g, h, V ) from Table 5.1.
, is a torus naturally acting on X = G * H V by G-automorphisms. Namely, we define the action morphism by (t 1 , For some triples (g, h, V ) the inequality rk e G (X) < rk G holds. These triples are presented in Table 5.7. The matrix in column 3 of the first row is the diagonal matrix, whose k + 1-th entry is (n − 1)x and the other entries are −x .
Table 5.7: The projections of l 0 e G,X to g N (g, h, V ) the projection of l 0 e G,X to g from assertion 3 of Proposition 3.2.1 that W (g, h, V ) = W (g, h 0 ), where h 0 denotes the s.s.g.p. of the h-module V . For h 0 one may take the following subalgebra Put m := t + g (ε 2 −ε 3 ) , q = b − + m. It is seen directly that s := h 0 ∩ m ∼ = sl 2 is a Levi subalgebra in h 0 and R u (h 0 ) ⊂ R u (q). The nontrivial part of the s-module q/h 0 is twodimensional. Applying Corollary 3.3.11 to Q and M, we see that s ε 2 −ε 3 ∈ W (g, h 0 ).

5.4.
Completing the proof of Theorem 5.1.2. To prove Theorem 5.1.2 it remains (1) To check that any W -quasiessential triple is W -essential and vice-versa.
(2) To show how to determine a W -essential part of a given admissible triple. The following lemma solves the both problems.
(2) Let (g, h, V ) be an admissible triple and h 0 a minimal ideal of h such that the condi- Assertion 2 of Lemma 5.4.1 implies that any W -essential triple is W -quasiessential. Moreover, a triple (g, h 0 , V /V h 0 ) from assertion 2 is either the trivial triple (g, 0, 0) or one of the triples from Table 5.1. Inspecting Table 5.1, we note that an ideal h 0 ⊂ h is determined uniquely. Therefore the triple (g, h 0 , V /V h 0 ) is a W -essential part of (g, h, V ).
Proof. Suppose an admissible triple (g, h, V ) is W -quasiessential but not W -essential. Then, by the definition of a W -essential triple, there exists a proper ideal h 0 ⊂ h such that W (g, h, V ) = W (g, h 0 , V ). On the other hand, there is α ∈ ∆(g) such that S (α) g T * (G * H V ), S (α) g T * (G * H 0 V ). Corollary 3.3.22 implies that s wα ∈ W (g, h 0 , V ) for all w ∈ W . By the computation of the previous subsection, there is w ∈ W such that s wα ∈ W (g, h, V ). Contradiction. Proceed to assertion 2. The triple (g, h 0 , V /V h 0 ) is W -quasiessential, thanks to the minimality condition for h 0 . If h 0 = {0}, then, by Corollary 3.3.22, W (g, h, V ) = W (g). If (g, h 0 , V /V h 0 ) is one of the triples NN28,32,33 from Table 5.1, then h = h 0 whence the equality of the Weyl groups. So we may assume that g = G 2 and that S (α) g T * (G * H V ) iff α is a long root. By Corollary 3.3.22, W (g, h, V ) contains all reflections corresponding to short roots. Assume, at first, that g is a classical Lie algebra. By Proposition 3.3.17, W (g, h, V ) is one of the subgroups listed in Table 3.3. But any of those groups containing all reflections corresponding to short roots is maximal among all proper subgroups of W (g) generated by reflections.
. It remains to consider the case g = F 4 . Analogously to the classical case, W (g, h, V ) contains a reflection corresponding to a long root. Otherwise, W (g, h, V ) is not large in W (g) (in Definition 3.3.16 take α = ε 1 − ε 2 , β = ε 2 − ε 3 ). Any subgroup in W (g) containing all reflections corresponding to short roots and some reflection corresponding to a long root is maximal. 5.5. Weyl groups of affine homogeneous spaces. In this subsection g is a reductive Lie algebra, h its reductive subalgebra.
Before stating the main result we make the following remark. The Weyl group W (g, h) depends only on the commutant [h, h], see Corollary 3.3.13. Therefore we may assume that h is semisimple. The strategy of the computation is similar to that above: we compute the Weyl groups only for so called W -essential subalgebras h ⊂ g and then show how to reduce the general case to this one.
Definition 5.5.1. A semisimple subalgebra h ⊂ g is called W -essential if any ideal h 0 ⊂ h such that a(g, h) = a(g, h 0 ), W (g, h) = W (g, h 0 ) coincides with h.
Proposition 5.5.2. Let g be a semisimple Lie algebra and h its semisimple subalgebra. Then the following claims hold.
(1) Let h be a W -essential subalgebra of g. Then the pair (g, h) is contained either in Table 5.1 or in Table 3.1. In the latter case W (g, h) coincides with the group indicated in Table 4.2.
(2) There is a unique minimal ideal h W −ess ⊂ h such that a(g, h W −ess ) = a(g, h), W (g, h W −ess ) = W (g, h) It coincides with a maximal ideal of h that is a W -essential subalgebra of g.
Proof. The case a(g, h) = t is clear. By definition, any a-essential semisimple subalgebra of g is W -essential. Now let h be a W -essential subalgebra in g with a(g, h) t. Let Denote by F the subgroup of G consisting of all elements leaving invariant the distinguished Borel and Cartan subalgebras in g. It follows from assertion 1 of Proposition 4.1.1 that X = G/H. By Theorem 3.3.10, W (g, h) = W (g, h) ⋋ F/T . From assertion 4 of Proposition 4.1.3 it follows that Table 4.1 and using Theorem 5.1.2, we see that for any subalgebra h 1 ⊂ g such that h ess is an ideal h 1 and a(g, h 1 ) = t the equality W (g, h) = W (g) holds. Thence W (g, h) = W (g, h ess ) = W (g) ⋋ F/T . In particular, if h ess = {0}, then h W −ess = h ess . 6. Computation of the Weyl groups for homogeneous spaces 6.1. Introduction. In this section we complete the computation of the groups W (g, h).
At first, note that Corollary 3.3.13 allows one to reduce the computation to the case when a maximal reductive subalgebra of h is semisimple. In this case G/H is quasiaffine, thanks to Sukhanov's criterium, see [Su]. Now recall that we have reduced the computation of W (g, h) to the case when a(g, h) = t (Theorem 3.3.10 and results of Section 4) and g is simple (Proposition 3.3.15).
The computation of W (g, h) for g = so 5 , G 2 is carried out in Subsection 6.2. Basically, it is built upon Proposition 3.3.1. In the beginning of Subsection 6.3 we show how to compute the space t W (g,h) . It turns out that for g of type A, D, E the group W (g, h) is determined uniquely by t W (g,h) . This is the main result of Subsection 6.3, its proof is based on the restrictions on W (g, h) obtained in Proposition 3.3.23. The computation for algebras of type C, B, F and rank bigger than 2 is carried out in Subsections 6.4-6.6. Remark 6.3.2 allows us to make an additional restriction on h: we require t W (g,h) = 0.
Remark 6.1.1. The computation of W (g, h) for all groups of rank 2 essentially does not use Proposition 3.3.23 and results of [Wa] on the classification of all wonderful varieties of rank 2. Let us note that the classification of all spherical varieties of rank 2 is not very difficult. The reductions described above allow to reduce the computation of the Weyl groups of varieties of rank 2 to the case when G itself has rank 2.
6.2. Types B 2 , G 2 . In this subsection G = SO 5 , G 2 and h denotes a subalgebra of g such that a(g, h) = t and a maximal reductive subalgebra h 0 of h is semisimple.
Suppose that R u (q) ⊂ h for some parabolic subalgebra q ⊂ g. We may assume that q is antistandard, let m be its standard Levi subalgebra. Then, by Corollary 3.3.14, W (g, h) = W (m, m ∩ h). Below in this subsection we suppose that h does not contain the unipotent radical of a parabolic.
Proposition 6.2.1. Let g = so 5 and a subalgebra h ⊂ g satisfy the above assumptions. Then the following conditions are equivalent.
Proof. We have already computed the groups W (g, h) for reductive subalgebras h. Suppose now that W (g, h) = W (g). Put Firstly, we consider the case when H is unipotent. Then dim h 4. Since h does not contain the unipotent radical of a parabolic, we see that dim h 3 and if dim h = 3, then h ∼ G h := Span C (e −ε 1 −ε 2 , e −ε 1 , e ε 2 −ε 1 + e −ε 2 ). The closure of G h in Gr(g, 3) contains G R u (q i ), i = 1, 2. Applying Corollary 3.3.14 and Proposition 3.3.1, we get s α 1 , s α 2 ∈ W (g, h).
If dim h = 1, then there exists a unipotent subalgebra h of dimension 2 containing h. Indeed n u 1 (h) = h, where u 1 denotes the maximal unipotent subalgebra of g containing h.
Let us check that W (g, h) = W (g) whenever h is a unipotent subalgebra of dimension 2. By Proposition 3.3.1, we may assume that Gh ⊂ Gr(g, 2) is closed or, equivalently, n g (h) is a parabolic subalgebra of g. From this one easily deduces that h ∼ G h 0 := Span C (e −ε 1 −ε 2 , e −ε 1 ). But h 0 is conjugate to a subalgebra in h whence, thanks to Corollary 3.2.2, W (g, h) = W (g).
It remains to consider the case h 0 ∼ = sl 2 . It is easily deduced from a(g, h) = t that h is semisimple.
Proposition 6.2.2. Suppose g = G 2 and h satisfies the above assumptions. Then the following conditions are equivalent: (1) W (g, h) = W (g).
The subalgebra h 1 coincides with the s.s.g.p. for the action G : G * A 2 C 3 and the equality for W (g, h 1 ) is deduced from Theorem 5.1.2. Now put h 2 := h 2 + Cα ∨ 1 . Clearly, h 2 ⊂ n g (h 2 ). The subalgebra h 2 is tamely contained in q 2 . Applying results of Subsection 3.2, we obtain a(g, h 2 ) = C(α 1 + α 2 ). Since h 2 does not contain a maximal unipotent subalgebra, we have W (g, h 2 ) = {1}. The required equality for W (g, h 2 ) stems from Corollary 3.3.13. Now suppose (1) holds. Firstly, we show that h 0 ∼ G g (α 2 ) implies h 0 = {0}. Assume the converse. Since a(g, h) = t, we have h 0 = A 2 . So if h 0 = 0, then h 0 contains an ideal conjugate to g (α 1 ) . We easily check that h is conjugate to a subalgebra in g (α 1 ) ⊕ g (3α 1 +2α 2 ) . The Weyl group of the latter coincides with W (g). So h 0 = {0}, in other words, h is unipotent. We As in the proof of Proposition 6.2.2, we get W (g, h) = W (g). Now let dim h 4. Let us check that W (g, h) = W (g). By Proposition 3.3.1, it is enough to check the claim only when n g (h) is a parabolic subalgebra of g. But in this case So it remains to consider the case h 0 = g (α 2 ) . By Theorem 5.1.2, h is not reductive. We may assume that h is tamely contained in q 2 . Since h ⊂ g (α 2 ) ⊕ g (α 2 +2α 1 ) and R u (q 2 ) ⊂ h, we get the required list of subalgebras h.
6.3. Types A, D, E. In this subsection g is a simple Lie algebra and h its subalgebra such that a(g, h) = t and a maximal reductive subalgebra of h is simple.
Firstly, we show how to compute the space t W (g,h) . Proposition 6.3.1. Let g, h be such as above and z a commutative reductive subalgebra of n g (h)/h containing all semisimple elements of z(n g (h)/h). Let h denote the inverse image of z in n g (h) under the natural epimorphism n g (h) ։ n g (h)/h. Then a(g, h) is the orthogonal complement to t W (g,h) in t. Remark 6.3.2. We use the notation of Proposition 6.3.1 and for z take the ideal of z(n g (h)/h) consisting of all semisimple elements. Suppose that a(g, h) = t. Then we can reduce the computation of W (g, h) = W (g, h) to the computation of W (g, h), where rk[g, g] < rk g, rk G/H = rk G, as follows.

Proof
There is a quasiaffine homogeneous space X of G such that there is a G-equivariant principal C × -bundle X → G/ H, see Subsection 7.2 for details. The stable subalgebra h 1 of X is naturally identified with h. By Proposition 3.3.2, W (g, h) ∼ = W ( g, h 1 ). Then we apply Theorem 3.3.10 to X.
However, for some algebras g the group W (g, h) is uniquely determined by t W (g,h) , see below, so the reduction described above is unnecessary. Proposition 6.3.3. Let g = sl n , n 3, so 2n , n 4 or E l , l = 6, 7, 8, and h be as above.
6.4. Type C l , l > 2. In this subsection g is a simple Lie algebra and h its subalgebra such that a(g, h) = t and a maximal reductive subalgebra of h is semisimple.
Proof. For two s-modules α−β) , Q := B − M, X := G/H. Let Z 0 denote a rational quotient for the action R u (Q) : X. Modifying Z 0 if necessary, one may assume that M acts regularly on Z 0 . By Proposition 8.2 from [Lo2], W M,Z 0 is generated by s α , s β . Thanks to assertion 4 of Proposition 3.2.1, W M,Z 0 = W (m, m z ) for a general point z ∈ Z 0 . From Proposition 6.2.1 it follows that m z is reductive and s := [m z , m z ] ∼ M m (α+β) . By [Lo2], Lemma 7.12, the action R u (Q) : G/H is locally free. Since Z 0 is a rational quotient for the action R u (Q) : G/H, we see that q x ∼ Q g (α+β) for x ∈ X in general position. Thence there is x ∈ X such that s ⊂ g x and T x X ∼ q/s. We may assume that x = eH. It follows that g/h ∼ q/s ∼ R u (q) + (C 2 ) ⊕2 . Note that g ∼ R u (q) ⊕2 ⊕ (C 2 ) ⊕2 ⊕ s whence (2).
Lemma 6.4.2. Let g, h satisfy the assumptions of the previous proposition and s ⊂ h possesses properties 1,2. Further, let Q be a parabolic subgroup of G such that h is tamely contained in q and M a Levi subgroup of Q such that h 0 := m ∩ h is a maximal reductive subalgebra of h containing s. Then there are simple ideals h 1 ⊂ h 0 , m 1 ⊂ m 0 such that s ⊂ h 1 ⊂ m 1 and S s h 1 m 1 /h 1 ⊕ R u (q)/ R u (h) ⊕ (R u (q)/ R u (h)) * . Proof. We preserve the notation of Proposition 6.4.1. Note that α + β is a long root. Therefore s is contained in a simple ideal m 1 ⊂ m and i(s, m 1 ) = 1, by (5.1). Moreover, s ∼ M m (γ) for some root γ ∈ ∆(m) (see Lemma 5.2.3). Analogously, s is contained in some simple ideal h 1 ⊂ h 0 . Since s ⊂ m 1 , we get h 1 ⊂ m 1 . Property 2 of Proposition 6.4.1 implies that the nontrivial parts of the s-modules (m/h 0 ) ⊕ V ⊕ V * and (h 1 /s) ⊕ (C 2 ) ⊕2 coincide, where V := R u (q)/ R u (h). Thence, see the proof of Lemma 5.2.5, Besides, we remark that the h 1 -module (m/h 0 ) ⊕ V ⊕ V * is orthogonal and its s.s.g.p is trivial. The latter follows easily from the observation that the s.s.g.p. for the action M : M * H (m/h ⊕ V ⊕ V * ) is trivial. One can list all such modules by using table from [AEV] and results from [E1]. It turns out that the nontrivial part of such a module is presented in Table 5.5.
Our algorithm for computing W (g, h) for g ∼ = sp 2n is based on the following proposition.
Proposition 6.4.3. Suppose g ∼ = sp 2n , n > 2. Assume, in addition, that t W (g,h) = 0. Further, suppose h is tamely contained in an antistandard parabolic subalgebra q ⊂ g and m ∩ h is a maximal reductive subalgebra of h, where m is the standard Levi subalgebra of q. Then the following conditions are equivalent.
Proof. Let us introduce some notation. Let I be a finite set. By V we denote the vector space with the basis ε i , i ∈ I. Let A I (resp., B I , C I , D I ) be a linear group acting on V as the Weyl group of type A #I (resp. B #I , C #I , D #I ).
Since W (g, h) is generated by reflections and t W (g,h) = 0, we see that there exists a partition I 1 , . . . , I k of {1, 2, . . . , n} such that W (g, h) = k i=1 Γ i , where Γ j = D I j or C I j . From Proposition 3.3.23 it follows that ε i +ε j ∈ Π(g, h) unless (i, j) = (n, n), (n, n−1) or (n−1, n). Thus k = 1. (2)⇒(1) stems from Corollary 3.3.11 applied to q and results of the previous section. Now assume that W (g, h) = W (g) whence W (g, h) = D n . The assumptions of Proposition 6.4.1 are satisfied. By that proposition, h 0 contains a subalgebra conjugate to g (αn) . Since α n is a long root, any subalgebra of m that is G-conjugate to g (αn) is contained in an ideal m 1 of the form g (α k ,...,αn) . Now (1)⇒(2) follows from Lemma 6.4.2 and assertion 1 of Proposition 5.2.1. 6.5. Type B l , l > 2. In this subsection, if otherwise is not indicated, g = so 2n+1 , n 3, and h ⊂ g is such that a(g, h) = t, t W (g,h) = {0} and the maximal reductive subalgebra of h is semisimple.
Proof. As in the proof of Proposition 6.4.3, there is a partition I 1 , . . . , I l of {1, 2 . . . , n} such that W (g, h) = l j=1 Γ j , where Γ j = B I j or D I j . Since ε i + ε j , ε k ∈ Π(g, h) for arbitrary i, j and k < n − 1 (Proposition 3.3.23), we get l = 2, Γ j = B I j . It follows that n ∈ I 1 , n − 1 ∈ I 2 . By Proposition 3.3.23, if k, k + 1 ∈ I j , j = 1, 2 for 1 < k < n, then k − 1 or k + 2 lies in I j . This implies the claim of the present proposition.
We remark that, by definition, 1 ∈ I.
Lemma 6.5.2. Let g ∼ = so 2n+1 , n 3 or F 4 . Suppose W (g, h) is a maximal proper subgroup generated by reflections in W (g) (for g = so 2n+1 we have checked this in Lemma 6.5.1). Let q, m, h 1 , m 1 be such as in Lemma 6.4.2. Then: (1) Subalgebras h 1 ⊂ h, m 1 ⊂ m are determined uniquely.
(2) Let h denote the subalgebra in h generated by h 1 and [h 1 , R u (h)]. Then W (g, h) = W (g, h).
Proof. It follows from assertion 1 of Proposition 5.2.1 that m 1 = m 2 for two different pairs (m 1 , h 1 ), (m 2 , h 2 ) satisfying the assumptions of Lemma 6.4.2. Thanks to Corollary 3.2.2, W (g, h) ⊂ W (g, h). The subalgebra h is tamely contatined in q. Applying Corollary 3.3.11 to q, we see that W (m 2 ) ⊂ W (g, h), W (m 2 ) ⊂ W (g, h). Since W (g, h) is maximal, this proves assertion 1. Analogously, W (m 1 ) ⊂ W (g, h), since the nontrivial parts of the h 1 -modules R u (q)/ R u (h), R u (q)/ R u (h) are the same. This proves assertions 2.
For indices i 1 , i 2 we write i 1 ∼ i 2 whenever i 1 , i 2 ∈ I or i 1 , i 2 ∈ J.
Summarizing results obtained above in this subsection, we see that it remains to compute W (g, h) under the following assumptions on (g, h): (a) a(g, h) = t, t W (g,h) = 0.
Proposition 6.5.4. Let h, h 1 , q, m 1 be such as in (a)-(e). The inequality W (g, h) = W (g) holds iff h is G-conjugate to a subalgebra from the following list: 1) h consists of all block matrices of the following form The sizes of blocks (from left to right and from top to bottom): k − 1, 1, 7, 1, k − 1; x ′ ij denotes the matrix −I p x T ij I q , where I p = (δ i+j,p+1 ) p i,j=1 , I q = (δ i+j,q+1 ) q i,j=1 , x 33 , x 51 are arbitrary elements of G 2 ֒→ so 7 , so k . In this case I = {n − 1}.
2) h consists of all block matrices of the following form  Here the sizes of blocks are k, n − k, 1, n − k, k; x 22 , x 51 are arbitrary elements of sl n−k , so k . We have I = {k + 2, k + 4, . . . , k + 2i, . . .}.
3) h consists of all block matrices of the following form: The sizes of blocks (from left to right and from top to bottom): k − 1, 1, 3, 1, 3, 1, k − 1; ι denotes an isomorphism of sl 3 -modules C 3 and 2 C 3 * ; x 33 , x 71 are arbitrary elements of sl 3 and so k−1 . Here I = {n − 1}. 4) h consists of all block matrices of the following form  Here the sizes of blocks are k, n − k, 1, n − k, k and n − k is even; x 22 , x 51 are arbitrary elements of sp n−k , so k , while x 32 lies in the nontrivial part of the sp n−k -module 2 C n−k . We have I = {k + 2, k + 4, . . . , k + 2i, . . .}. Here I = {k + 2, k + 4, . . . , n}.

5)
h consists of all block matrix of the following form The sizes of blocks are k − 1, 1, r, 1, r, 1, k − 1; x ′ ij have the same meaning as in (6.1); ι denotes an isomorphism of sp r -modules C r , C * r ; x 71 ∈ so k−1 , x 33 ∈ sp r , x 53 lies in the nontrivial part of the sp r -module 2 C r * , and ω is an appropriate nonzero element in ( 2 C r * ) h 1 . We have I = {k + 1, k + 3, . . . , n − 1}.
Note that in all five cases the equality t W (g,h) = 0 can be checked by using Proposition 6.3.1 (compare with the proof of Proposition 6.6.2 below).
The case (sl r , sl r , 2 C r * ⊕ C r ). Let us check that this triple cannot occur. Since r > 3, we see that R u (q) contains a unique m 1 -submodule isomorphic to 2 C r * . It is spanned by e −ε i −ε j , k + 1 i k + r. On the other hand, any m 1 -submodule of R u (q) isomorphic to C * r is contained in h. Vectors of the form [ξ, η], where ξ, η lie in the C r * -isotypical component of the m 1 -module R u (q), generate a submodule of R u (q) isomorphic to 2 C r * .
The case (sl r , sl r , 2 C r * ⊕ C r * ), r 3. Let V 0 denote the m 1 -submodule of R u (q) spanned by e −ε i −ε j , k + 1 i < j k + r. The isotypical component of type C r * in the m 1 -module R u (q) is the direct sum of Span C (e −ε i ), Span C (e −ε i −ε j ), j k, Span C (e −ε i ±ε j ), j > k+r, where i ranges from k+1 to k+r. If k+r = n−1, then, conjugating h by an element from G (αn) , one may assume that Span But if r > 3, then V 0 is a unique submodule of R u (q) isomorphic to 2 C r * . So k + r = n provided r > 3. Further, if k + r = n and V 0 ⊂ h, then Span C (e −ε i −ε j , i = k + 1, k + r) is the isotypical component of type C r * in the h 1 -module R u (h). Otherwise, the commutators of elements from the isotypical C r * -component in R u (h) generate V 0 .
One easily verifies that (6.11) The other commutators of V 1 , V ± 2i , V ± 3 vanish. Let us check that e −ε 1 −ε 2 ∈ R u (h). Indeed, according to (6.11), one may consider the commutator on V 2 as a nondegenerate skew-symmetric form. The intersection R u (h) ∩ V 2 is 6-dimensional whence not isotropic.
There are seven T 0 -orbits of submodules U ⊂ V 1 ⊕ V + 22 ⊕ V + 3 . To the T 0 -orbit of U we associate the triple (x 1 , x 2 , x 3 ) consisting of 0 and 1 by the following rule: V 1 ⊂ U (resp. V + 22 ⊂ U, V + 3 ⊂ U) iff x 1 = 0 (resp., x 2 = 0, x 3 = 0). Abusing the notation, we write h (x 1 ,x 2 ,x 3 ) instead of h U . The T 0 -orbit corresponding to (x 1 , x 2 , x 3 ) lies in the closure of that corresponding to (y 1 , y 2 , y 3 ) iff x i y i . Consider the seven possible variants case by case.
The case (1, 1, 1). Since h (1,1,0) ∈ T 0 h, we have Π(g, h) = {α 1 , α 2 , α 2 + α 3 , α 3 + α 4 }. The case (0, 1, 1). Analogously to the case (1, 1, 0), one can show that t W (g,h) = {0}. Since h ∈ T 0 h (1,1,1) , we see that Π(g, h) = {α 1 , α 2 , α 2 + α 3 , α 3 + α 4 }. Case 2. Here X = G * H V is an affine homogeneous vector bundle and π : G * H V → G/H is the natural projection. Applying the algorithm of case 1 to G/H, we compute the space a(g, h) and find a point x in the distinguished component of (G/H) L 0 G,G/H . Applying the following algorithm to the group L 0 := L • 0 G,G/H and the module V = π −1 (x), we compute a L 0 ,V . Algorithm 7.1.1. Let G be a connected reductive algebraic group and V a G-module. Put G 0 = G, V 0 = V . Assume that we have already constructed a pair (G i i is a maximal unipotent subalgebra of g i normalized by T and opposite to b i and the superscript 0 means the annihilator. Put G i+1 = Z G i (α). The group G i+1 is connected and L 0 G i ,V i = L 0 G i+1 ,V i+1 . Note that rk[g i+1 , g i+1 ] rk[g i , g i ] with the equality iff α ∈ V [g i ,g i ] . Thus if [g i , g i ] acts non-trivially on V , then we may assume that rk[g i+1 , g i+1 ] < rk[g i , g i ]. So V = V [g k ,g k ] for some k. Here L 0 G,V = L 0 G k ,V k coincides with the unit component of the inefficiency kernel for the action G k : V k .