V Grines
email: grines@vmk.unn.ru.

F Laudenbach
email: francois.laudenbach@univ-nantes.fr.

O Pochinka
email: olga-pochinka@yandex.ru.

Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds

, where gradient-like case was considered. We introduce a kind of Morse-Lyapunov function, called dynamically ordered, which fits well dynamics of diffeomorphism. The paper is devoted to finding conditions to the existence of such an energy function, that is, a function whose set of critical points coincides with the non-wandering set of the considered diffeomorphism. We show that the necessary and sufficient conditions to the existence of a dynamically ordered energy function reduces to the type of embedding of one-dimensional attractors and repellers, each of them is a union of zero-and one-dimensional unstable (stable) manifolds of periodic orbits of a given Morse-Smale diffeomorphism on a closed 3-manifold.

Introduction and formulation of the results

Let M be a closed jrientable 3-manifold and f : M → M be a preserving orientation Morse-Smale diffeomorphism, that is: its nonwandering set Ω f is finite, hence consists of periodic points; f is hyperbolic along Ω f and the stable and unstable manifolds have transverse intersections.

Definition 1 A Morse function ϕ : M → R is said to be a Lyapunov function for f if:

1) ϕ(f (x)) < ϕ(x) for every x ∈ Ω f ; 2) ϕ(f (x)) = ϕ(x) for every x ∈ Ω f .

Sometimes we shall speak of a Lyapunov function even when it is only defined on some domain N ⊂ M, meaning that the above conditions 1), 2) hold only for points x ∈ N such that f (x) ∈ N.

Let us recall that a C 2 -smooth function ϕ : M → R is called a Morse function if all its critical points are non-degenerate. Using ideas from [START_REF] Smale | On gradient dynamical systems[END_REF] it is possible to construct Lyapunov functions for f . For this aim, one considers the suspension of f , a 4-dimensional manifold which is fibered over the circle and is endowed with a Morse-Smale vector field X transverse to the fibration. The method introduced by S. Smale in [START_REF] Smale | On gradient dynamical systems[END_REF] for constructing Lyapunov function for Morse-Smale vector fields without closed orbits can be extended to suspension and allows one to construct a Lyapunov function Φ for X. The restriction of Φ to the base fibre, identified with M, is a Lyapunov function for diffeomorphism f .

According to statement 5 below the periodic points of f are critical points of its Lyapunov function ϕ and the index of ϕ at p ∈ Ω f equals the dimension of W u p . At the same time any periodic point p is a maximum of the restriction of ϕ to the unstable manifold W u p and a minimum of its restriction to the stable manifold W s p . If these extrema are non-degenerate then the invariant manifolds of p are transversal to all regular level sets of ϕ in some neighborhood of the point p. This local property is useful for the construction of a (global) Lyapunov function. Next definition was introduced in [START_REF] Grines | Self-indexing energy function for Morse-Smale diffeomorphisms on 3-manifolds[END_REF].

Definition 2 A Lyapunov function ϕ : M → R for the Morse-Smale diffeomorphism f : M → M is called a Morse-Lyapunov function if every periodic point p is a non-degenerate maximum (resp. minimum) of the restriction of ϕ to the unstable (resp. stable) manifold W u p (resp. W s p ).

Among the Lyapunov functions of f those which are Morse-Lyapunov form a generic set in the C ∞ -topology (see [START_REF] Grines | Self-indexing energy function for Morse-Smale diffeomorphisms on 3-manifolds[END_REF], theorem 1). In general, a Morse-Lyapunov function may have critical points which are not periodic points of f .

Definition 3 A Morse-Lyapunov function ϕ is called an energy function for a

Morse-Smale diffeomorphism f if the set of critical points of ϕ coincides with Ω f . D. Pixton in [START_REF] Pixton | Wild unstable manifolds[END_REF] established an existence of energy function for any Morse-Smale diffeomorphisms given on closed smooth two-dimensional manifold and constructed a gradient-like diffeomorphism on S 3 which has no energy function. According to S. Smale [START_REF] Smale | On gradient dynamical systems[END_REF] any Morse-Smale flow without closed trajectories (gradient-like flow) given on closed smooth manifold of any dimension possesses by an energy function. Thus there is an actual problem a finding of conditions to an existence of energy function for Morse-Smale diffeomorphisms. First step in this direction was made by the authors for gradient-like diffeomorphisms in the papers [START_REF] Grines | An energy function for gradientlike diffeomorphisms on 3-manifolds[END_REF], [START_REF] Grines | Self-indexing energy function for Morse-Smale diffeomorphisms on 3-manifolds[END_REF].

Let us recall that a Morse-Smale diffeomorphism f : M → M is called gradient-like if for any pair of periodic points x, y (x = y) the condition W u x ∩ W s y = ∅ implies dim W s x < dim W s y . It follows from the definition that a Morse-Smale diffeomorphism is gradient-like if and only if there are no heteroclinic points that is, intersection points of two-dimensional and onedimensional invariant manifolds of different saddle points. Notice that twodimensional invariant manifolds of different saddle points of a gradient-like diffeomorphism may have a non-empty intersection along the so-called heteroclinic curves (see figure 1).

In [START_REF] Grines | An energy function for gradientlike diffeomorphisms on 3-manifolds[END_REF], [START_REF] Grines | Self-indexing energy function for Morse-Smale diffeomorphisms on 3-manifolds[END_REF] (Theorem 4) we gave necessary and sufficient conditions to the existence of a self-indexing energy function for a Morse-Smale diffeomorphism f : M → M and showed that a non gradient-like diffeomorphisms do not possess a self-indexing energy function. Here self-indexing means ϕ(p) = dim W u p for every point p ∈ Ω f .

In the present paper we introduce the notion of dynamically ordered Morse-Lyapunov function for an arbitrary Morse-Smale diffeomorphism on 3-manifold. By using the above-mentioned arguments, such a function will exist easily if it is not required to be an energy function. We will show that the existence of such an energy function depends on how the one-dimensional attractors (and repellers) embed into the ambient manifold. More details are given below.

Let f : M → M be a Morse-Smale diffeomorphism. Following to S. Smale we introduce a partial order ≺ on the set of periodic orbits of f in the following way:

O p ≺ O r ⇐⇒ W s Op ∩ W u Or = ∅ .
This definition means intuitively that all wandering points flow down along unstable manifolds to smaller elements. A sequence of different periodic orbits

O p = O p 0 , O p 1 , . . . , O p k = O r (k ≥ 1) such that O p 0 ≺ O p 1 ≺ . . . ≺ O p k is called a chain of length k connecting O r to O p .
The maximum length of such chains is called, by J. Palis in [START_REF] Palis | On Morse-Smale dynamical systems[END_REF], the behaviour of O r relative to O p and is denoted by beh

(O r |O p ). For completeness it is assumed beh(O r |O p ) = 0 if W u Or ∩ W s Op = ∅.
For each q ∈ {0, 1, 2, 3}, denote Ω q the subset of periodic points r such that dim W u r = q and denote k q the number of periodic orbits in the set Ω q . Set k f = k 0 + k 1 + k 2 + k 3 the number of all periodic orbits. For each periodic orbit O r we set q Or = dim W u

Or and b Or = max

p∈Ω 0 {beh(O r |O p )}.
Definition 4 A numbering of the periodic orbits: Notice that any dynamical numbering preserves the partial order ≺ (that is,O i ≺ O j implies i ≤ j). Indeed, as the intersection

O 1 , . . . , O k f is called dy- namical if it satisfies to following conditions: 1) if q O i < q O j then i < j; 2) if q O i = q O j and b O i < b O j then i < j.
W s O i ∩ W u O j is transverse, the condition O i ≺ O j implies the inequality dim W s O i + dim W u O j -3 ≥ 0. Then 3 -q O i + q O j -3 ≥ 0 and, hence, q O i ≤ q O j . If q O i < q O j then i < j due to 1). If q O i = q O j then the condition O i ≺ O j implies or O i = O j and, hence, i = j, either b O i < b O j and, hence, i < j due to 2).
On figure 1 it is represented a phase portrait of a Morse-Smale diffeomorphism f : S 3 → S 3 with Ω f consisting of fixed points which are dynamically numerated.

Definition 5

Let O 1 , . . . , O k f be a dynamical numbering of the periodic orbits of f . A Morse-Lyapunov function ϕ for f is said to be dynamically ordered when ϕ(O i ) = i for i ∈ {1, . . . , k f }.

For each i = 1, . . . , k 1 , set A i = i j=1 W u O j . It is known that the set A i is an attractor, that is it has a trapping neighborhood M i , which is a compact set such that f (M i ) ⊂ int M i (M i is f -compressed) and k≥0 f k (M i ) = A i (see, for
example, [START_REF] Robinson | Dynamical Systems: stability, symbolic dynamics, and chaos[END_REF]). Denote by r i the number of saddles, by s i the number of sinks and by c i the number of connected components in A i . Set

g i = c i + r i -s i .
Let us recall that a smooth compact orientable three-dimensional manifold is called a handlebody of a genus g ≥ 0 if it is diffeomorphic to a manifold which is obtained from a closed 3-ball by an orientation reversing identification of g pairs of pairwise disjoint closed 2-discs in its boundary. The boundary of such a handlebody is an orientable surface of genus g.

Definition 6

A trapping neighborhood M i of the attractor A i is called a handle if: M i consists of c i handlebodies. The sum g M i of genera of all connected components of M i is called genus of the handle neighborhood.

Notice that for each i = 1, . . . , k 0 , the number g i equals 0, the attractor A i is zero-dimensional (as it consists of the sink orbits) and has a handle neighborhood M i of genus g i = 0 consisting of c i pairwise disjoint 3-balls (it follows, for example, from statement 6 below). For each i = k 0 + 1, . . . , k 1 the attractor A i contains an one-dimensional connected component, therefor we will say (taking liberty) that A i is one-dimensional attractor.

Proposition 1 Each one-dimensional attractor

A i of Morse-Smale diffeomor- phism f : M → M has a handle trapping neighborhood M i with genus g M i ≥ g i . Definition 7 A handle neighborhood M i of one-dimensional attractor A i is said to be tight if: 1) g M i = g i ; 2) W s σ ∩ M i consists of exactly one two-dimensional closed disc for each saddle point σ ∈ O i .
A one-dimensional attractor A i possessing tight trapping neighborhood M i is said to be tightly embedded.

By definition a repeller for f is an attractor for f -1 . Moreover, dynamical numbering of the orbits O 1 , . . . , O k f of a diffeomorphism f induces a dynamical numbering of the orbits Õ1 , . . . , Õk f of a diffeomorphism f -1 next way: Õi = O k f -i . Then a one-dimensional repeller for f is said to be tightly embedded if it is such an attractor for f -1 according to induced numbering.

Notice that the property for a one-dimensional attractor (repeller) to be tightly embedded gives a topological information about the embedding of the unstable manifolds of its saddle periodic points. In the example which was constructed by D. Pixton in [START_REF] Pixton | Wild unstable manifolds[END_REF] the unique one-dimensional attractor A 3 = cl W u σ has the following property: g 3 = 0 but any 3-ball around cl W u σ intersects W s σ at more than one 2-disc (see figure 2, where are drawn the phase portrait of Pixton's diffeomorpfism f and a 3-ball). Hence, this one-dimensional attractor is not tightly embedded.

Our main results are the following theorems.

A i is called strongly tight if M i \ A i is diffeomorphic to ∂M i × (0, 1].
A onedimensional attractor A i possessing a strongly tight trapping neighborhood M i is said to be strongly tightly embedded.

Theorem 2 Let f be a Morse-Smale diffeomorphism on a closed 3-manifold M. If all one-dimensional attractors and repellers of f are strongly tightly embedded, then f possesses a dynamically ordered energy function.

Notice that the condition in the last theorem is not necessary. For example in section 5 of paper [START_REF] Grines | Self-indexing energy function for Morse-Smale diffeomorphisms on 3-manifolds[END_REF] there was constructed a diffeomorphism on S 2 × S 1 possessing a dynamically ordered energy function, but whose one-dimensional attractor and repeller are not strongly tightly embedded.

The next theorem states a criterion for the existence of some dynamically ordered energy function for a Morse-Smale diffeomorphism without heteroclinic curves given on S 3 . Methods from [START_REF] Bonatti | Classification of the Morse-Smale diffeomorphisms with the finite set of heteroclinic orbits on 3-manifolds[END_REF] for realizing Morse-Smale diffeomorphisms show that this class is not empty. Moreover, it contains diffeomorphisms with chains of intersections of saddle invariant manifolds of arbitrary length (see figure 3, where it is represented a phase portrait of a diffeomorphism from the class under consideration). The criterion is based on paper [START_REF] Bonatti | Three-manifolds admitting Morse-Smale diffeomorphisms without heteroclinic curves[END_REF], where it is specified interrelation between topology of the ambient 3-manifold M and structure of the non-wandering set of a Morse-Smale diffeomorphism without heteroclinic curves given on M. In particular, for any diffeomorphism of S 3 without heteroclinic curves, the number r of all saddles and the number l of all sinks and sources satisfy the equality r = l -2 (see statement 7 below). This equality implies that g i = 0 for any one-dimensional attractor A i . Thus, tightly embedded attractor A i is strong tightly embedded. Applying results of theorems 1, 2 we get next criterion.

Theorem 3 A Morse-Smale diffeomorphism f : S 3 → S 3 without heteroclinic curves possesses a dynamically ordered energy function if and only if each onedimensional attractor and repeller is tightly embedded. ACKNOWLEDGMENTS V. Z. Grines and O. V. Pochinka acknowledge the support of the grant of government of Russian Federation no. 11.G34.31.0039 for partial financial support. F. Laudenbach is supported by the French program ANR "Floer power".

Auxiliary facts

In this section, we recall some statements that we need in the proof and give references.

Statement 1 (λ-lemma, [8]). Let p be a hyperbolic fixed point of a diffeomorphism

f : M n → M n , dim W u p = ℓ, 0 < ℓ < n, Let B u ⊂ W u p and B s ⊂ W s p
be small ℓ-disc and (nℓ)-disc respectively centered at p. Let V := B u × B s be their product in a chart about p. Let B be an ℓ-disc transverse to W s p at x. Then, for any ε > 0, there exists a positive integer k 0 such that the connected component of

f k (B) ∩ V containing f k (x) is ε-C 1 -close to B u for each k ≥ k 0 .
Definition 9 Let F be a compact smooth surface properly embedded in a 3manifold W (that is, ∂F ⊂ ∂W ). Then F is called compressible in one from two following cases:

1) there is a non contractible simple closed curve c ⊂ int F and smoothly embedded 2

-disk D ⊂ int W such that D ∩ F = ∂D = c; 2) there is a 3-ball B ⊂ int W such that F = ∂B. The surface F is said to be incompressible 1 in W if it is not compressible in W .
Statement 2 ([13], corollary 3.2) Let S g be an orientable surface of genus g ≥ 1 and let F be an incompressible orientable surface properly embedded in

S g × [0, 1] such that ∂F ⊂ S g × {1}. Then there is a surface F 1 ⊂ S g × {1} which is homeomorphic to F , such that ∂F = ∂F 1 and F ∪ F 1 bounds domain ∆ in S g × [0, 1] such that cl∆ is homeomorphic to F × [0, 1]
, where cl(•) stands for the closure.

A particular case of statement 2 is the following fact.

Corollary 1 ([5], theorem 3.3) Let S g be a closed orientable surface of genus g ≥ 1 and let surface F ⊂ int(S g × [0, 1]) be a closed surface which has genus g and does not bound a domain in

S g × [0, 1]. Then F is incompressible in S g × [0, 1] and the closure of each connected component of S g × [0, 1] \ F is homeomorphic to S g × [0, 1]. Proof: According to the preceding statement, it is sufficient to check that F is incompressible in S g × [0, 1]. If F is compressible,
there exists some incompressible surface F ′ whose genus g ′ is less than g and which still does not bound a domain in S g × [0, 1]. So F ′ is not a sphere and g ′ > 0. As F ′ is incompressible, the preceding statement tells us that

F ′ is diffeomorphic to S g . Contradiction. ⋄ Statement 3 ([4], lemma 3.3) For any Morse-Smale diffeomorphism f : M → M we have that 1 + |Ω 1 | -|Ω 0 | = 1 + |Ω 2 | -|Ω 3 |, where | • | stands for the cardinality.
Statement 4 ([7], theorem 5.2) Let M n be a closed manifold, ϕ : M n → R be a Morse function, C q be the number of all its critical points with index q, β q (M n ) be the q-th Betti number and χ(M n ) be the Euler characteristic.

Then

β q (M n ) ≤ C q and χ(M n ) = n q=0 (-1) q C q . Statement 5 Let ϕ : M n → R be a Lyapunov function for a Morse-Smale diffeomorphism f : M n → M n . Then 1) -ϕ is Lyapunov function for f -1 ; 2) if p is a periodic point of f then ϕ(x) < ϕ(p) for every x ∈ W u p \ p and ϕ(x) > ϕ(p) for every x ∈ W s p \ p; 3) if p is a periodic point of f then p is a critical point of ϕ whose index is dim W u p . Statement 6 ([4], lemma 2.2) Let f : M n → M n be a Morse-Smale diffeomor-
phism on an n-dimensional manifold and let O be a periodic orbit. For p ∈ O, set q = dim W u p . Then, there is some neighborhood U and an energy function

ϕ : U → R for f such that (W u p ∩ U) ⊂ Ox 1 . . . x q , (W s p ∩ U) ⊂ Ox q+1 . . . x n for Morse coordinates x 1 , . . . , x n of ϕ near p. Statement 7 ([2]
, theorem) Let M be a three-dimensional closed, connected, orientable manifold. Let f : M → M be any Morse-Smale diffeomorphism without heteroclinic curves whose non-wandering set consists of r saddles and l nodes (sinks and sources). Then m = r-l+2 2 is non negative integer and following facts hold:

1) if m = 0, then M is the 3-sphere;

2) if m > 0, then M is the connected sum of m copies S 2 × S 1 . Conversely, for any non negative integers r, l, m such that m = r-l+2 2 is non negative integer, there exists 3-manifold M and some Morse-Smale diffeomorphism f : M → M with following properties:

a) M is 3-sphere if m = 0 and M is the connected sum of m copies S 2 × S 1 if m > 0;
b) the non-wandering set of f consists of r saddles and l sinks and sources, the wandering set of f has no heteroclinic curves.

On one-dimensional attractors Proof of proposition 1

For each i = k 0 +1, . . . , k 1 let us prove the existence of handle neighborhood M i for one-dimensional attractor A i with g M i ≥ g i . Proof: According to statement 6, there is a neighborhood

U A k 0 ⊂ W s A k 0 of zero-dimensional attractor A k 0 and an energy function ϕ A k 0 : U A k 0 → R for f such that ϕ A k 0 (A k 0 ) = 0 and for small ε > 0 each connected component of set
M k 0 = ϕ -1 A k 0 ((-∞, ε]) reads {(x 1 , x 2 , x 3 ) ∈ U A k 0 : x 2 1 + x 2 2 + x 2 3 ≤ ε} in local coordinates x 1 , x 2 , x 3 .
Then M k 0 is trapping neighborhood of zero-dimensional attractor A k 0 , which is a union of c k 0 pairwise disjoint 3-balls. By induction on i = k 0 + 1, . . . , k 1 we construct a handle trapping neighborhood M i for A i .

Let i = k 0 + 1. Set S k 0 = ∂M k 0 . Without loss of generality we can suppose that S k 0 intersects W u O k 0 +1 transversely; let n k 0 be the number of intersection points. Set

V k 0 = W s Ω f ∩A k 0 \ A k 0 . Then the quotient Vk 0 = V k 0 /f is made of the cobordism M k 0 \ int f (M k 0 )
by gluing its boundaries by f . Hence, Vk 0 is smooth orientable 3-manifold without boundary and natural projection

p k 0 : V k 0 → Vk 0 is cover. Then p k 0 (W u O k 0 +1
) is a pair of knots which intersects p k 0 (S k 0 ) transversely at n k 0 points. Thus, there is a tubular neighborhood

T u k 0 +1 ⊂ Vk 0 of p k 0 (W u O k 0 +1 ) such that T u k 0 +1 ∩ p k 0 (S k 0 ) consists of n k 0 2-discs. Set T u k 0 +1 = p -1 k 0 ( T u k 0 +1
). According to the λ-lemma (see statement 1),

T u k 0 +1 ∪ W s O k 0 +1 is a neighborhood of O k 0 +1
. According to statement 6, there are some neighborhood

U O k 0 +1 ⊂ (T u k 0 +1 ∪ W s O k 0 +1 ) of O k 0 +1 and an energy function ϕ O k 0 +1 : U O k 0 +1 → R for f with ϕ O k 0 +1 (O k 0 +1 ) = 0. When ε > 0 is small enough, each connected component of H k 0 +1 = ϕ -1 O k 0 +1 ((-∞, ε]) reads {(x 1 , x 2 , x 3 ) ∈ U O k 0 +1 : -x 2 1 + x 2 2 + x 2 3 ≤ ε} in local coordinates x 1 , x 2 , x 3 . According to the λ-lemma, when k ∈ N is large enough, f -k (S k 0 ) intersects both H k 0 +1 and f (H k 0 +1
), its intersection with these domains consists of n k 0

2-discs and f (H

k 0 +1 ) \ int f -k (M k 0 ) ⊂ int H i (see figure 4). Thus, M k 0 +1 = f -k (M k 0 ) ∪ H k 0 +1 is a union of handlebodies, as it is obtained from the union of 3-balls f -k (M k 0 ) by gluing one-handles H k 0 +1 \ int f -k (M k 0 ). Let us show that f (M k 0 +1 ) ⊂ int M k 0 +1 . Indeed, it is true for a point x ∈ f -k (M k 0 ) as f -k (M k 0 ) is f -compressed and it is true for a point x ∈ (H k 0 +1 \ f -k (M k 0 )) as f (H k 0 +1 ) \ int f -k (M k 0 ) ⊂ int H k 0 +1 .
Let us prove the equality

k≥0 f k (M k 0 +1 ) = A k 0 +1 . As A k 0 +1 ⊂ M k 0 +1 and f k (A k 0 +1 ) = A k 0 +1 for k ∈ Z then A k 0 +1 ⊂ k≥0 f k (M k 0 +1 ). Let us set A ′ k 0 +1 = k≥0 f k (M k 0 +1 ) and show that A ′ k 0 +1 = A k 0 +1 . Assume contrary: there is a point x ∈ (A ′ k 0 +1 \ A k 0 +1 ). Due to theorem 2.3 in [11] there is a point p ∈ (Ω f \ A k 0 +1 ) such that x ∈ W u p . As the set A ′ k 0 +1 is closed and invariant then cl (O x ) ⊂ A ′ k 0 +1 and, hence, p ∈ A ′ k 0 +1
. This is a contradiction with the fact

A ′ k 0 +1 ⊂ W s A k 0 +1 ∩Ω f .
Recall that we denote by r i the number of saddles, by s i the number of sinks, by c i the number of connected components of the attractor A i and set

g i = c i + r i -s i . By the construction M k 0 +1 consists of c k 0 +1 3-balls with 1-handles 2 . Denote by g M k 0 +1 the sum of genera of connected components of M k 0 +1 . Let us show that g M k 0 +1 ≥ g k 0 +1 .
In denotation above, the number of points in the orbit

O k 0 +1 equals r k 0 +1 - r k 0 . As A k 0 +1 = A k 0 ∪ W u O k 0 +1 and cl W u O k 0 +1 \ W u O k 0 +1 ⊂ A k 0 then c k 0 +1 ≤ c k 0 . Denote by l k 0 +1 the number of connected components of the set H k 0 +1 \ int f -k (M k 0 )
. By the construction each of them is 1-handle and removing of (l k 0 +1 -(c k 0c k 0 +1 )) 1-handles from M k 0 +1 gives the set with the same c k 0 +1 connected components. Then the sum of genera g M k 0

of M k 0 can be calculate by formula

g M k 0 = g M k 0 +1 -(l k 0 +1 -(c k 0 -c k 0 +1 )). By the construction l k 0 +1 ≥ (r k 0 +1 -r k 0 ), hence g M k 0 ≤ g M k 0 +1 -(r k 0 +1 -r k 0 -(c k 0 -c k 0 +1 )) and g M k 0 +1 ≥ g M k 0 + r k 0 +1 -r k 0 -c k 0 + c k 0 +1 . As g M k 0 = g k 0 then g M k 0 +1 ≥ c k 0 + r k 0 -s k 0 + r k 0 +1 -r k 0 -c k 0 + c k 0 +1 = c k 0 +1 + r k 0 +1 -s k 0 . As s k 0 = s k 0 +1 then g M k 0 +1 ≥ g k 0 +1 .
A smoothing of the set M k 0 +1 is the required handle trapping neighborhood. Assuming that handle neighborhood for attractor A i-1 already constructed, repeating construction above (changing k 0 by i-1), we construct f -compressed set

M i = f -k (M i-1 ) ∪ H i , being a union of handle neighborhood f -k (M i-1 ) with 1-handles H i \ int f -k (M i-1 ). It is similar proved that M i is required handle neighborhood. ⋄ Proposition 2
The one-dimensional attractor A k 1 is connected.

Proof: Firstly, let us prove that any trapping neighborhood M k 1 of A k 1 is connected. Let us assume the contrary: M k 1 is a union of pairwise disjoint closed sets B 1 and B 2 . As M k 1 is f -compressed then without loss of generality we can suppose that f

(B i ) ⊂ int B i , i = 1, 2. By construction, U 1 = k>0 f -k (int B 1 ), U 2 = k>0 f -k (int B 2 ) are pairwise disjoint open sets and U 1 ∪ U 2 = W s Ω 0 ∪Ω 1 . On the other hand W s Ω 0 ∪Ω 1 = M \ W s Ω 2 ∪Ω 3 and, hence, W s Ω 0 ∪Ω 1 is connected as dim M = 3 and dim W s Ω 2 ∪Ω 3 ≤ 1. This is a contradiction. Thus A i is connected as intersection of nested connected compact sets M i ⊃ f (M i ) ⊃ . . . ⊃ f k (M i ) ⊃ . . .. ⋄
4 Necessary condition for existence of dynamically ordered energy function

Proof of theorem 1

Let us prove that if a Morse-Smale diffeomorphism f : M → M has a dynamically ordered energy function, then its one-dimensional attractors and repellers are tightly embedded. Proof: Notice that f and f -1 possess dynamically ordered energy functions simultaneously. Indeed, if ϕ : M → R is such function for f then -ϕ : M → R is an energy function for f -1 (see statement 5) and φ = k f + 1ϕ : M → R is dynamically ordered energy function for f -1 . Therefore, it is enough to prove the fact for attractors.

Let ϕ : M → R be a dynamically ordered energy function for f :

M → M, i = k 0 + 1, . . . , k 1 and M i = ϕ -1 ([1, i + ε i ]), ε i > 0.
It follows from properties of dynamically ordered energy function and statement 5 that any orbit O j with number j ≤ i belongs to M i . Due to statement 5,

W u O j ⊂ M i . Thus A i ⊂ M i . It follows from definition of Lyapunov function that f (M i ) ⊂ int M i . Similar to proposition 1 it is proved equality k≥0 f k (M i ) = A i . Thus, M i is trapping neighborhood of attractor A i .
Then M i has the same number of connected components as A i . Let us prove that there is

ε i > 0 such that M i is a tight neighborhood of A i .
As ϕ is a Morse-Lyapunov function then there is ε i > 0 such that W s σ ∩ M i consists of exactly one closed 2-disc for each saddle point σ ∈ O i . It follows from properties of dynamically ordered energy function and statement 5 that ϕ| M i has exactly r i +s i critical points, among of them s i points have index 0 and r i points have index 1. According to Morse theory, M i is a union of of s i 3-balls with gluing of r i 1-handles and hence is a union of c i handlebodies. Denote by g M i the sum of genus of handlebodies from M i . According to statement 4, χ(M i ) = s ir i . It follows from Morse theory that M i has the homotopy type of a cellular complex consisting of s i zero-dimensional and r i one-dimensional cells, then -g M i + c i = m ir i or g M i = g i . ⋄

5 Construction of a dynamically ordered energy function for f

Now f is a Morse-Smale diffeomorphism on a closed 3-manifold M and its one-dimensional attractors and repellers are strongly tightly embedded. Construction of a dynamically ordered energy function for f is based on technical lemmas of next section.

Recall that, by assumption of theorem 2, each one-dimensional attractor A i , i = k 0 + 1, . . . , k 1 is strongly tightly embedded and, hence, has a handle neighborhood M i of genus g i such that M i \ A i is homeomorphic to S i × (0, 1], where S i = ∂M i , and for each point

σ ∈ O i the intersection W s σ ∩ M i consists of exactly one 2-disk. Set D i = M i ∩ W s O i .
According to statement 6, for each zero-dimensional attractor A i , i = 1, . . . , k 0 , there is a handle neighborhood of genus g i = 0 which is a union of c i 3-balls, we will denote it by M i and set

S i = ∂M i . For i = 1, . . . , k 1 set K i = M i \ int f (M i ), N i = W s A i ∩Ω f and V i = N i \ A i .

According to ring hypothesis and corollary 1,

K i is diffeomorphic to S i × [0, 1]. As V i = n∈Z f n (K i ) then V i is diffeomorphic to S i × R.

Extension of Lyapunov functions

Definition 10 Let D be a subset of M which is diffeomorphic to product S × [0, 1] for some (possibly non connected) surface S. Then D is said to be an (f, S)-compressed product when there is a diffeomorphism g : D → S × [0, 1] such that g -1 (S ×{t}) bounds an f -compressed domain in M for any t ∈ [0, 1].

Proposition 3 Let D be an (f, S)-compressed product. Then for any values

d 0 < d 1 there is an energy function ϕ D : D → R for f | D such that ϕ D (g -1 (S × {0})) = d 0 and ϕ D (g -1 (S × {1})) = d 1 . Proof: The desired function ϕ D : D → R is defined by formula ϕ D (x) = d 0 + t(d 1 -d 0 ) for x ∈ g -1 (S × {t}), t ∈ [0, 1]. ⋄ Lemma 1 Let i ∈ {1, .
. . , k 1 } and P i , Q i be handle neighborhoods of genus g i of the attractor A i . If there is a dynamically ordered energy function ϕ Q i :

Q i → R for f with S Q i = ∂Q i
as a level set then there is a dynamically ordered energy function ϕ P i : P i → R for f with S P i = ∂P i as a level set.

Proof: We follow to scheme of the proof of lemma 4.2 from [START_REF] Grines | An energy function for gradientlike diffeomorphisms on 3-manifolds[END_REF]. Give some remarks.

Without loss of generality we assume that Q i ⊂ int P i (in the opposite case, instead pair

(Q i , ϕ Q i ) we can use pair (f n (Q i ), ϕ f n (Q i ) ), where f n (Q i ) ⊂ int P i and ϕ f n (Q i ) = ϕ Q i f -n ).
As V i is diffeomorphic to S i × R then, according to ring hypothesis and corollary 1, G i = P i \ int Q i is a product. As handle neighborhoods f n (Q i ) and f n (P i ) contain the attractor A i for each n ∈ Z then the surfaces f n (S Q i ) and f n (S P i ) do not bound domains in V i and, hence, are incompressible due to corollary 1. Now let us construct the function ϕ P i , for this aim we consider two cases: 1)

S P i ∩ ( n>0 f -n (S Q i )) = ∅ and 2) S P i ∩ ( n>0 f -n (S Q i )) = ∅.
In case 1), let m be the first positive integer such that f m (P

i ) ⊂ int Q i . If m = 1, then G i is (f, S i )-

compressed product and proposition 3 yields the required function as extension of the function ϕ

Q i to G i .
If m > 1, the surfaces f (S P i ), f 2 (S P i ), . . . , f m-1 (S P i ) are mutually "parallel", that is: two by two they bound a product cobordism (according to ring hypothesis and corollary 1). Therefore they subdivide G i in (f, S i )-compressed products and, hence, proposition 3 yields the required function as extension of the function

ϕ Q i to f m-1 (P i )\int Q i , f m-2 (P i )\int f m-1 (P i ), . . . , P i \int f (P i ) in series.
In case 2), without loss of generality we may assume that S P i is transverse to n>0 f -n (S Q i ), which implies that there is a finite family C of intersection curves.

We are going to describe a process of decreasing the number of intersection curves by an isotopy of Q i among handle neighborhoods of genus g i possessing a dynamically ordered energy function for f which is constant on the boundary of the neighborhood.

Firstly we consider all intersection curves from C which are homotopic to zero in S P i . Let c be an innermost such curve. Then there is a disc δ ⊂ S P i which is bounded by c and such that int δ contains no curves from the family

C. As c ⊂ f -n (S Q i ) for some n and f -n (S Q i ) is incompressible in V i , then c bounds a disc d ⊂ f -n (S Q i ). Then 2-sphere δ ∪ d is embedded and bounds a 3-ball b in N i (when the component of f -n (S Q i ) containing d is a 2-sphere,

replace d by the complementary disc if necessary).

There are two occurrences: (a)

f n (b) ⊂ Q i and (b) f n (b) ⊂ f -1 (Q i ). We define Q ′ i as cl(Q i \ f n (b)) in the case (a) and Q i ∪ f n (b) in the case (b). The fact that c is an innermost curve implies f (Q i ) ⊂ Q ′ i ⊂ Q i in case (a) and Q i ⊂ Q ′ i ⊂ f -1 (Q i ) in case (b). In both cases there is a smooth approximation Qi of Q ′ i such that f (Q i ) ⊂ int Qi ⊂ Q i if (a), Q i ⊂ int Qi ⊂ f -1 (Q i ) if (b)
, and the number of intersection curves in

S P i ∩ ( n>0 f -n (∂ Qi )) is less than the cardinality of C. In case (a), ϕ f (Q i ) = ϕ Q i f -1 : f (Q i ) → R is a dynamically ordered energy
function which is constant on the boundary. Therefore Qi \ int f (Q i ) is an (f, S i )-compressed product and, hence, due to proposition 3, there is a similar function on Qi . Similarly in case (b), Qi is equipped with a dynamically ordered energy function which is constant on the boundary as Qi \ int Q i is an (f, S i )-compressed product. We will repeat this process until getting a handle neighborhood Qi of genus g i for the attractor A i such that S P i ∩ ( n>0 f -n (∂ Qi )) does not contain curves which are homotopic zero in S P i . Thus we may assume that

S P i ∩( n>0 f -n (S Q i ))
does not contain intersection curves which are homotopic to zero in S P i .

We denote by m the largest integer such that f m (S

P i )∩S Q i = ∅. Let F be a connected component of f m (S P i ) ∩ G i . We have ∂F ⊂ ∂S Q i . Let us show that F is incompressible in G i . Indeed, if δ is a disc in G i with boundary γ ⊂ F then γ bounds 2-disk δ ⊂ f m (S P i ) as f m (S P i ) is incompressible surface in V i .
By assumption the components of ∂F are not homotopic to zero in f m (S P i ), then ∂F ∩ δ = ∅ and, hence, δ ⊂ F .

Therefore, according to statement 2 there is some surface

F 1 ⊂ S Q i diffeo- morphic to F , with ∂F = ∂F 1 , and F ∪ F 1 bounds a domain ∆ in G i which, up to smoothing of the boundary, is diffeomorphic to F × [0, 1]. We then define Qi as Q i ∪ ∆ up to smoothing. By the choice of m, Qi is f -compessed as f (∆) ⊂ Q i .
As Qi is obtained by an isotopy supported in a neighborhood of ∆ from Q i then Qi \ int Q i is an (f, S i )-compressed product. Thus we get a dynamically ordered energy function on Qi with ∂ Qi as a level set. Arguing recursively, we are reduced to case 1).

⋄

Lemma 2 Let i ∈ {k 0 + 1, . . . , k 1 }, M i be strongly tight neighborhood of the attractor A i , D i = M i ∩ W s O i and N(D i ) ⊂ M i be a tubular neighborhood of D i such that N(D i )∩A i-1 = ∅ and the set P i-1 = M i \int N(D i ) is f -compressed.
Then P i-1 is a handle neighborhood of genus g i-1 for the attractor A i-1 .

Proof: Similar to proposition 1 it is proved the equality k≥0 f k (P i-1 ) = A i-1 .

Thus P i-1 is a trapping neighborhood of the attractor A i-1 and, hence, the set P i-1 consists of c i-1 connected components. Each of them is handlebody, as it is obtained from M i removing (r ir i-1 ) 1-handles, which are the set N(D i ). As in proof of proposition 1, a sum g P i-1 of genera P i-1 is calculated by formula

g P i-1 = g i -((r i -r i-1 ) -(c i-1 -c i )). Then g P i-1 = c i + r i -s i - ((r i -r i-1 ) -(c i-1 -c i )) = c i-1 + r i-1 -s i . As s i-1 = s i then g P i-1 = g i-1 . ⋄

Global construction

We divide a construction of the dynamically ordered energy function for f : M → M on steps.

Step 1. By induction on i = 1, . . . , k 1 let us prove the existence of a dynamically ordered energy function ϕ M i on M i of the attractor A i with level set S i .

For i = 1 the attractor A 1 coincides with sink orbit O 1 of the diffeomorphism f . According to statement 6 there is a neighborhood

U O 1 ⊂ M of the orbit O 1 , equipped by an energy function ϕ O 1 : U O 1 → R for f and such that ϕ O 1 (O 1 ) = 1. Moreover, for each connected component U ω , ω ∈ O 1 of the set U O 1 there are Morse coordinates (x 1 , x 2 , x 3 ) such that ϕ O 1 (x 1 , x 2 , x 3 ) = 1 + x 2 1 + x 2 2 + x 2 3 .
Then there is a value

ε 1 > 0 such that set Q 1 = ϕ -1 O 1 (1 + ε 1 ) consists of f -compressed union of c 1 3-balls.
Thus Q 1 is a handle neighborhood of genus 0 for the attractor A 1 . As g 1 = 0 then, according to lemma 1, there is a dynamically ordered energy function ϕ M 1 on the neighborhood M 1 for the attractor A 1 with level set S 1 .

Let, by assumption of the induction, there is a dynamically ordered energy function ϕ M i-1 on the neighborhood M i-1 of the attractor A i-1 with level set S i-1 . Let us construct the function ϕ M i . There is two cases: a

) i ≤ k 0 ; b) i > k 0 .
In the case a) the neighborhood M i consists of a handle neighborhood of genus 0 for the attractor A i-1 (denote it P i-1 ) and a trapping neighborhood of the orbit O i , consisting from 3-balls (denote it Q i ). By assumption of the induction and lemma 1 there is a dynamically ordered energy function ϕ P i-1 on P i-1 with level set ∂P i-1 . Similar to case i = 1 it is shown the existence of a dynamically ordered energy function ϕ Q i on Q i with level set ∂Q i . The required function ϕ M i is formed from ϕ P i-1 and ϕ Q i . b) we follow to scheme of proof from section 4.3 of paper [START_REF] Grines | An energy function for gradientlike diffeomorphisms on 3-manifolds[END_REF]. According to statement 6, the orbit O i has a neighborhood U O i ⊂ M endowed with an energy function ϕ O i :

U O i → R of f with ϕ O i (O i ) = i. Moreover, each connected component U σ , σ ∈ O i of U O i is endowed with Morse coordinates (x 1 , x 2 , x 3 ) such that ϕ O i (x 1 , x 2 , x 3 ) = i + x 2 1 + x 2 2 + x 2 3
, the x 1 -axis is contained in the unstable manifold and the (x 2 , x 3 )-plane is contained in the stable manifold of σ.

It follows from properties of strongly tight neighborhood M i and λ-lemma that there is a tubular neighborhood

N(D i ) ⊂ M i of D i = M i ∩ W s O i such that N(D i ) ∩ A i-1 = ∅, set P i-1 = M i \ int N(D i ) is f -compressed and surface ∂P i-1 transversal intersects each connected component of the set ϕ -1 O i (i)
\ O i at one closed curve. By lemma 2, the set P i-1 is a handle neighborhood of genus g i-1 for the attractor A i-1 . By assumption of induction and lemma 1 there is a dynamically ordered energy function ϕ P i-1 on P i-1 with level set ∂P i-1 .

For 5). Notice that P ε i = P i-1 and, hence,

ε i ∈ (0, 1), t ∈ [-ε i , ε i ] set P t = ϕ -1 P i-1 ([1, ϕ P i-1 (∂P i-1 ) -ε i + t]), H t = {x ∈ U O i : ϕ O i (x) ≤ i + t} and E ε i = (P ε i \ int P -ε i ) ∩ (H ε i \ int H -ε i ) (see figure
f (P ε i ) ⊂ int P ε i . As ϕ O i is a Lyapunov function for f | U O i then ϕ O i (f -1 (ϕ -1 O i (i) \ O i )) > i and, hence,
(H 0 \ O i ) ⊂ int f -1 (H 0 \ O i ).
This and conditions of choice of N(D i ) implies the existence of a value ε i with following properties:

(1) f (P ε i ) ⊂ int P -ε i ;

(2) for each t ∈ [-ε i , ε i ] surface ∂P t transversal intersects each connected component of the set ∂H t \ D i at one closed curve;

(

3) f -1 (E ε i ) ∩ H ε i = ∅. For t ∈ [-ε i , ε i ] set Q t = P t H t . By the construction the set Q t , t = 0 is f -compressed. Moreover, Q -ε i
after smoothing is a handle neighborhood of genus g i-1 of the attractor A i-1 and Q ε i after smoothing is strongly tight neighborhood of the attractor A i . By assumption of induction and lemma 1 there is a dynamically ordered energy function ϕ

Q -ε i on Q -ε i , which is a constant on ∂Q -ε i . As ϕ Q -ε i (A i-1 ) ≤ i -1 then, due to proposition 3, we can suppose that ϕ Q -ε i (Q -ε i ) = i -ε i . Define function ϕ Qε i : Q ε i → R on the set Q ε i by formula: ϕ Qε i (x) = ϕ Q -ε i (x), x ∈ Q -ε i ; i + t, x ∈ Q t .
Let us check that ϕ Qε i is a dynamically ordered energy function f , then the existence of required function ϕ M i : M i → R will follow from lemma 1.

Represent the set Q ε i as a union of subsets with pairwise disjoint interiors:

Q ε i = A ∪ B ∪ C, where A = Q -ε i , B = P ε i \ Q -ε i and C = Q ε i \ (P ε i ∪ Q -ε i ).
along trajectories of f . If x ∈ A then f (x) ∈ A and ϕ Qε i (f (x)) < ϕ Qε i (x), as ϕ Qε i | A is a Lyapunov function. If x ∈ B then, due to condition (1) of choice of ε i , f (x) ∈ and, hence, ϕ Qε i (x) > i -ε i , ϕ Qε i (f (x)) < i -ε i , therefor ϕ Qε i (f (x)) < ϕ Qε i (x). If x ∈ C then, due to condition (3) of choice of ε i , either f (x) ∈ A
and decreasing is proved as for x ∈ B, or f (x) ∈ C and decreasing follows from the fact that ϕ Qε i | C is a Lyapunov function.

Step 2. In this step we delive a construction similar to step 1 for diffeomorphism f -1 . For this aim we recall that dynamical numbering of the orbits O 1 , . . . , O k f of the diffeomorphism f induces dynamical numbering of the orbits Õ1 , . . . , Õk f of the diffeomorphism f -1 following way: Õi = O k f -i . Denote by Ãi the attractors of the diffeomorphism f -1 , by Mi thir neighborhood and by gi a number, defined by formula gi = ci + risi , wher ci the number of the connected components of the attractor Ãi , ri the number of the saddle pointd and si the number of the sink points of the diffeomorphism f -1 , belonging to Ãi .

Set k1 = k fk 1 and consider the attrator Ãk 1 for the diffeomorphism f -1 (which, recall, is a repeller for the diffeomorphism f ). Similar to step 1 we construct a a dynamically ordered energy function φ Mk 1 for f -1 on the neighborhood Mk 1 with level set Sk 1 = ∂ Mk 1 .

Step 3. In this step we show that set P k 1 = M \ int Mk 1 is a handle neighborhood of genus g k 1 of the attractor A k 1 , this implies the existence of the required function ϕ. Indeed, by lemma 1, the existence of a dynamically ordered energy function ϕ M k 1 on the neighborhood M k 1 of the attractor A k 1 implies the existence of a dynamically ordered energy function for ϕ P k 1 on P k 1 with level set ∂P k 1 . According to proposition 3 the function ϕ P k 1 we can construct such that ϕ P k 1 ( Sk 1 ) = k f + 1φ Mk 1 ( Sk 1 ). As ∂P k 1 = Sk 1 then required function ϕ is defined by formula ϕ(x) = ϕ P k 1 (x), x ∈ P k 1 ;

k f + 1 -φ Mk 1 (x), x ∈ Mk 1 .
Thus, let us prove that the set P k 1 = M \ int Mk 1 is a handle neighborhood of genus g k 1 of the attractor A k 1 . Set Ñk 1 = W s Choose n ∈ N such that f n (M k 1 ) ⊂ int P k 1 . Then, according to ring hypothesis and corollary 1, manifold K = P k 1 \ int f n (M k 1 ) is diffeomorphic to S k 1 × [0, 1]. By the construction f n (M k 1 ) is a handle neighborhood of genus g k 1 of the attractor A k 1 and P k 1 = f n (M k 1 ) ∪ K. This implies that P k 1 also is handle neighborhood of genus g k 1 of the attractor A k 1 .

6 Dynamically ordered energy function for diffeomorphisms on 3-sphere

In this section f : S 3 → S 3 is a Morse-Smale diffeomorphism without heteroclinic curve.

Proof of theorem 3

Let us prove that diffeomorphism f possesses a dynamically ordered energy function if and only if all its one-dimensional attractors and repellers are tightly embedded. Proof: The necessity of conditions of the theorem follows from 1, let us proof the sufficiency.

Let i = k 0 + 1, . . . , k 1 . Then A i is one-dimensional attractor, consisting of c i connected components, containing r i saddles, s i sinks and for which a number g i can be calculated by formula g i = c i + r is i . Firstly prove that g i = 0 for each i = k 0 + 1, . . . , k 1 .

We start from g k 1 . According to proposition 2, the attractor A k 1 is connected that is m k 1 = 1 and, hence, g Indeed, g i+1g i = (c i+1c i ) + (r i+1r i ) -(s i+1s i ). At the same time (c ic i+1 ) ≤ (r i+1r i ), s i+1 = s i and, hence, g i+1 ≥ g i .

Thus, g i = 0 for each i = k 0 + 1, . . . , k 1 . Then

K i = M i \ int f (M i ) is a union of 3-dimensional annulus S 2 × [0, 1]. As M i \ A i = k≥0 f k (M i ) then
M i \A i is diffeomorphic to ∂M i ×(0, 1]. Thus, the attractor A i is strongly tight embedded. Similar fact has place for repellers. The, according to theorem 2, f possesses a dynamically ordered energy function.

⋄

Figure 1 :

 1 Figure 1: Phase portrait of a Morse-Smale diffeomorphism f : S 3 → S 3 with dynamical numbering of the periodic orbits

Figure 2 :Theorem 1

 21 Figure 2: One-dimensional attractor of Pixton's example is not tightly embedded

Figure 3 :

 3 Figure 3: A Morse-Smale diffeomorphism without heteroclinic curves given on S 3

Figure 4 :

 4 Figure 4: Construction of a handle neighborhood for one-dimensional attractor

Figure 5 :

 5 Figure 5: Illustration to step 1

Ãk 1 ∩Ω f - 1 and Ṽk 1 = Ñk 1 \

 1111 Ãk 1 .Notice that the open sets V k 1 and Ṽk 1 are coincide, as both are obtained from M by removing of A k 1 and Ãk 1 . It follows from proof of proposition 2 that each of next setsA k 1 , Ãk 1 , M k 1 , Mk 1 , N k 1 , Ñk 1 , V k 1 , Ṽk 1 is connected. Then g k 1 = 1 + |Ω 1 | -|Ω 0 | and gk 1 = 1 + |Ω 2 | -|Ω 3 |.From statement 3 we get g k 1 = gk 1 . Thus the handle neighborhoods M k 1 and Mk 1 have the same genera and their boundaries S k 1 and Sk 1 belong to the set V k 1 , which is diffeomorphic to S k 1 × R.

k 1 = 1 +

 11 |Ω 1 | -|Ω 0 |. Due to statement 3, we have g k 1 = gk 1 , where gk 1 = 1 + |Ω 2 | -|Ω 3 |. According to statement 7, 2 + |Ω 1 ∪ Ω 2 | -|Ω 0 ∪ Ω 3 | = 0for any Morse-Smale diffeomorphism without heteroclinic curves on S 3 . Thus g k 1 + gk 1 = 0 and, hence, g k 1 = gk 1 = 0. Further let us show that g i ≤ g i+1 for each i = k 0 , . . . , k 1 -1.

  By the construction ϕ Qε i | A a dynamically ordered energy function for f , ϕ Qε i (∂A) = iε i , the function ϕ Qε i | B has no critical points and function ϕ Qε i | C coincides with function ϕ O i | C . Let us check decreasing property of ϕ Qε i

It is well known to topologists that a bicollared surface, different from sphere, is incompressible if and only if the inclusion S ֒→ W induces an injection of fundamental groups.

Recall that a

3-dimensional 1-handle is the product of an interval with a 2-disc.


Paolo Bravi

Jacopo Gandini

Andrea Maffei

Alessandro Ruzzi

NORMALITY AND NON-NORMALITY OF GROUP COMPACTIFICATIONS IN SIMPLE PROJECTIVE SPACES

If G is a complex simply connected semisimple algebraic group and if λ is a dominant weight, we consider the compactification X λ ⊂ P End(V (λ)) obtained as the closure of the G × Gorbit of the identity and we give necessary and sufficient conditions on the support of λ so that X λ is normal; as well, we give necessary and sufficient conditions on the support of λ so that X λ is smooth.

Introduction

Consider a semisimple simply connected algebraic group G over an algebraically closed field k of characteristic zero. If λ is a dominant weight (with respect to a fixed maximal torus T and a fixed Borel subgroup B ⊃ T ) and if V (λ) is the simple G-module of highest weight λ, then End V (λ) is a simple G × G-module. Let I λ ∈ End V (λ) be the identity map and consider the variety X λ ⊂ P End(V (λ)) given by the closure of the G × G-orbit of [I λ ]. In [START_REF] Kannan | Projective normality of the wonderful compactification of semisimple adjoint groups[END_REF], S. Kannan studied for which λ this variety is projectively normal, and this happens precisely when λ is minuscule. In [START_REF] Timashev | Equivariant compactifications of reductive groups[END_REF], D. Timashev studied the more general situation of a sum of irreducible representations, giving necessary and sufficient conditions for the normality and smoothness of these compactifications; however the conditions for normality are not completely explicit. In this paper we give an explicit characterization of the normality of X λ , which allows to simplify the conditions for the smoothness as well.

To explain our results we need some notation. Let ∆ be the set of simple roots (w.r.t. T ⊂ B) and identify ∆ with the vertices of the Dynkin diagram. Define the support of λ as the set Supp(λ) = {α ∈ ∆ : λ, α ∨ = 0}.

Theorem A (see Theorem 12). The variety X λ is normal if and only if λ satisfies the following property:

(⋆) For every non-simply laced connected component ∆ ′ of ∆, if Supp(λ) ∩ ∆ ′ contains a long root, then it contains also the short root which is adjacent to a long simple root.

In particular, if the Dynkin diagram of G is simply laced then X λ is normal, for all λ. In the paper we will prove the theorem in a more general form, for simple (i.e. with a unique closed orbit) linear projective compactifications of an adjoint group (see section 1.4). We will make use of the wonderful compactification of G ad , the adjoint group of G, and of the results on projective normality of these compactifications proved by S. Kannan in [START_REF] Kannan | Projective normality of the wonderful compactification of semisimple adjoint groups[END_REF]. These results hold in the more general case of a symmetric variety; however our method does not apply to this more general situation (see section 4.2).

Theorem B (see Theorem 21). The variety X λ is smooth if and only if λ satisfies property (⋆) of

Theorem A together with the following properties: i) For every connected component ∆ ′ of ∆, Supp(λ) ∩ ∆ ′ is connected and, in case it contains a unique element, then this element is an extreme of ∆ ′ ; ii) Supp(λ) contains every simple root which is adjacent to three other simple roots and at least two of the latter; iii) Every connected component of ∆ Supp(λ) is of type A.

Theorem B can be generalized to any simple and normal adjoint symmetric variety. Following a criterion of Q-factoriality for spherical varieties given by M. Brion in [START_REF] Brion | Variétés sphériques et théorie de Mori[END_REF], properties i) and ii) characterize the Q-factoriality of the normalization of X λ (see Proposition 19), while property iii) arises from a criterion of smoothness given by D. Timashev in [START_REF] Timashev | Equivariant compactifications of reductive groups[END_REF] in the case of a linear projective compactification of a reductive group.

As a corollary of Theorem B, we get that X λ is smooth if and only if its normalization is smooth. The paper is organized as follows. In the first section we introduce the wonderful compactification of G ad and the normalization of the variety X λ . In the second section we prove Theorem A, and in the third section Theorem B. In the last section we discuss some possible generalizations of our results.

1. Preliminaries 1.1. Notation. Recall that G is semisimple and simply connected. Fix a Borel subgroup B ⊂ G, a maximal torus T ⊂ B and let U denote the unipotent radical of B. Lie algebras of groups denoted by upper-case latin letters (G, U, L, . . .) will be denoted by the corresponding lower-case german letter (g, u, l, . . .). Let Φ denote the set of roots of G relatively to T and ∆ ⊂ Φ the basis associated to the choice of B. For all α ∈ ∆ let e α , α ∨ , f α be an sl(2)-triple of T -weights α, 0, -α. Let Λ denote the weight lattice of T and Λ + the subset of dominant weights. For all α ∈ ∆, denote by ω α the corresponding fundamental weight.

If λ ∈ Λ, recall the definition of its support:

Supp(λ) = {α ∈ ∆ : λ, α ∨ = 0}.
If I ⊂ ∆, define its border ∂I, its interior I • and its closure I as follows:

∂I = {α ∈ ∆ I : ∃ β ∈ I such that β, α ∨ = 0}; I • = I ∂(∆ I); I = I ∪ ∂I.
For λ ∈ Λ, denote by L λ the line bundle on G/B whose T -weight in the point fixed by B is -λ. For λ dominant, V (λ) = Γ(G/B, L λ ) * is an irreducible G-module of highest weight λ; when we deal with different groups we will use the notation V G (λ).

Denote by Π(λ) the set of weights occurring in V (λ) and set Π + (λ) = Π(λ) ∩ Λ + . Let λ → λ * be the linear involution of Λ defined by (V (λ)) * ≃ V (λ * ), for any dominant weight λ.

The weight lattice Λ is endowed with the dominance order defined as follows: µ λ if and only if λµ ∈ N∆. If β = α∈∆ n α α ∈ Z∆, define its support over ∆ (not to be confused with the previous one) as follows:

Supp ∆ (β) = {α ∈ ∆ : n α = 0}.
We introduce also some notations about the multiplication of sections. Notice that, for all λ, µ ∈ Λ, L λ ⊗ L µ = L λ+µ . Therefore, if λ, µ are dominant weights and n ∈ N, the multiplication of sections defines maps as follows:

m λ,µ : V (λ) × V (µ) → V (λ + µ) and m n λ : V (λ) → V (nλ).
We will also write uv for m λ,µ (u, v) and u n for m n λ (u). Since G/B is irreducible, m λ,µ and m n λ induce the following maps at the level of projective spaces:

ψ λ,µ : P(V (λ)) × P(V (µ)) → P(V (λ + µ)) and ψ n λ : P(V (λ)) → P(V (nλ)).
The following lemma is certainly well known; however we do not know any reference.

Lemma 1. Let λ, µ be dominant weights.

i) If Supp(λ) ∩ Supp(µ) = ∅, then the map ψ λ,µ : P(V (λ)) × P(V (µ)) → P(V (λ + µ)) is a closed embedding. ii) For any n > 0, the map ψ n λ :

P(V (λ)) → P(V (nλ)) is a closed embedding. Proof. i). Fix highest weight vectors v λ ∈ V (λ), v µ ∈ V (µ) and v λ+µ = v λ v µ ∈ V (λ + µ).
If V is irreducible, then P(V ) has a unique closed orbit, namely the orbit of the highest weight vector. Consequently, since P(V (λ)) × P(V (µ)) has a unique closed orbit, in order to prove the claim it suffices to prove that ψ λ,µ is smooth in x = ([v λ ], [v µ ]) and that the inverse image of [v λ+µ ] is x. The second claim is clear for weight reasons.

In order to prove that ψ λ,µ is smooth in

x, consider T -stable complements U ⊂ V (λ), V ⊂ V (µ) and W ⊂ V (λ + µ) of k v λ , k v µ and k v λ+µ .
So in a neighbourhood of x the map ψ λ,µ can be described as

ψ : U × V -→ W where ψ(u, v) = uv µ + v λ v + uv.
The differential of ψ λ,µ in x is then given by the differential of ψ in (0, 0), thus it is described as follows:

dψ x (u, v) = uv µ + v λ v.
Suppose that dψ x is not injective. Since it is T -equivariant, consider a maximal weight η ∈ Π(λ + µ) {λ+ µ} such that there exists a couple of non-zero T -eigenvectors (u, v) ∈ ker dψ x with weights respectively ηµ and ηλ. Suppose that ηµ ∈ Π(λ) {λ} is not maximal and take α ∈ ∆ such that ηµ + α ∈ Π(λ) {λ} and e α u = 0: then

(e α u)v µ + v λ (e α v) = e α (uv µ + v λ v) = 0
and η + α ∈ Π(λ + µ) {λ + µ}, against the maximality of η. Thus ηµ is maximal in Π(λ) {λ} and similarly ηλ is maximal in Π(µ) {µ}. Therefore, on one hand it must be ηµ = λα with α ∈ Supp(λ), while on the other hand it must be ηλ = µβ with β ∈ Supp(µ). Since Supp(λ) ∩ Supp(µ) = ∅, this is impossible and shows that, if (u, v) ∈ ker dψ x , then it must be u = 0 or v = 0. Suppose now that (u, 0) ∈ ker dψ x : then uv µ = 0 and by the irreducibility of G/B also u = 0. A similar argument applies if v = 0.

ii). Suppose that v, w ∈ V (λ) are such that v n = w n : then v = tw for some t ∈ k. Thus ψ n λ is injective. Let us show now that ψ n λ is smooth; it is enough to show it in x = [v λ ] where v λ ∈ V (λ) is a highest weight vector. Let V ⊂ V (λ) be the T -stable complement of kv λ , identified with the tangent space T x P(V (λ)). If v ∈ V , the differential d(ψ n λ ) x is described as follows

d(ψ n λ ) x (v) = nv n-1 λ v.
Thus d(ψ n λ ) x is injective and ψ n λ is smooth.

1.2. The variety X λ . If λ is a dominant weight, denote by E(λ) the G × G-module End(V (λ)) and set X λ the closure of the G × G-orbit of [I λ ] ∈ P(E(λ))
. More generally if λ 1 , . . . , λ m are dominant weights we define

X λ1,...,λm = G × G([I λ1 ], . . . , [I λm ]) ⊂ P(E(λ 1 )) × • • • × P(E(λ m )).
Since E(λ) is an irreducible G × G-module of highest weight (λ, λ * ), as a consequence of Lemma 1 we get that if λ and µ have non-intersecting supports and if n ∈ N then

X λ+µ ≃ X λ,µ and X nλ ≃ X λ .
As a consequence we get the following proposition:

Proposition 2. Let λ, µ be dominant weights. Then X λ ≃ X µ as G × G-varieties if and only if λ and µ have the same support. Moreover, if Supp(λ) = {α 1 , . . . , α m } then

X λ ≃ X ωα 1 ,...,ωα m .
Proof. By the discussion above we have to prove only that the condition is necessary. This follows by noticing that if X λ and X µ are G × G-isomorphic then also their closed G × G-orbits are isomorphic, which is equivalent to the fact that λ and µ have the same support.

1.3. The wonderful compactification of G ad and the normalization of X λ . When λ is a regular weight (i.e. Supp(λ) = ∆) the variety X λ is called the wonderful compactification of G ad and it has been studied by C. De Concini and C. Procesi in [START_REF] Concini | Complete symmetric varieties, Invariant Theory[END_REF]. We will denote this variety by M : it is smooth and the complement of its open orbit is the union of smooth prime divisors with normal crossings whose intersection is the closed orbit. Moreover, following the general theory of spherical varieties (see [START_REF] Knop | The Luna-Vust theory of spherical embeddings[END_REF]), M dominates every simple normal completion of a quotient of G ad . The closed orbit of M is isomorphic to G/B ×G/B and the restriction of line bundles determines an embedding of Pic(M ) into Pic(G/B ×G/B), that we identify with Λ × Λ as before; the image of this map is the set of weights of the form (λ, λ * ). Therefore Pic(M ) is identified with Λ and we denote by M λ a line bundle on M whose restriction to

G/B × G/B is isomorphic to L λ ⊠ L λ * . If D ⊂ M is a G × G-
s ν injects Γ(M, M λ-ν ) in Γ(M, M λ ).
If λ is a dominant weight, the map G ad -→ P(E(λ)) extends to a map q λ : M -→ P(E(λ)) whose image is X λ and such that M λ = q * λ (O P(E(λ)) [START_REF] Bourbaki | Eléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitres IV[END_REF]). If we pull back the homogeneous coordinates of P(E(λ)) to M , we get then a submodule of Γ(M, M λ ) which is isomorphic to E(λ) * ; by abuse of notation we will denote this submodule by E(λ) * .

If λ ∈ Λ, in [START_REF] Concini | Complete symmetric varieties, Invariant Theory[END_REF]Theorem 8.3] the following decomposition of Γ(M, M λ ) is given:

Γ(M, M λ ) = µ∈Λ + : µ λ s λ-µ E(µ) * .
Consider the graded algebra A(λ) = ∞ n=0 A n (λ), where A n (λ) = Γ(M, M nλ ), and set X λ = Proj A(λ). We have then a commutative diagram as follows:

M p λ / / / / q λ A A A A A A A A X λ r λ X λ
In [START_REF] Kannan | Projective normality of the wonderful compactification of semisimple adjoint groups[END_REF], it has been shown that A(λ) is generated in degree 1 and in [START_REF] Concini | Normality and non normality of certain semigroups and orbit closures, Algebraic transformation groups and algebraic varieties[END_REF] that r = r λ is the normalization of X λ . Notice that the projective coordinate ring of X λ ⊂ P(E(λ)) is given by the graded subalgebra

B(λ) = ∞ n=0 B n (λ) of A(λ) generated by E(λ) * ⊂ Γ(M, M λ ).
1.4. The variety X Σ . We consider now a generalization of the variety X λ . Let Σ be a finite set of dominant weights and denote

E(Σ) = µ∈Σ E(µ); let x Σ = [(I µ ) µ∈Σ ] ∈ P(E(Σ)) and define X Σ as the closure of the G × G-orbit of x Σ in P(E(Σ)). If Σ = λ, then we get the variety X λ , while if Σ = Π + (λ) we get its normalization X λ .
We say that Σ is simple if there exists λ ∈ Σ such that Σ ⊂ Π + (λ) or equivalently if Σ contains a unique maximal element with respect to the dominance order . Notice that, if Σ is simple, then X Σ is simple: indeed by previous description, if λ ∈ Σ is the maximal element, then we get

X λ r / / X Σ / / X λ In particular, it follows that r = r Σ : X λ -→ X Σ is the normalization of X Σ .
We say that a projective G × G-variety X is linear if there exists an equivariant embedding X ⊂ P(V ) where V is a finite dimensional rational G × G-module. All compactifications considered in this paper are linear, in particular by [START_REF] Knop | Local properties of algebraic group actions, Algebraische Transformationsgruppen und Invariantentheorie[END_REF]Corollary 2.6] all normal projective compactifications are linear.

If Σ is contained in a coset of Λ modulo Z∆, then X Σ is simple if and only if Σ is simple. Indeed, since X Σ is linear, by [12, Proposition 1], the normalization X Σ → X Σ induces a bijection on the orbit sets: hence, if X Σ is simple, then X Σ is simple as well and we get a morphism M -→ X Σ ⊂ P(E(Σ)). If M λ ∈ Pic(M ) is the pullback of the hyperplane bundle, it follows then λ ∈ Σ and Σ ⊂ Π + (λ).

Notice also that if X is any simple linear compactification of a quotient of G ad , then the same argument shows that X ≃ X Σ for some simple Σ.

If Σ is simple, denote

B(Σ) = ∞ n=0 B n (Σ) the projective coordinate ring of X Σ ⊂ P(E(Σ)): it is the subalgebra of A(λ) generated by E(Σ) * ⊂ Γ(M, M λ ).

Normality

In this section we determine for which simple Σ the variety X Σ is normal, proving in particular Theorem A. In the following, by λ we will always denote the maximal element of Σ.

Let ϕ λ ∈ E(λ) * be a highest weight vector and set X • Σ ⊂ X Σ the open affine subset defined by the non-vanishing of ϕ λ . In particular, we set

X λ = X Π + (λ) and notice that X • λ = r -1 (X • Σ ). Notice that X • Σ is B × B-
stable and, since it intersects the closed orbit, it intersects every orbit: therefore

X Σ is normal if and only if X • Σ is normal if and only if the restriction r X • λ : X • λ → X • Σ is an isomorphism. Denote by B(Σ) the coordinate ring of X • Σ and by Ā(λ) the coordinate ring of X • λ ; then we have Ā(λ) = { ϕ ϕ n λ : ϕ ∈ A n (λ)} ⊃ { ϕ ϕ n λ : ϕ ∈ B n (Σ)} = B(Σ) and X Σ is normal if and only if Ā(λ) = B(Σ). The rings Ā(λ) and B(Σ) are not G × G-modules, however since X •
Σ is an open subset of X Σ we still have an action of the Lie algebra g ⊕ g on them. By [START_REF] Kannan | Projective normality of the wonderful compactification of semisimple adjoint groups[END_REF], Ā(λ) is generated by the elements of the form ϕ/ϕ λ with ϕ ∈ A 1 (λ). In particular we have the following lemma. Lemma 3. The variety X Σ is normal if and only if for all µ ∈ Λ + such that µ λ there exists n > 0 such that

s λ-µ E(µ + (n -1)λ) * ⊂ B n (Σ).
Proof. Let ϕ µ ∈ s λ-µ E(µ) be a highest weight vector and suppose that X Σ is normal. Then, by the descriptions of Ā(λ) and B(Σ), for every dominant weight µ λ there exist n > 0 and ϕ

∈ B n (Σ) such that ϕ/ϕ n λ = ϕ µ /ϕ λ or equivalently ϕ = ϕ µ ϕ n-1 λ ∈ B n (Σ). Since ϕ is a highest weight vector of the module s λ-µ E(µ + (n -1)λ) * the claim follows.
Conversely assume that for every dominant weight µ λ there exists n such that

s λ-µ E(µ + (n -1)λ) * ⊂ B n (Σ); in particular ϕ = ϕ µ ϕ n-1 λ ∈ B n (Σ). Let's prove that ϕ/ϕ λ ∈ B(Σ)
for every ϕ ∈ s λ-µ E(µ) * ; this implies the thesis since Ā(λ) is generated in degree one. If ϕ = ϕ µ this is clear. Using the action of the Lie algebra g ⊕ g on B(Σ), let's show that if ϕ/ϕ λ ∈ B(Σ) then f α (ϕ)/ϕ λ ∈ B(Σ): indeed we have

f α (ϕ) ϕ λ = f α ( ϕ ϕ λ ) + ϕ ϕ λ • f α (ϕ λ ) ϕ λ and the claim follows since f α (ϕ λ ) ∈ E(λ) * ⊂ B 1 (Σ).
We can describe the set B n (Σ) more explicitly. Indeed, as in [START_REF] Concini | Normality and non normality of certain semigroups and orbit closures, Algebraic transformation groups and algebraic varieties[END_REF] or in [START_REF] Kannan | Projective normality of the wonderful compactification of semisimple adjoint groups[END_REF], it is possible to identify sections of a line bundle on M with functions on G and use the description of the multiplication of matrix coefficients. Recall that as a G × G-module we have k

[G] = λ∈Λ + E(λ) * ≃ λ∈Λ + V (λ) * ⊗ V (λ). More explicitly if V is a representation of G, define c V : V * ⊗ V -→ k[G] as usual by c V (ψ ⊗ v)(g) = ψ, gv . If we multiply functions in k[G] of this type then we get c V (ψ ⊗ v) • c W (χ ⊗ w) = c V ⊗W (ψ ⊗ χ) ⊗ (v ⊗ w) :
in particular we get that the image of the multiplication E(λ

) * ⊗ E(µ) * -→ k[G] is the sum of all E(ν) * with V (ν) ⊂ V (λ) ⊗ V (µ).
As a consequence we obtain the following Lemma:

Lemma 4 ([7, Lemma 3.1] or [START_REF] Concini | Normality and non normality of certain semigroups and orbit closures, Algebraic transformation groups and algebraic varieties[END_REF]). Let ν, ν ′ be dominant weights, then the image of E(ν

) * ⊗ E(ν ′ ) * in Γ(M, M ν+ν ′ ) via the multiplication map is V (µ)⊂V (ν)⊗V (ν ′ ) s ν+ν ′ -µ E(µ) * .
Proof. Indeed let π : G → M be the map induced by the inclusion G ad ⊂ M . Then any line bundle on G can be trivialized so that the image of π * :

E(λ) * ⊂ Γ(M, M ν ) -→ k[G] is the image of c V (λ)
and the claim follows from previous remarks.

Together with Lemma 3, this gives the following Proposition 5. The variety X Σ is normal if and only if, for every µ ∈ Λ + such that µ λ, there exist n > 0 and λ 1 , . . . , λ n ∈ Σ such that

V (µ + (n -1)λ) ⊂ V (λ 1 ) ⊗ • • • ⊗ V (λ n ).
2.1. Remarks on tensor products. By Proposition 5, in order to establish the normality (or the non-normality) of X Σ , we need some results on tensor product decomposition.

Lemma 6. Let λ, µ, ν be dominant weights and let ∆ ′ ⊂ ∆ be such that Supp ∆ (λ + µν) ⊂ ∆ ′ ; let L ⊂ G be the standard Levi subgroup associated to ∆ ′ . If π ∈ Λ + , denote by V L (π) the simple L-module of highest weight π. Then

V (ν) ⊂ V (λ) ⊗ V (µ) ⇐⇒ V L (ν) ⊂ V L (λ) ⊗ V L (µ).
Proof. If a is any Lie algebra, denote U(a) the corresponding universal enveloping algebra. Suppose that

V L (ν) ⊂ V L (λ) ⊗ V L (µ); fix maximal vectors v λ ∈ V L (λ) and v µ ∈ V L (µ) for the Borel subgroup B ∩ L ⊂ L and fix p ∈ U(l ∩ u -) ⊗ U(l ∩ u -) such that p (v λ ⊗ v µ ) ∈ V L (λ) ⊗ V L (µ) is a maximal vector of weight ν. Since V L (λ) ⊗ V L (µ) ⊂ V (λ) ⊗ V (µ), we only need to prove that p (v λ ⊗ v µ ) is a maximal vector for B too. If α ∈ ∆ ′ then we have e α p (v λ ⊗ v µ ) =
0 by hypothesis. On the other hand, if α ∈ ∆ ∆ ′ , notice that e α commutes with p, since by its definition p is supported only on the f α 's with α ∈ ∆ ′ . Since v λ ⊗ v µ is a maximal vector for B, then we get

e α p (v λ ⊗ v µ ) = p e α (v λ ⊗ v µ ) = 0; thus p (v λ ⊗ v µ ) generates a simple G-module of highest weight ν. Assume conversely that V (ν) ⊂ V (λ) ⊗ V (µ) and fix p ∈ U(u -) ⊗ U(u -) such that p (v λ ⊗ v µ ) ∈ V (λ) ⊗ V (µ) is a maximal vector of weight ν. Since Supp ∆ (λ + µ -ν) ⊂ ∆ ′ ,
we may assume that the only f α 's appearing in p are those with α ∈ ∆ ′ ; therefore p (v λ ⊗ v µ ) ∈ V L (λ) ⊗ V L (µ) and it generates a simple L-module of highest weight ν.

Lemma 7. Fix λ, µ, ν ∈ Λ + such that V (ν) ⊂ V (λ) ⊗ V (µ).
Then, for any ν ′ ∈ Λ + , it also holds

V (ν + ν ′ ) ⊂ V (λ + ν ′ ) ⊗ V (µ). Table 1. type of Φ highest short root A r α 1 + • • • + α r = ω 1 + ω r B r α 1 + • • • + α r = ω 1 C r α 1 + 2(α 2 + • • • + α r-1 ) + α r = ω 2 D r α 1 + 2(α 2 + • • • + α r-2 ) + α r-1 + α r = ω 2 E 6 α 1 + 2α 2 + 2α 3 + 3α 4 + 2α 5 + α 6 = ω 2 E 7 2α 1 + 2α 2 + 3α 3 + 4α 4 + 3α 5 + 2α 6 + α 7 = ω 1 E 8 2α 1 + 3α 2 + 4α 3 + 6α 4 + 5α 5 + 4α 6 + 3α 7 + 2α 8 = ω 8 F 4 α 1 + 2α 2 + 3α 3 + 2α 4 = ω 4 G 2 2α 1 + α 2 = ω 1
Proof. Fix a maximal vector v ν ′ ∈ V (ν ′ ) and consider the U -equivariant map

φ : V (λ) ⊗ V (µ) -→ V (λ + ν ′ ) ⊗ V (µ) w 1 ⊗ w 2 -→ m λ,ν ′ (w 1 , v ν ′ ) ⊗ w 2
The claim follows since, if

v ν ∈ V (λ) ⊗ V (µ) is a U -invariant vector of weight ν, then φ(v ν ) ∈ V (λ + ν ′ ) ⊗ V (µ) is a U -invariant vector of weight ν + ν ′ .
We now describe some more explicit results. When we deal with explicit irreducible root systems, unless otherwise stated, we always use the numbering of simple roots and fundamental weights of Bourbaki [START_REF] Bourbaki | Eléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitres IV[END_REF].

In order to describe the simple subsets Σ ⊂ Λ + which give rise to a non-normal variety X Σ , we will make use of following lemma.

Lemma 8.

(1) Let G be of type B r . Then, for any n, V ((n -1)ω 1 ) ⊂ V (ω 1 ) ⊗n .

(2) Let G be of type G 2 . Then, for any n, V (ω 1 + (n -1)ω 2 ) ⊂ V (ω 2 ) ⊗n .

Proof. We consider only the first case, the second is similar. Fix a highest weight vector v 1 ∈ V (ω 1 ). If α is any simple root and if 1 s r, notice that f α acts non-trivially on

f αs-1 • • • f α1 v 1 if and only if α = α s . The T -eigenspace of weight 0 in V (ω 1 ) is spanned by v 0 = f αr • • • f α1 v 1 , and similarly the T -eigenspace of weight (n -1)ω 1 in V (ω 1 ) ⊗n is spanned by v ⊗i-1 1 ⊗ v 0 ⊗ v ⊗n-i 1
, where 1 i n. Since the vectors

e αr (v ⊗i-1 1 ⊗ v 0 ⊗ v ⊗n-i 1 ) = v ⊗i-1 1 ⊗ (e αr v 0 ) ⊗ v ⊗n-i 1
are linearly independent, there exists no maximal vector of weight (n -1)ω 1 in V (ω 1 ) ⊗n . Dual results will be needed to describe the subsets Σ which give rise to a normal variety X Σ , but before we need to introduce some further notation.

If Φ is an irreducible root system and ∆ is a basis for Φ we will denote by η the highest root if Φ is simply laced or the highest short root if Φ is not simply laced. For the convenience of the reader we list the highest short root of every irreducible root system in Table 1.

Recall the condition (⋆) defined in the introduction: a dominant weight λ satisfies (⋆) if, for every non-simply laced connected component ∆ ′ ⊂ ∆, if Supp(λ) ∩ ∆ ′ contains a long root then it contains also the short root which is adjacent to a long simple root. Definition 9. If ∆ ′ ⊂ ∆ is a non-simply laced connected component, order the simple roots in ∆ ′ = {α 1 , . . . , α r } starting from the extreme of the Dynkin diagram of ∆ ′ which contains a long root and denote α q the first short root in ∆ ′ . If λ is a dominant weight such that α q ∈ Supp(λ) and such that Supp(λ)∩∆ ′ contains a long root, denote α p the last long root which occurs in Supp(λ)∩∆ ′ ; for instance, if ∆ ′ is not of type G 2 , then the numbering is as follows:

q q q q q q ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ q q α 1 αp αq αr
The little brother of λ with respect to ∆ ′ is the dominant weight

λ lb ∆ ′ = λ - q i=p α i = λ -ω 1 + ω 2 if G is of type G 2 λ + ω p-1 -ω p + ω q+1 otherwise
where ω i is the fundamental weight associated to α i if 1 i r, while ω 0 = ω r+1 = 0. The set of the little brothers of λ will be denoted by LB(λ); notice that LB(λ) is empty if and only if λ satisfies condition (⋆) of Theorem A. For convenience, define LB(λ) = LB(λ) ∪ {λ}, while if ∆ is connected and non-simply laced set λ lb = λ lb ∆ .

Lemma 10. Assume G to be simple and let λ ∈ Λ + {0}. Denote η the highest root of Φ if the latter is simply laced or the highest short root otherwise.

(1) If λ satisfies the condition (⋆) then

V (λ) ⊂ V (η) ⊗ V (λ).
(2) If λ does not satisfy the condition (⋆) and if λ lb is the little brother of λ then

V (λ) ⊂ V (η) ⊗ V (λ lb ).
Proof. If ∆ is simply laced, then V (η) ≃ g is the adjoint representation: in this case the claim follows straightforward by considering the map g ⊗ V (λ) → V (λ) induced by the g-module structure on V (λ), which is non-zero since λ is non-zero. Suppose now that ∆ is not simply laced. If λ satisfies condition (⋆), then by Lemma 7 it is enough to study the case λ = ω α where α is a short simple root:

Type B r : V (ω r ) ⊂ V (ω 1 ) ⊗ V (ω r ). Type C r : V (ω i ) ⊂ V (ω 2 ) ⊗ V (ω i ), with i < r. Type F 4 : V (ω 3 ) ⊂ V (ω 4 ) ⊗ V (ω 3 ) and V (ω 4 ) ⊂ V (ω 4 ) ⊗ V (ω 4 ). Type G 2 : V (ω 1 ) ⊂ V (ω 1 ) ⊗ V (ω 1 ).
If λ does not satisfy condition (⋆), by Lemma 7 we can assume that λ = ω α with α a long root:

Type B r : V (ω i ) ⊂ V (ω 1 ) ⊗ V (ω i-1 ), if 1 < i < r, and V (ω 1 ) ⊂ V (ω 1 ) ⊗ V (0). Type C r : V (ω r ) ⊂ V (ω 2 ) ⊗ V (ω r-2 ). Type F 4 : V (ω 1 ) ⊂ V (ω 4 ) ⊗ V (ω 4 ) and V (ω 2 ) ⊂ V (ω 4 ) ⊗ V (ω 1 + ω 4 ). Type G 2 : V (ω 2 ) ⊂ V (ω 1 ) ⊗ V (ω 1 ).
The above mentioned inclusion relations for tensor products are essentially known: let us treat the case of type C r with λ = ω i and i < r, the other cases are easier or can be checked directly.

Let v 0 be a highest weight vector of V (ω 2 ) and w 0 be a highest weight vector of V (ω i ). Let f be the following product (in the universal enveloping algebra U(u -))

f = f αi • • • f α1 • f αi+1 • • • f αr-1 • f αr • • • f α2 ,
and consider all the factorizations f = p • q such that p, q ∈ U(u -). If β 1 , . . . , β j ∈ ∆, set

r (f β1 • • • f βj ) = (-1) j 2 δ f βj • • • f β1 ,
where δ equals 0 (resp. 1) if α i occurs an even (resp. odd) number of times in {β 1 , . . . , β j }. Then it is easy to check that the vector

p•q=f p.v 0 ⊗ r q.w 0 is a U -invariant vector in V (ω 2 ) ⊗ V (ω i ) of T -weight ω i .
If the Dynkin diagram of G is not simply laced we will need some further properties of tensor products. If ∆ is connected but not simply laced, we will denote by α S the short simple root that is adjacent to a long simple root α L ; moreover, we will denote the associated fundamental weights by ω S and ω L . Finally, define ζ as the sum of all simple roots and notice that ω S + ζ is dominant. Lemma 11. Let λ be a non-zero dominant weight.

(1) If G is of type F 4 or C r (r 3) and if Supp(λ) contains a long root then

V (λ + ω S ) ⊂ V (ζ + ω S ) ⊗ V (λ).
(2) If G is of type G 2 and if λ does not satisfy (⋆) then

V (λ + ω 1 ) ⊂ V (ω 2 ) ⊗ V (λ lb ). (3) If G is of type G 2 and if α S ∈ Supp(λ) then V (λ + ω 1 ) ⊂ V (ω 2 ) ⊗ V (λ).
Proof. By Lemma 7 it is enough to check the statements for λ = ω α with α a long root in the first two cases and α = α S in the last case.

Type C r : by Lemma 7 it is enough to check that

V (ω r-1 ) ⊂ V (ω 1 ) ⊗ V (ω r ). Type F 4 : we have λ = ω 1 or λ = ω 2 and ω S + ζ = ω 1 + ω 4 .
Type G 2 : we have λ = ω 2 and λ lb = ω 1 in point ( 2) and λ = ω 1 in point (3).

2.2.

Normality and non-normality of X Σ . We are now able to state the main theorem.

Theorem 12. Let Σ be a simple set of dominant weights and let λ be its maximal element. The variety X Σ is normal if and only if Σ ⊃ LB(λ).

Theorem A stated in the introduction follows immediately by considering the case Σ = {λ}. The remaining part of this section will be devoted to the proof of Theorem 12. The general strategy will be based on Proposition 5 and will proceed by induction on the dominance order of weights. The ingredients of this induction will be the results proved in section 2.1 together with the description of the dominance order given by J. Stembridge in [START_REF] Stembridge | The partial order of dominant weights[END_REF]: the dominance order between dominant weights is generated by pairs which differ by the highest short root for a subsystem of the root system.

If K is a subset of ∆, denote Φ K ⊂ Φ the associated root subsystem and, in case K is connected, denote by η K the corresponding highest short root. Moreover, if β = α∈∆ n α α, set β| K = α∈K n α α. The result of [START_REF] Stembridge | The partial order of dominant weights[END_REF] that we will use is the following.

Lemma 13 ([11, Lemma 2.5]). Let λ, µ be two dominant weights with λ > µ; set I = Supp ∆ (λµ). Let Φ K be an irreducible subsystem of Φ I (where K ⊂ I).

(a) If (λ -µ) K , α ∨ 0 for all α ∈ K ∩ Supp(µ), then µ + η K λ. (b) If in addition µ + η K , α ∨ 0 for all α ∈ I K, then µ + η K ∈ Λ + .
The next two lemmas are the main steps of our induction.

Lemma 14. Suppose that Φ is irreducible; let λ, µ ∈ Λ + such that λ > µ and Supp ∆ (λµ) = ∆.

Assume that either Φ is simply laced, or there exists a short root α ∈ Supp(λ) such that λµ, α ∨ 0, or α S / ∈ Supp(µ). Then there exists a connected subset

K of ∆ such that i) µ + η K λ; ii) µ + η K ∈ Λ + ; iii) K ∩ Supp(λ) = ∅. Proof. Set K 1 = {α ∈ ∆ : λ -µ, α ∨ 0}.
Since λ > µ we have that K 1 ∩ Supp(λ) is non-empty. Notice also that Supp(µ) ⊃ ∆ K 1 . Define K as follows: a) If Φ is simply laced, let K be a connected component of K 1 which intersects Supp(λ). b) If α ∈ Supp(λ) is a short root such that λµ, α ∨ 0 let K be the connected component of K 1 containing α. c) If Φ is not simply laced and there does not exist a short root α as in b), let K be a connected component of K 1 which intersects Supp(λ).

Properties i) and iii) are then easily verified by Lemma 13(a) and by construction.

To prove ii) notice that, if Φ is not simply laced, by the construction of K it follows that if α L ∈ K then α S ∈ K as well: indeed, K is a connected component of K 1 and if there is no short root α as in b) then α S ∈ Supp(µ) implies α S ∈ K 1 . By the description of highest short roots in Table 1 we deduce that, if α ∈ K K • , then the respective coefficient in η K is 1: hence η K , α ∨ = -1 for all α ∈ ∂K and, since Supp(µ

) ⊃ ∆ K 1 ⊃ ∂K, we get µ + η K ∈ Λ + .
In order to proceed with the induction, in the next lemma we will need to consider the condition (⋆) also for a Levi subgroup of G. If K ⊂ ∆ let L K be the associated standard Levi subgroup; we say that λ ∈ Λ + satisfies condition (⋆ K ) if, for every non-simply laced connected component K ′ of K such that Supp(λ) ∩ K ′ contains a long root, Supp(λ) ∩ K ′ contains also the short root adjacent to a long root. Notice that if λ satisfies (⋆) then it also satisfies (⋆ K ) for all K ⊂ ∆.

Similarly we can also define the little brother of a dominant weight w.r.t. the Levi subgroup L K : if K ′ is a connected component of K such that λ does not satisfy (⋆ K ′ ), define the little brother λ lb K ′ w.r.t. K ′ as in Definition 9 and denote by LB K (λ) the set of little brothers of λ constructed in this way. Notice that if K ′ is a connected component of K such that λ does not satisfy (⋆ K ′ ) and if ∆ ′ is the connected component of ∆ containing K ′ , then λ does not satisfy (⋆ ∆ ′ ) as well and

λ lb K ′ = λ lb ∆ ′ . In particular LB K (λ) ⊂ LB(λ).
Lemma 15. Assume G to be simple and let λ, µ be two dominant weights such that λ > µ and Supp ∆ (λµ) = ∆. Then there exist µ ′ ∈ Λ + and λ ′ ∈ LB(λ) such that µ < µ ′ λ and

V (µ + λ) ⊂ V (µ ′ ) ⊗ V (λ ′ ).
Proof. Suppose first that either Φ is simply laced or α S / ∈ Supp(µ) or there exists a short root α in Supp(λ) such that λµ, α ∨ 0. Take K as in Lemma 14 and set µ ′ = µ + η K : then by Lemma 10 together with Lemma 7 we get

V L K (µ + λ) ⊂ V L K (µ ′ ) ⊗ V L K (λ ′ )
with λ ′ ∈ LB K (λ). The claim follows by Lemma 6 together with the inclusion LB K (λ) ⊂ LB(λ).

Suppose now that Φ is not simply laced, that α S ∈ Supp(µ) and that there is no short root α ∈ Supp(λ) such that λµ, α ∨ 0. Since λ > µ there exists α ∈ Supp(λ) such that λµ, α ∨ 0: therefore, Supp(λ) contains at least a long root. Set µ ′ = µ + ζ; notice that µ ′ λ and that µ ′ is dominant. The claim follows then by Lemma 10 and by Lemma 7 if Φ is of type B, while if Φ is of type C, F 4 or G 2 it follows by Lemma 11 and by Lemma 7.

Proof of Theorem 12. We prove first that the condition is necessary. Assume that there exists a little brother µ = λ lb ∆ ′ of λ which is not in Σ. We prove that for every positive n and for every choice of weights λ 1 , . . . , λ n ∈ Σ the module

V (µ + (n -1)λ) is not contained in V (λ 1 ) ⊗ • • • ⊗ V (λ n ).
We proceed by contradiction. Assume there exist weights λ 1 , . . . , λ n as above and notice that any of them satisfies µ λ

i λ: indeed, λ -µ = nλ -(µ + (n -1)λ) nλ -λ i λ -λ i for every i. Therefore Supp ∆ ( λ i -(µ + (n -1)λ)) ⊂ Supp ∆ (λ -µ)
. By Definition 9 together with Lemma 6, it is enough to analyse the case G of type B r and Supp(λ) = {α 1 } or G of type G 2 and Supp(λ) = {α 2 }. We analyse these two cases separately.

Type B r : we have λ = aω 1 , µ = (a -1)ω 1 and µ + (n -1)λ = (na -1)ω 1 . If a = 1 we notice that there are no dominant weights between λ and µ. So the only possibility is λ i = λ = ω 1 for all more explicit the criterion of smoothness given in [START_REF] Timashev | Equivariant compactifications of reductive groups[END_REF] in the case of a linear projective compactification of a reductive group.

3.1. The colored cone of X λ . Let X be a simple normal compactification of G ad , call Y its unique closed orbit. Set D(G ad ) the set of B × B --stable prime divisors of G ad and D(X) ⊂ D(G ad ) the set of divisors whose closure in X contains Y . Let N (X) be the set of G × G-stable prime divisors of X, so that the set of B × B --stable prime divisors of X is identified with D(G ad ) ∪ N (X).

Let T ad ⊂ G ad be the image of T ; then the character group X (T ad ) coincides with the root lattice Z∆, while the cocharacter group X ∨ (T ad ) coincides with the coweight lattice Λ ∨ . If V is a simple G × Gmodule denote by V (B×B -) the subset of B × B --eigenvectors. Notice that k(G ad ) (B×B -) /k * ≃ Z∆ and define a natural map ρ : D(G ad ) ∪ N (X) → Λ ∨ by associating to a B × B --stable prime divisor of X the cocharacter associated to the rational discrete valuation induced by D. If D ∈ N (X), then ρ(D) is the opposite of a fundamental coweight, while if D ∈ D(G ad ), then ρ(D) is a simple coroot; moreover, ρ is injective and ρ(D(G ad )) = ∆ ∨ (see [12, § 7]).

Let C(X) be the convex cone in Λ ∨ Q generated by ρ D(X) ∪ N (X) ; by the general theory of spherical embeddings we have that C(X) is generated by ρ(D(X)) together with the negative Weyl chamber of Φ. The colored cone of X is then the couple C(X), D(X) : up to equivariant isomorphisms, it uniquely determines X as a G × G-compactification of G ad .

In the case of the compactification X λ , then ρ(D(X)) = ∆ ∨ Supp(λ) ∨ (see [START_REF] Timashev | Equivariant compactifications of reductive groups[END_REF]Theorem 7]).

3.2. Q-factoriality. In order to give a necessary and sufficient condition for the Q-factoriality of X λ we need to determine the set of extremal rays of the associated cone C( X λ ).

Lemma 16. If α ∈ ∆ Supp(λ), then α ∨ generates an extremal ray of C( X λ ).

Proof. If a simple coroot α ∨ ∈ C( X λ ) does not generate an extremal ray, then we can write

α ∨ = β∈∆ {α} a β β ∨ - β∈∆ b β ω ∨ β ,
with a β , b β 0 for every β: this yields a contradiction since then it would be α, α ∨ 0.

Recall that a convex cone is said to be simplicial if it is generated by linearly independent vectors; the following proposition is a particular case of a characterization of Q-factoriality that M. Brion gave in [START_REF] Brion | Variétés sphériques et théorie de Mori[END_REF] in the general case of a spherical variety. We recall it in the case of our interest.

Proposition 17 (see [START_REF] Brion | Variétés sphériques et théorie de Mori[END_REF]Proposition 4.2]). The variety

X λ is Q-factorial if and only if C( X λ ) is simpli- cial.
Therefore, since C( X λ ) has maximal dimension, X λ is Q-factorial if and only if the number of extremal rays of the associated cone equals the rank of G. To describe such rays we need to introduce some more notation; the description will be slightly more complicated if Φ is of type D or E.

Denote ∆ e the set of extremal roots of ∆ and set ∆ Supp(λ) = n i=1 I i the decomposition in connected components. Denote

I e = Ii =I de Ii∩∆ e =∅ I i ,
where I de is defined as follows. If ∆ is of type D or E, denote γ de the unique simple root which is adjacent to other three simple roots and, if it exists, denote I de ⊂ ∆ Supp(λ) the unique connected component such that γ de ∈ I de and |I de ∩ ∆ e | = 1, otherwise define I de to be the empty set. Denote I * de ⊂ I de the minimal connected subset such that γ de ∈ I * de and I * de ∩ ∆ e = ∅, or define it to be the empty set otherwise. Finally define J(λ) = ∆ (I e ∪ I * de ) ∪ ∆ e Supp(λ) .

Lemma 18. The extremal rays of C( X λ ) are generated by the simple coroots α ∨ with α ∈ ∆ Supp(λ) and by the opposite of fundamental coweights -ω ∨ α with α ∈ J(λ).

Proof. Recall that C( X λ ) is generated by the simple coroots α ∨ with α ∈ ∆ Supp(λ) together with the fundamental coweights -ω ∨ α with α ∈ ∆ and that every coroot α ∨ with α ∈ ∆ Supp(λ) generates an extremal ray of C( X λ ).

A coweight -ω ∨ α does not generate an extremal ray if and only if it can be written as follows

-ω ∨ α = β∈K a β β ∨ - β∈H b β ω ∨ β
with a β > 0 for every β ∈ K and with b β > 0 for every β ∈ H, for suitable non-empty subsets K ⊂ ∆ Supp(λ) and H ⊂ ∆ {α}. Since the right member of the equality is negative against every simple root in ∂K, we get ∂K = {α}.

Notice that K is connected. Indeed if K ′ ⊂ K is a connected component then ∂K ′ = {α} and β∈K ′ a β α, β ∨ < 0: therefore if K contains two connected components it must be β∈K a β α, β ∨ -2.
On the other hand α, ω ∨ β = 0 for every β ∈ H, therefore if K is not connected it follows

-1 = -α, ω ∨ α = β∈K a β α, β ∨ -2.
Since ∂K is one single root, K contains an extreme of ∆, thus we get K ⊂ I e ∪ I de . Suppose that γ de ∈ K ⊂ I de : then we get a contradiction since it would be |∂K| = 2. Therefore we get K ⊂ I e ∪ (I * de {γ de }) and α ∈ I e ∪ I * de . Such a subset K cannot exist if α ∈ ∆ e Supp(λ), otherwise it would be K = ∆ {α} which intersects Supp(λ). We get then that every -ω ∨ α with α ∈ J(λ) generates an extremal ray of C( X λ ).

Suppose conversely that α ∈ J(λ). Then we can construct a connected subset K ⊂ I e ∪ (I * de {γ de }) such that ∂K = {α}. If γ ∈ K ∩ ∆ e , consider the fundamental coweight (ω K γ ) ∨ associated to γ in the irreducible root subsystem associated to K: then we get

(ω K γ ) ∨ = β∈K a β β ∨ = ω ∨ γ -mω ∨ α ,
where a β > 0 are rational coefficients and where m > 0 is an integer. Therefore -ω ∨ α does not generate an extremal ray of C( X λ ).

Proposition 19. The variety X λ is Q-factorial if and only if the following conditions are fulfilled: i) Supp(λ) is connected; ii) If Supp(λ) contains a unique element, then this element is an extreme of ∆; iii) If ∆ is of type D or E, then Supp(λ) contains γ de and at least two simple roots adjacent to γ de .

Proof. By Proposition 17 together with Lemma 18

X λ is Q-factorial if and only if | Supp(λ)| = |J(λ)|. Suppose that X λ is Q-factorial. Consider the dominant weight λ ′ = α ∈I e ∪I * de ω α : then J(λ ′ ) = J(λ) and |∆| = |J(λ)| + |∆ Supp(λ)| |J(λ ′ )| + |∆ Supp(λ ′ )| |∆|,
which implies Supp(λ) = Supp(λ ′ ). This shows ∆ Supp(λ) = I e ∪ I * de , and we get the following decomposition of J(λ): J(λ) ∩ Supp(λ) = ∆ (I e ∪ I * de ), J(λ) Supp(λ) = ∆ e Supp(λ).

If I de = ∅, set I de ∩ ∆ e = {α de }. Define a surjective map F : J(λ) {α de } -→ Supp(λ) as follows: F is the identity on J(λ) ∩ Supp(λ), while if α ∈ J(λ) Supp(λ) consider the connected component K ⊂ ∆ Supp(λ) containing α and define F (α) by the relation ∂K = {F (α)}: since α = α de , it must be |∂K| = 1. Therefore F is well defined and it is surjective since Supp(λ) J(λ) = ∂I e . Therefore ∆ Supp(λ) = I e and we get i). Being surjective, F has to be injective as well; this easily implies both ii) and iii).

Suppose conversely that Supp(λ) is connected, or equivalently that ∆ Supp(λ) = I e : then ii) and iii) imply |∆ e Supp(λ)| = |∂I e |. This shows that X λ is Q-factorial, since then |J(λ)| + |∆ Supp(λ)| = |∆|.

Corollary 20. If X λ is Q-factorial, the extremal rays of C( X λ ) are generated by: i) the coroots α ∨ with α ∈ ∆ Supp(λ), ii) the coweights -ω ∨ α with α ∈ Supp(λ) • ∪ ∆ e Supp(λ) .

3.3. Smoothness. Suppose that Σ = {λ, λ 1 , . . . , λ s } is a simple set of dominant weights, where λ is the maximal one. In this section we will prove the following generalization of Theorem B.

Theorem 21. The variety X Σ is smooth if and only if X λ is normal, Q-factorial and every connected component of ∆ Supp(λ) has type A.

Corollary 22. X Σ is smooth if and only if X λ is smooth.

To prove Theorem 21, we will make use of a characterization of smoothness for arbitrary group compactifications given by D. Timashev in [START_REF] Timashev | Equivariant compactifications of reductive groups[END_REF]. For convenience, we will use a generalization of it which can be found in [START_REF] Ruzzi | Smooth projective symmetric varieties with Picard number equal to one[END_REF] in the more general context of symmetric spaces. We recall it in the case of a simple group compactification.

Theorem 23 (see [START_REF] Ruzzi | Smooth projective symmetric varieties with Picard number equal to one[END_REF]Theorem 2.2], [START_REF] Timashev | Equivariant compactifications of reductive groups[END_REF]Theorem 9]). The variety X λ is smooth if and only if the following conditions are fulfilled: i) All connected components of ∆ Supp(λ) are of type A and there are no more than | Supp(λ)| of them. ii) The cone C( X λ ) is simplicial and it is generated by a basis of the coweight lattice Λ ∨ . iii) One can enumerate the simple roots in order of their positions at Dynkin diagrams of connected components I k = {α k 1 , . . . , α k n k } of ∆ Supp(λ) , k = 1, . . . , n, and partition the basis of the free semigroup C( X λ ) ∨ ∩ Z∆ into subsets {π k 1 , . . . , π k n k +1 }, k = 1, . . . , p, p n, in such a way that π k j , (α h i ) ∨ = δ i,j δ h,k and π k j -j n k +1 π k n k +1 is the j-th fundamental weight of the root system generated by {α k 1 , ..., α k n k } for all j, k.

Proof of Theorem 21. First, we prove that the conditions are necessary; since we only have to prove that X λ is normal, we may assume that ∆ is non-simply laced. By Theorem 23 i), Supp(λ) contains at least one of the two simple roots α S , α L ; suppose that Supp(λ) contains α L but not α S . Denote K = {α 1 , . . . , α l } ⊂ ∆ Supp(λ) the connected component which contains α S and number its simple roots starting from α S : therefore α 1 = α S and α l ∈ ∆ e , moreover K is either of type C l+1 or of type G 2 .

Set ω ∨ = (l + 1)(ω K l ) ∨ , where (ω K l ) ∨ is the fundamental coweight associated to α l in the root subsystem Φ K associated to K; then

ω ∨ = l i=1 iα ∨ i = (l + 1)ω ∨ α l -mω ∨ α L . where m = 2 if K is of type C l+1 (with l 1) and m = 3 if K is of type G 2 .
If K is not of type B 2 , then ∆ is either of type C r (with r > 2) or of type F 4 or of type G 2 and every simple coroot β ∨ ∈ ∆ ∨ is a primitive element in Λ ∨ (i.e. there does not exist π ∨ ∈ Λ ∨ which satisfies tπ ∨ = β ∨ with t > 1): therefore by Lemma 18 together with Theorem 23 ii) {α ∨ 1 , . . . , α ∨ l , ω ∨ α l } is part of a basis of Λ ∨ and we get a contradiction since then the equality above would imply ω ∨ α L ∈ Λ ∨ . Otherwise K is of type B 2 , thus ∆ is of type B r and 1 2 α ∨ S ∈ Λ ∨ : then we get a contradiction since by Theorem 23 iii) there exists π ∈ C( X λ ) ∨ ∩ Z∆ such that π, α ∨ S = 1.

Let's prove now that conditions of Theorem 23 are verified if X λ is normal, Q-factorial and ∆ Supp(λ) has type A. Set N = C( X λ ) ∩ Λ ∨ the monoid generated by the primitive elements of the extremal rays of C( X λ ).

To prove condition i), it is enough to notice as in Proposition 19 that, since Supp(λ) is connected, we have ∆ Supp(λ) = I e and the number of its connected components equals |∆ e Supp(λ)| |J(λ)| = | Supp(λ)|.

To prove condition ii), let's show that, if β ∈ ∆ J(λ) = I e ∆ e , then -ω ∨ β ∈ N . Denote I = {α 1 , . . . , α l } ⊂ ∆ Supp(λ) the connected component which contains β in its closure and number its simple roots starting from the extreme of I which is not an extreme of ∆; therefore α l ∈ ∆ e . Let j be such that β = α j or set j = 0 if β ∈ Supp(λ). Set K = {α j+1 , . . . , α l } and set ω ∨ = (lj + 1)(ω K l ) ∨ , where (ω K l ) ∨ is the fundamental weight associated to α l in the root subsystem Φ K associated to K; then

ω ∨ = l-j i=1 iα ∨ j+i = (l -j + 1)ω ∨ α l + β, α ∨ j+1 ω ∨ β .
Since X λ is normal, by Theorem A we get β, α ∨ j+1 = -1; therefore by Corollary 20 -ω ∨ β ∈ N . Finally let's show that condition iii) holds. Suppose that K = {α 1 , . . . , α l } ⊂ ∆ Supp(λ) is a connected component, where the simple roots in K are numbered starting from the extreme of K which is not an extreme of ∆, and define

π K i = (α ∨ i ) * if i l (-ω ∨ α l ) * if i = l + 1
where, if {v 1 , . . . , v r } is a basis of Λ ∨ , {v * 1 , . . . , v * r } denotes the dual basis of Λ. Therefore, if ω K j is the j-th fundamental weight of Φ K , we have ω K j = π K j -j l+1 π K l+1 .

Remarks and generalizations

In this section we will consider the more general situation of compactifications of symmetric varieties. Let G be as before and σ : G → G an involution of G. We denote by H • the subgroup of points fixed by σ and by H its normalizer. The notation is not completely coherent with those of previous sections: G plays now the role that G × G played before, while H • has now the role that the diagonal of G × G had before.

Let Ω + be the set of dominant weights λ such that V (λ) has a non-zero vector fixed by H • and Ω the sublattice of Λ generated by Ω + . The monoid Ω + (resp. the lattice Ω) is in a natural way the set of dominant weights (resp. the set of weights) of a (possibly non-reduced) root system Φ, which is called the restricted root system. For λ ∈ Ω + we can consider the (unique) point x λ ∈ P(V (λ)) fixed by H and define X λ as the closure of the G-orbit of x λ in P(V (λ)).

Proposition 2 generalizes to this more general situation without any further comment.

4.1. Normality of X λ and the closure of a maximal torus orbit. Let T ⊂ G be a maximal torus such that the dimension of T H is maximal and let Z λ = T x λ ⊂ X λ . In [START_REF] Ruzzi | Smooth projective symmetric varieties with Picard number equal to one[END_REF], it is proved that when X λ is normal then Z λ also is normal. The converse of this result does not hold in general. Indeed Z λ is always normal in the case of the G × G-compactification of G ad .

4.2. Generalization to symmetric varieties: normality. The wonderful compactification has been defined in the more general situation of symmetric varieties and the description of the normalization of X λ generalizes thanks to the results contained in [START_REF] Chirivì | Projective normality of complete symmetric varieties[END_REF] and [START_REF] Chirivì | On normality of cones over symmetric varieties[END_REF] (which generalize [START_REF] Kannan | Projective normality of the wonderful compactification of semisimple adjoint groups[END_REF] and [START_REF] Concini | Normality and non normality of certain semigroups and orbit closures, Algebraic transformation groups and algebraic varieties[END_REF]). In particular, Lemma 3 holds here in general. However, in the case of symmetric varieties we do not have a clear description of the multiplication of sections as in Lemma 4. In particular, we have no analogue of Proposition 5.

  stable prime divisor then the line bundle defined by D is of the form M α D , where α D is a simple root. The map D → α D defines a bijection between the set of G × G-stable prime divisors and ∆, and we denote by M α the prime divisor which corresponds to a simple root α. We denote by s α a section of M α whose associated divisor isM α ; notice that such a section is G × G-invariant. More generally if ν = α∈∆ n α α ∈ N∆, set s ν = α∈∆ s nα α ∈ Γ(M, M ν ).Then, given any λ ∈ Λ, the multiplication by

i and this is in contradiction with Lemma 8. If a > 1, notice that there is only one dominant weight between λ and µ, namely ν = λα 1 = (a -2)ω 1 + ω 2 ; hence for all i it must be λ i = λ or λ i = ν. Since λ i µ + (n -1)λ, at most one λ i can be equal to ν; therefore V (µ + (n -1)λ) ⊂ V (λ) ⊗n or V (µ + (n -1)λ) ⊂ V (ν) ⊗ V (λ) ⊗(n-1) . In the first case we obtain V ((na -1)ω 1 ) = V (µ + (n -1)λ) ⊂ V (λ) ⊗n ⊂ V (ω 1 ) ⊗na , against Lemma 8. In the second case we notice that V (ω

) ⊗a and we can conclude as in the first case.

Type G 2 : we have λ = aω 2 , µ = ω 1 + (a -1)ω 2 and we proceed as in the previous case.

We now prove that the condition is sufficient, showing that for every dominant weight µ λ there exist n > 0 and weights λ 1 , . . . ,

To do this, we proceed by decreasing induction with respect to the dominance order.

If µ = λ then the claim is clear, so we assume µ < λ.

. By Lemma 15 applied to the semisimple part of the Levi L = L K associated to K, there exists a weight µ ′ which is dominant with respect to K such that µ < µ ′ µ + β 1 and there exists ν ∈ LB K (µ + β 1 ) which satisfy

) and we may apply the induction on µ ′ λ. Therefore there exist weights λ 1 , . . . , λ n ∈ LB(λ) such that

Finally by Lemma 7 we conclude

Smoothness

In this section we will study the variety X λ ; in particular we will give necessary and sufficient conditions on Supp(λ) for its Q-factoriality and for its smoothness.

Thanks to Lemma 1, we may assume that G is a simple group. Indeed suppose ∆ = ∪ n i=1 ∆ i is the decomposition in connected components and write λ = λ 1 +. . .+λ n with Supp(λ i ) ⊂ ∆ i : correspondingly we get a decomposition X λ = X λ1 × . . . × X λn , and every X λi is an embedding of the corresponding simple factor of G ad if λ i = 0 or a point if λ i = 0. From now on, we will therefore assume that Φ is an irreducible root system.

By the Bruhat decomposition, the group G ad has an open B × B --orbit; therefore it is a spherical G × G-homogeneous space. Following the general theory of spherical embeddings (see [START_REF] Knop | The Luna-Vust theory of spherical embeddings[END_REF]), its simple normal embeddings are classified by combinatorial data called the colored cones. Here we will skip an overview of such theory, and we will simply recall the definition of the colored cone in the particular case of a simple normal embedding of G ad .

Recall that a normal variety X is said Q-factorial if, given any Weil divisor D of X, there exists an integer n = 0 such that nD is a Cartier divisor. In subsection 3.1, we will explicitly describe the colored cone of X λ ; then in subsection 3.2 we will study Q-factoriality of X λ following [START_REF] Brion | Variétés sphériques et théorie de Mori[END_REF]. Finally, in subsection 3.3, we will use Theorem 12 together with the description of the colored cone of X λ to make One may wonder whether the normality of X λ is equivalent to the analogous combinatorial condition on the weight λ, that is, λ satisfies condition (⋆) w.r.t. the root system Φ; here is a counterexample.

Let G be of type B 2 and let σ be the involution of type B I: thus G/H ≃ SO(5)/S O(3) × O(2) and ∆ = 2∆. Consider λ = 2ω 1 ∈ Ω + ; then X λ is a normal embedding of G/H.

Denote by σ the dominance order w.r.t. the root system Φ and suppose that X λ is normal. Then λ satisfies for all µ ∈ Ω + such that µ σ λ there exists n ∈ N such that V (µ + (n -1)λ) ⊂ S n (V (λ)).

If one assumes that the multiplication map is as generic as possible, then also the converse is true. 4.3. Generalization to symmetric varieties: smoothness. In the setting of normal compactifications of symmetric varieties G/H • , fix a maximal torus T such that T H • has maximal dimension and a Borel subgroup B ⊃ T such that BH • ⊂ G is dense. If X is a simple normal compactification of G/H, denote D(X) the set of B-stable and not G-stable prime divisors of X which contain the closed orbit. Denote ρ : D(X) → Ω ∨ the map defined by the evaluation of functions; by [13, Proposition 1] ρ(D(X)) is a basis of the restricted coroot system Φ ∨ . Since the map ρ is not always injective, following the criterion of Q-factoriality in [START_REF] Brion | Variétés sphériques et théorie de Mori[END_REF] in order to generalize Proposition 19 we only need to assume that ρ is injective on D(X), and the proof is the same. Such proposition is true also for compactifications of G/H • , and not only of G/H, since Q-factoriality concerns no integrality questions.

Theorem 21 also can be generalized to this setting with the same proof, but we do not have anymore the equivalence between property (⋆) and the normality of X λ . Thus the theorem has to be reformulated as follows (recall that a simple normal spherical variety is always quasi-projective).

Theorem 24. A simple normal compactification X of G/H is smooth if and only if it is Q-factorial, ∆ ρ(D(X)) satisfies (⋆) and every connected component of ρ(D(X)) has type A.

Current address: Dipartimento di Matematica "Guido Castelnuovo", "Sapienza" Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy E-mail address: bravi@mat.uniroma1.it; gandini@mat.uniroma1.it; amaffei@mat.uniroma1.it; ruzzi@mat.uniroma1.it