On exceptional completions of symmetric varieties

Let G be a simple group with an exceptional involution σ having H as fixed point set. We study the embedding of G/H in the projective space P(V ) for a simple G–module V with a line fixed by H but having no nonzero vector fixed by H . For a certain class of such modules V we describe the closure of G/H proving in particular that it is a smooth variety. Mathematics Subject Index 2000: 14M17, 14L30.


Introduction
Let G be a simple and simply connected algebraic group and σ an involution of G with set of fixed point G σ . We denote by H the normalizer of G σ . In this paper we describe some special completions of the symmetric variety G/H . If V is an irreducible representation of G we say that it is quasi-spherical if there exists a line in V stable by the action of H . If V is quasi-spherical and h V ∈ P(V ) is a point fixed by H we have a map from G/H to P(V ) defined by gH → g · h V . We denote the closure of the image of this map by X V (as shown in [2] the line h V is unique so X V depends only on V ). These varieties are of some interest: one may ask, for example, whether they are smooth or normal (see [3]).
We say that an involution is exceptional if there exists an irreducible representation V of G and a vector v in V such that H stabilizes the line through v but v is not fixed by G σ . As shown in the first section, in this situation H is equal to G σ and it is a Levi corresponding to a maximal parabolic associated to a simple root which appears with multiplicity 1 in the highest root θ . Let ω be the fundamental weight corresponding to this simple root and consider the quasi-spherical irreducible representation V of highest weight nω + θ , with n a positive integer. We give a description of X V proving in particular that it is a smooth variety.

Exceptional involutions
Let g be a simple Lie algebra over an algebraically closed field k of characteristic zero, and let σ be an order 2 automorphism of g. Denote by h the subalgebra of fixed points of σ in g.
Let G be a connected and simply connected group with Lie algebra g. The involution σ induces an involution on G that we still denote by σ . Let G σ be the set of points fixed by σ in G and H be the normalizer of G σ . It is well known that G σ is reductive and connected and that H is the maximal subgroup of G having G σ as identity component (see [1]).
An irreducible representation V is called spherical if V has a nonzero vector fixed by G σ and it is called quasi-spherical if there is a point in P(V ) fixed by H . It is easy to see that if V is spherical then the line pointwise fixed by G σ is unique (see [1]). Notice also that a spherical representation V is quasi-spherical. Indeed if v ∈ V is a nonzero vector fixed by G σ and h ∈ H = N (G σ ), then h −1 ghv = v , hence also hv is a vector fixed by G σ . But, as noted, the space of vectors fixed by G σ is at most one-dimensional, this shows that the point [v] in P(V ) is fixed by H . Hence V is quasi-spherical.
We say that the involution σ is exceptional if there exists a quasi-spherical representation which is not spherical. (There are other equivalent definitions of exceptional involution, see for example [2].) Let V be such an irreducible representation and let v be a nonzero vector which spans a line that is stable under the action of H but not pointwise fixed by G σ . In particular G σ acts on k v by a one-dimensional character. This implies that the center of G σ contains a non trivial torus Z ; we denote by z its Lie algebra and by χ the character of Z such that c · v = χ(c)v for c ∈ Z .
Notice that g is spherical and, since we know that the vector fixed by G σ is unique up to scalar, we have that z is one-dimensional. Now we want to choose a suitable positive root system, this is done in two steps.
First step. We begin choosing a maximal toral subalgebra of h containing z. It is known that we can extend this subalgebra to a σ -stable maximal toral subalgebra t of g, let Φ be its associated root system, and we can take a σstable system of positive roots. Let ρ ∨ the sum of positive coroots and notice that σ(ρ ∨ ) = ρ ∨ . We denote by T the torus corresponding to t and we notice that Z is a subtorus of T , hence we can choose a non zero element z ∈ z that is real when evaluated on the roots.
Step two. Now we can choose the positive system as the subset Φ + of roots which are positive on z + ρ ∨ for a small positive . So we have σ(Φ + ) ⊂ Φ + and (z, β) ≥ 0 for all positive roots β . In particular the first condition implies that σ(β) is a simple root if β is a simple root The choice of Φ + ⊂ Φ determines simple roots, fundamental weights and dominant weights that we will consider fixed from now on. In particular if α is a root we denote by g α ⊂ g the root space corresponding to α and if α is a simple root we denote by ω α the corresponding fundamental weight. Also if λ is a dominant weight we denote by V λ the irreducible representation of highest weight λ and we denote by v λ a highest weight vector of this module. In what follows θ will be the highest root of Φ + and w 0 the longest element of the Weyl group We have the following characterization of exceptional involutions.
If σ is exceptional then H is connected (hence it is equal to G σ ) and it is the Levi subgroup of a maximal parabolic corresponding to a simple root α which appears with multiplicity 1 in the highest root θ , moreover Conversely if α is a simple root which appears with multiplicity 1 in the highest root θ and w 0 (ω α ) = −ω α then there exists an involution σ such that H , the normalizer of G σ , is equal to the Levi of the maximal parabolic corresponding to α; in particular σ is exceptional.

Proof.
We notice first that if L is a Levi of a maximal parabolic corresponding to a simple root α then Conversely assume w 0 (ω α ) = −ω α and let g ∈ N G (L). Since T is a maximal torus of L we can assume that g ∈ N G (T ). Consider the element w defined by g in the Weyl group, since g ∈ N G (L) we must have that w preserves the space orthogonal to the roots associates to L so w(ω α ) = ±ω α . Now if w(ω α ) = ω α we can assume that w permutes the simple roots different from α, up to multiplying g by a suitable element in Now we prove the second claim of the Proposition, so suppose that α is a simple root which appears with multiplicity 1 in the highest root θ and let Ψ be the root subsystem generated by the simple roots different from α. Consider the involution σ defined by Using that α appears with multiplicity 1 in θ it is easy to see that σ is a well defined involution of Lie algebras and that it has the claimed properties. Finally notice that the representation V ωα is quasi-spherical but not spherical since the only line fixed by H in V ωα is the line spanned by the highest weight vector, which is not fixed pointwise. In particular this shows that σ is exceptional. Conversely suppose that σ is exceptional and let V be an irreducible module with a nonzero vector v which spans an H -fixed line not G σ -pointwise fixed. We are now going to use all objects introduced above Proposition 1.1: the subtorus Z , it Lie algebra z, the "real" non zero element z ∈ z, the positive system Φ + .
We begin the proof of the first claim of the Proposition by proving that h = Z g (z); the inclusion h ⊂ Z g (z) is clear so we have to prove the other inclusion.
Notice that by our choice of Φ + , Z g (z) is the Levi subalgebra of g generated by the root vectors x β of weight β for β simple root such that (z, β) = 0; so our claim is equivalent to σ(x β ) = x β for all such simple roots β .
In particular σ(x β ) = ±x β for all roots β , since σ is an involution. Now assume that there exists β simple and different from α such that σ(x β ) = x β . We may choose β as close as possible to α and let, as above, β = α 1 , α 2 , . . . , α m = α be a simple root connected minimal string from β to α; further let γ be the sum of these roots. By minimality σ(x α k ) = x α k for all k = 2, . . . , m − 1 and also σ(x α ) = −x α . But, on one hand we have ], hence σ(x γ ) = x γ and on the other hand [z, x γ ] = 0 and this is impossible since h ⊂ Z g (z).
So we have proved that Z g (z) = h as claimed. In particular h is the Levi subalgebra of the maximal parabolic associated to the simple root α and σ must be defined as in equations (1). Now the fact that it is a morphism of algebras implies that α appears with multiplicity 1 in θ .
By the remark at the beginning of this proof it remains to prove only that H is connected or equivalently that H is in the centralizer of Z . First notice that H is the normalizer of G σ and Z is the identity component of the center of G σ , hence H normalizes also Z . So if we take elements g ∈ H and c ∈ Z we know that gcg −1 ∈ Z , and what we want to show is gcg −1 = c. But Z being a one dimensional torus and χ a nontrivial character, our claim is equivalent to χ(gcg −1 ) = χ(c). By our hypothesis on v we know that g −1 v is in the line spanned by v , hence χ(gcg −1 )v = gc(g −1 v) = gχ(c)g −1 v = χ(c)v and the proof is finished.
As a direct consequence of the Proposition 1.1 above, we notice that if σ is an exceptional involution then the root system of G is simply laced since w 0 = −1, hence ω α is a minuscule weight since the simple root α appears with coefficient 1 in the highest root θ .

Exceptional symmetric varieties
From now on we fix an exceptional involution σ and we denote by α the corresponding simple root and P α the associated maximal parabolic as in Proposition 1.1. Also we denote by p α the Lie algebra of P α and by ω the fundamental weight ω α dual to α ∨ . We also keep the notation introduced in the proof of Proposition 1.1. So z ⊂ h is the center of h and we recall that N G (z) = H .
In the irreducible module V ω of highest weight ω , kv ω is the unique line fixed by P α . Notice that if we take the natural G-equivariant map Chirivì and Maffei 43 the image h of the line z ⊗ v ⊗n ω is a line fixed by H ; we want to study the variety X nω+θ := Gh ⊂ P(V nω+θ ) proving the following theorem: If σ is an exceptional involution (of a simple group) then the variety X nω+θ is smooth and the morphism  : G/H −→ X nω+θ defined by gH → g · h is an open immersion.
We begin considering the variety P of the parabolic subalgebras in g conjugated to p α . Let Y be the subvariety in P × P(g) consisting of pairs (p, l) with l a line in the solvable radical p r of p. It is clear that so that in particular, Y is a smooth variety. We are going to show that for each n ≥ 1, X nω+θ is G-isomorphic to Y . Let us start with some preliminary observations about the structure of Y . First of all notice that the unipotent radical n α of p α is a hyperplane in p r α complementary to z. Since n α is an ideal in p α , Y contains the G-stable divisor D := G × Pα P(n α ) which is just the variety of pairs (p, l) with l a line in the nilpotent radical of p. The root space g θ is contained in n α and we shall also consider the G-orbit O ⊂ D of the pair (p α , g θ ).

Lemma 2.2.
(1) Y D is the G-orbit of (p α , z); (2) O is the unique closed G-orbit in Y .

Proof.
(1) In order to show our claim it suffices to see that P α (z) equals {l ∈ P(g) | l ⊂ p r α and l ⊂ n α }, which choosing a non zero element z ∈ z we can identify with z + n α .
(2) Notice that in n α ⊂ g there is a unique line fixed by B , namely g θ ; hence the B -variety P(n α ) has a unique point fixed by B . So our claim follows at once by Borel fixed point Theorem.
We now want to construct a morphism ϕ : Y → X nω+θ . As usual we identify P with the G-orbit of the highest weight line kv nω in P(V nω ). It follows that Y ⊂ G/P α ×P(g) ⊂ P(V nω )×P(g) ⊂ P(V nω ⊗g) where the last inclusion is given by the Segre embedding. In this way Y is identified with the closure of the G-orbit of the line v nω ⊗z. Denote by W the unique G-stable complement of V nω+θ in V nω ⊗g and consider the rational G-equivariant projection π : P(V nω ⊗ g) → P(V nω+θ ) which is defined on the complement U of P(W ). We have Lemma 2.3.
(1) U is a G-stable open set in P(V nω ⊗ g), so we only need to show that π is defined on the point (p α , g θ ) whose orbit is the unique closed G-orbit in Y . This point maps to v nω ⊗ g θ ∈ P(V nω ⊗ g) which in turn is mapped to the point representing the highest weight line in P(V nω+θ ). Our claim is proved.
(3) By the definition of D we need to show that the subspace v nω ⊗ n α is contained in V nω+θ . We know that v nω ⊗ g θ ⊂ V nω+θ .
Since ω is minuscule, it easily follows that for each g β ⊂ n α we can find a sequence of positive roots γ 1 , . . . , γ m not having α in their support with the property that g β = f γ 1 · · · f γm (g θ ), f γ denoting a non zero element in g −γ .
On the other hand recall that for each i, Let us denote by ϕ : Y → X nω+θ the restriction of π to Y . We have, ϕ is an isomorphism.

Proof.
First we claim that ϕ(D) does not intersect the orbit ϕ(Y D). Indeed, if we suppose otherwise ϕ(D) would contain that orbit and, by Lemma 2.3(3), D = G × Pα P(n α ) would contain a point fixed by H . Then, since the projection of D to P(g) is contained in the projectification of the nilpotent cone, we would get the existence of a H -fixed line consisting of nilpotent elements in g. But H ⊃ T so that such a line is a root space g β for some root β . Now notice that ω being minuscule immediately implies that there exists a simple root γ = α such that either [g γ , g β ] = 0 or [g −γ , g β ] = 0. This gives a contradiction.
Since we have seen that the restriction of ϕ to D is an isomorphism, the fact that ϕ(D) ∩ ϕ(Y D) = ∅ clearly implies that ϕ is finite and that it is an isomorphism if and only if its differential dϕ y in the point y = (p α , g θ ) ∈ O is injective.
Notice that we can identify T y P(V nω ⊗ g) with the unique T -stable complement of the line v nω ⊗ g θ in V nω ⊗ g and similarly we can identify T y P(V nω+θ ) with the unique T -stable complement of the highest weight line in V nω+θ , i.e. with the intersection V nω+θ ∩ T y P(V nω ⊗ g). Using these identifications we get where n − α denotes the nilpotent radical of the parabolic opposite to p α andp r α the unique T -stable complement of g θ in p r α . Furthermore dϕ y is just the restriction to T y Y of the natural G-equivariant projectionπ : Writeñ α =p α ∩ n α andp r α = z ⊕ñ α . Notice that and that T y Y = T y D ⊕ v nω ⊗ z. By Lemma 2.3 dϕ y | TyD is injective, so the only thing we need to show is that dϕ y (v nω ⊗ z) / ∈ dϕ y (T y D). Since the differential is T -equivariant and z has weight zero, it suffices to show that dϕ y (v nω ⊗ z) / ∈ dϕ y (T y D[nω]), where T y D[nω] denotes the subspace of weight nω in T y . We have T y D[nω] = g −θ · (v nω ⊗ g θ ), so we are reduced to show that the two lines are distinct.
On the other hand notice that, α being minuscule, β is a sum of simple roots different from α, hence g β is contained in h. From the fact that g β consists of nilpotent elements and the lineπ(v nω ⊗ z) is preserved by H it follows that g βπ (v nω ⊗ z) = 0 proving our claim.
We can now prove our Theorem Proof.
[of Theorem 2.1] The smoothness of X nω+θ follows by Lemma 2.4. To prove that  is injective we observe that it is an equivariant morphism and that the stabilizer of z ∈ P(g) is H , hence also that of (p α , z) is H .
As the referee pointed out to us Theorem 2.1 holds in more generality. If G σ has a non trivial center one may replace H by G σ ; indeed in such case G σ has all the properties stated in Proposition 1.1 for H but now w 0 (ω α ) = −ω α , whereas in our setting w 0 (ω α ) = −ω α . However all the proofs remain valid since we have never used this last condition in Section 2.
Moreover notice that, as recalled in the Introduction, the H -fixed line in a simple quasi-spherical module is unique whereas the line fixed by G σ is not, in general, unique if G σ has a non trivial center. However the explicit construction of X nω+θ given before the statement of Theorem 2.1, does'nt use this uniqueness property.