Michael Bulois
email: michael.bulois@univ-st-etienne.fr

Lucy Moser-Jauslin
email: lucy.moser-jauslin@u-bourgogne.fr

Ronan Terpereau
email: ronan.terpereau@u-bourgogne.fr

REAL STRUCTURES ON NILPOTENT ORBIT CLOSURES

We determine the equivariant real structures on nilpotent orbits and the normalizations of their closures for the adjoint action of a complex semisimple algebraic group on its Lie algebra.

With this terminology, Question 1.1 can be rephrased as follows.

Question 1.2. What are the (G, σ)-equivariant real structures on the nilpotent orbit closures in g and on their normalizations? If X is an almost homogeneous G-variety, that is, if X contains a dense open G-orbit U , then the condition ( ) implies that µ(U ) = U , and thus µ |U is a (G, σ)equivariant real structure on U . Therefore, a natural strategy to determine the equivariant real structures on nilpotent orbit closures and their normalizations is in two steps. First, we determine the equivariant real structures on nilpotent orbits. Second, we determine which ones extend to their closures and their normalizations.

Denoting by dσ e : g → g the differential of σ : G → G at the identity element, we check (see Example 2.8) that dσ e is a (G, σ)-equivariant real structure on g, viewed as a G-variety for the adjoint action. In particular, if O is a nilpotent orbit in g, then dσ e induces a (G, σ)-equivariant real structure on O if and only if dσ e (O) = O. However, there are also equivariant real structures on nilpotent orbits that are not obtained by differentiating a real group structure on G (Example 1.3), nor even by restricting an equivariant real structure from the Lie algebra g (Example 1.4).

Example 1.3. Let g be a semisimple Lie algebra, and assume that dσ e (O) = O. Let θ ∈ R. Then µ θ : O → O, v → e iθ dσ e (v) is a (G, σ)-equivariant real structure on O which is not obtained by differentiating a real group structure on G when θ / ∈ 2πZ (because, in this case, µ θ does not preserve the Lie bracket).

Example 1.4. Let G = SL 3 (C) with σ(g) = g for all g ∈ G (here g denotes the complex conjugate of g), and let O reg be the regular nilpotent orbit in sl 3 (i.e. the unique nilpotent orbit whose closure contains all the other nilpotent orbits). Then the map µ defined by

Oreg → Oreg, g •   0 1 0 0 0 1 0 0 0   →   σ(g)   1 i 0 0 1 0 0 0 1     •   0 1 0 0 0 1 0 0 0   = σ(g) •   0 1 i 0 0 1 0 0 0  
is a (G, σ)-equivariant real structure on O reg that does not extend to a (G, σ)equivariant real structure on sl 3 (see Section 6 for a proof of this claim).

The following theorem provides a complete answer to the second part of Question 1.2. More precisely, we determine the equivariant structures on the nilpotent orbits and the normalizations of their closures. In the cases where the closure is itself not normal, it remains an open question to determine all the equivariant real structures. This will be discussed in Section 5.

Main Theorem. Let G be a complex semisimple algebraic group endowed with a real group structure σ. Let g be the Lie algebra of G, let O be a nilpotent orbit in g, let O be the closure of O in g, and let O be the normalization of O. Then the following are equivalent: The Main Theorem will be proved in several steps as follows: Let us make a few comments related to Question 1.2 and our Main Theorem:

• (a) ⇔ (b) is Proposition 3.1. • (b) ⇔ (c)
(1) In Section 3, we provide the list of nilpotent orbits that admit a (G, σ)-equivariant real structure. It turns out that, except for a few cases in type D 2n (with n ≥ 2), every nilpotent orbit O in g admits a (G, σ)-equivariant real structure when g is simple.

(2) For every nilpotent orbit O in a complex simple Lie algebra g, we have O dσe = O ∩ g dσe , which is a real manifold (possibly empty) whose G 0 (R)-orbits, usually called real nilpotent orbits, are classified (see [CM93, Section 9]). Let us mention that a study of some properties of the complexification process for these real nilpotent orbits can be found in [START_REF] Crooks | Properties of nilpotent orbit complexification[END_REF].

( Notation. In this article we work over the field of real numbers R and over the field of complex numbers C. We denote by

) 3
Γ := Gal(C/R) = {1, γ}
the Galois group of the field extension C/R. We will always denote by G a complex algebraic group, by Dyn(G) its Dynkin diagram (when G is semisimple), and by G 0 a real algebraic group such that

G 0 × Spec(R) Spec(C)
G. When X is a complex algebraic variety, the group of automorphisms of X over Spec(C) is denoted by Aut C (X). Moreover, when X is equipped with a regular G-action, the subgroup of G-equivariant automorphisms is denoted by Aut G C (X). We denote by G m,C the multiplicative group over C. The reader is referred to [START_REF] Collingwood | Nilpotent orbits in semisimple Lie algebras[END_REF] for the general background on Lie theory and nilpotent orbits in semisimple Lie algebras.

Preliminaries

2.1. Real group structures.

2.1.1. We start by recalling the notions of real forms and real group structures for a given complex algebraic group G, and how these two notions are related.

Definition 2.1.

1. A real form of G is a pair (G 0 , Θ) with G 0 a real algebraic group and Θ : G → G 0 × Spec(R) Spec(C) an isomorphism of complex algebraic groups. (Most of the time, one drops the isomorphism Θ and simply write that G 0 is a form of G.) 2. A real group structure σ on G is a scheme involution on G such that the diagram

G σ / / G Spec(C) Spec(z →z) / / Spec(C) commutes and ι G • σ = σ • ι G and m G • (σ × σ) = σ • m G ,
where ι G : G → G is the inverse morphism and m G : G × G → G is the multiplication morphism. 3. Two real group structures σ 1 and σ 2 on G are equivalent if there exists a complex algebraic group automorphism ψ ∈ Aut gr (G) such that

σ 2 = ψ • σ 1 • ψ -1 .
There is a correspondence between real group structures on G and real forms of G given as follows (see [BS64, Section 2.12] for a general statement of Galois descent, and [FSS98, Section 1.4] for the particular case of algebraic groups).

• To a real group structure σ on G, one associates the real algebraic group G 0 := G/ σ , where γ acts on G through σ, and the isomorphism Θ is given by (q, f ), where q : G → G 0 is the quotient morphism and f : G → Spec(C) is the structure morphism. • To a real form (G 0 , Θ) of G, one associates the real group structure σ that makes the following diagram commutes: (i) σ(G i ) = G i and σ |Gi is a real group structure on G i ; or (ii) there exists j = i such that σ(G i ) = G j , then G i G j and σ |Gi×Gj is equivalent to (g 1 , g 2 ) → (σ 0 (g 2 ), σ 0 (g 1 )) for an arbitrary real group structure σ 0 on G i G j .

G Θ / / σ G 0 × Spec(R) Spec(C) Id×(λ →λ) G Θ / / G 0 × Spec(R) Spec(C) Moreover,
Real group structures on complex simply-connected simple algebraic groups are well-known (see e.g. [GW09, Section 1.7.2] or [Kna02, Section VI.10]). Therefore, all real group structures on complex (simply-connected) semisimple algebraic groups can be determined from Lemma 2.3. 2.1.4. Let G be a complex semisimple algebraic group endowed with a real group structure σ. In order to describe which nilpotent orbits in g admit a (G, σ)equivariant real structure (in Section 3), we need to recall the notions of split real group structures, as well as the notions of inner and outer twists. (with the notation of Lemma 2.2), then σ is called split.

For c ∈ G, we write inn

c : G → G, g → cgc -1 .
We say that inn c is an inner automorphism of G. An automorphism of G that is not inner is called outer. If σ 1 and σ 2 are two real group structures on G, we may write σ 2 = ϕ • σ 1 with ϕ ∈ Aut gr (G), and we say that σ 2 is an inner (resp. an outer ) twist of σ 1 if ϕ is an inner (resp. an outer) automorphism of G. Note that the relation of being inner twists is an equivalence relation on the set of real group structures on G.

According to [Con14, Proposition 7.2.12], every real group structure on G is an inner twist of a unique (up to equivalence) quasi-split real group structure on G.

Moreover, in types

A 1 -B -C -E 7 -E 8 -F 4 -G 2 (i.e.
when Aut(Dyn(G)) is trivial), any quasi-split real group structure is equivalent to a split one, and in types A n (n ≥ 2) -D -E 6 (i.e. when Aut(Dyn(G)) contains exactly one conjugacy class of elements of order 2), there are exactly two equivalence classes of quasi-split real group structures; this follows for instance from [Con14, Propositions 7.2.2 and 7.2.12].

2.1.5. Finally, we recall how the choice of a real group structure σ on a complex semisimple algebraic group G induces a Γ-action on Dyn(G). We refer to [Con14, Remark 7.1.2] or [MJT21, Section A.2] and references therein for details.

Let B ⊆ G be a Borel subgroup, and let T ⊆ B be a maximal torus. (We say that (T, B) is a Borel pair in G.) Then (T , B ) := (σ(T ), σ(B)) is another Borel pair in G, and so there exists c σ ∈ G (unique up to left multiplication by an element of T ) such that

c σ T c -1 σ = T and c σ B c -1 σ = B. Let θ := inn cσ • σ that is an antiregular automorphism of G. Since θ(T ) = T and θ(B) = B,
the automorphism θ induces a lattice automorphism of the character group X(T ) := Hom gr (T, G m,C ) as follows:

∀χ ∈ X(T ), χ → γ χ := σ 0 • χ • θ -1 ,
where σ 0 (t) = t is the complex conjugation on G m,C . One can check that this lattice automorphism is of order ≤ 2, that it does not depend on the choice of c σ , and that it stabilizes the sets of roots R ⊆ X(T ) and of simple roots S ⊆ R associated with the triple (G, B, T ).

Definition 2.5. By the previous discussion, the choice of σ induces an action Γ → Aut(Dyn(G)), usually referred as the -action induced by σ. This action does not depend on the choice of the Borel pair (T, B) in G. We will denote by σ D ∈ Aut(Dyn(G)) the image of the non-trivial element γ ∈ Γ.

Remark 2.6. It follows from the definition of the -action that if σ is an inner twist of σ , then the -actions induced by σ and σ coincide. In particular, if σ is an inner twist of a split real group structure on G, then σ D is trivial. On the other hand, if σ is an outer twist of a split real group structure on G, then it follows from [Con14, Section 1.5, second half of p. 41] that σ D is non-trivial.

2.2. Equivariant real structures. We fix a complex algebraic group G, a real group structure σ on G, and we denote by G 0 = G/ σ the corresponding real form.

2.2.1. We start by recalling the notions of real forms and equivariant real structures for complex G-varieties, and how they are related, as we did for complex algebraic groups in Section 2.1.1. Definition 2.7. Let X be a complex G-variety.

1. An (R, G 0 )-form of X is a pair (X 0 , Ξ) with X 0 a real G 0 -variety and Ξ : X → X 0 × Spec(R) Spec(C) an isomorphism of complex G-varieties. (As for real forms of complex algebraic groups, one usually drops the isomorphism Ξ and simply write that X 0 is a form of X.) 2. A (G, σ)-equivariant real structure on X is an antiregular involution µ on X, that is, a scheme involution on X such that the following diagram commutes

X µ / / X Spec(C) Spec(z →z) / / Spec(C)
and satisfying the condition ( ) stated in the introduction. 3. Two (G, σ)-equivariant real structures µ and µ on X are equivalent if there exists a G-equivariant automorphism ϕ ∈ Aut G C (X) such that µ = ϕ • µ • ϕ -1 . We now suppose that X is quasi-projective. (Later in this article, we will in fact restrict to the case where X is quasi-affine.) Under this condition, as for real group structures (see Section 2.1.1), there is a correspondence between (G, σ)-equivariant real structures on X and (R, G 0 )-forms of X (see [Bor20, Section 5]). Moreover, this correspondence induces a bijection between isomorphism classes of (R, G 0 )-forms of X (as real algebraic G 0 -varieties) and equivalence classes of (G, σ)-equivariant real structures on X.

Example 2.8. Considering the Lie algebra g of G as a G-variety for the adjoint action, we observe that the antilinear Lie algebra involution dσ e , obtained by differentiating σ at the identity element, is also a (G, σ)-equivariant real structure on g. Indeed, for a fixed g ∈ G, if we denote inn g : G → G, h → ghg -1 , then we have

inn σ(g) • σ = σ • inn g ,
and the condition ( ) follows from computing the differential at the identity element on both sides of this equality. 2.2.2. The condition ( ) implies that an equivariant real structure µ on a Gvariety X maps G-orbits to G-orbits, and so a given G-orbit is either fixed by µ or exchanged with another G-orbit. In particular, if X has a dense open orbit U , then U must be fixed by µ, and so µ |U is an equivariant real structure on U . Let us note that a given equivariant real structure µ 0 on U need not extend to X, but if µ 0 extends to X, then this extension is unique by [Har77, Chapter I, Lemma 4.1]. 2.2.3. In the case of homogeneous spaces, we have the following criterion for the existence of an equivariant real structure.

Lemma 2.9. ([MJT21, Lemma 2.4]) Let X = G/H be a homogeneous space. Then X admits a (G, σ)-equivariant real structure if and only if there exists g 0 ∈ G such that the following two conditions hold:

1. (G, σ)-compatibility condition: σ(H) = g 0 Hg -1 0 2. involution condition: σ(g 0 )g 0 ∈ H in which case one real structure is given by

∀k ∈ G, µ(kH) = σ(k)g 0 H.
2.2.4. The classical way to determine the number of equivalence classes for real structures, assuming that such a structure exists, is via Galois cohomology (see [START_REF] Serre | Galois cohomology[END_REF] for a general reference on Galois cohomology).

Lemma 2.10. ([MJT21, Lemma 2.11], see also [Wed18, Section 8]) Let X be a G-variety endowed with a (G, σ)-equivariant real structure µ 0 . Let the Galois group Γ act on Aut G C (X) by conjugation by µ 0 , and denote by H 1 (Γ, Aut G C (X)) the corresponding first Galois cohomology set. Then the map

H 1 (Γ, Aut G C (X)) → {equivalence classes of (G, σ)-equivariant real structures on X} ϕ → ϕ • µ0
is a bijection that sends the identity element to the equivalence class of µ 0 .

2.3. Nilpotent orbits in semisimple Lie algebras. In this section we recall some classical facts concerning nilpotent orbits in semisimple Lie algebras that we will need to prove the Main Theorem.

Let G be a complex semisimple algebraic group, and let g be its Lie algebra. We denote by N the nilpotent cone of g, that is, the G-stable closed subset of g formed by the elements v ∈ g such that ad v : g → g, u → [u, v], is a nilpotent endomorphism of the C-vector space g. Then the following hold: using the fact that each nilpotent orbit is a decomposition class). (N7) Since the Lie algebra g is semisimple, it is isomorphic to a direct sum of simple Lie algebras i∈I g i on which G acts componentwise through the adjoint group G/Z(G). Then the nilpotent orbits in g identify with the products i∈I O i , where each O i is a nilpotent orbit in g i . The same holds for the nilpotent orbit closures in g and their normalizations.

(N5) If v ∈ N , then (G • v) ∩ z(g v ) ⊆ g is a dense open subset of z(g v ), where g v := {u ∈ g | [u, v] = 0} and z(g v ) is the center of g v (see [

Existence of real structures on nilpotent orbits

Let G be a complex semisimple algebraic group endowed with a real group structure σ. Let (T, B) be a Borel pair in G. Let g be the Lie algebra of G, and let O be a nilpotent orbit in g.

In this section we prove the equivalences (a) ⇔ (b)(Proposition 3.1) and (a) ⇔ (d) (Corollary 3.3) of the Main Theorem. In particular, the equivalence (b) ⇔ (d) gives a combinatorial criterion for the existence of an equivariant real structure on a given nilpotent orbit, and we then use this criterion to determine which nilpotent orbits admit an equivariant real structure (see Proposition 3.5 and the discussion before). Since dσ e is a (G, σ)-equivariant real structure on g, the condition ( ) holds for every element in g; in particular, the condition ( ) applied to v implies that σ(H) = H . Let us assume that O admits a (G, σ)-equivariant real structure. By Lemma 2.91., there exists g 0 ∈ G such that σ(H) = g 0 Hg -1 0 , and so H and H = σ(H) are conjugate. But, according to (N6), the subgroups Proof. We keep the notation of Section 2.1.5. Let t ⊆ g be the Cartan subalgebra associated to T ⊆ G. Given a nilpotent orbit O ⊆ g, we recall the construction of the corresponding weighted Dynkin diagram w(O) (see [CM93, Section 3.5] for details). There exists an sl 2 -triple (x, t, y) with x ∈ O, t ∈ t, and such that for every simple root α ∈ S, we have dα e (t) ∈ N, where dα e ∈ t * is the differential of the simple root α at the identity element. In fact, for a given O, there is a unique t ∈ t satisfying this last condition, and one can furthermore show that dα e (t) ∈ {0, 1, 2}. The following result is a straightforward consequence of Proposition 3.2 together with (N4). We now determine which nilpotent orbits in g admit an equivariant real structure (i.e. the nilpotent orbits O such that dσ e (O) = O). Applying Lemma 2.3, we reduce to the case where either (A) g is simple; or (B) g l ⊕ l with l = Lie(L) a simple summand, and σ :

L × L → L × L is (g 1 , g 2 ) → (σ (g 2 ), σ (g 1 ))
, where σ is any real group structure on L.

In Case (B), the nilpotent orbits of g stabilized by dσ e are the orbits of the form (O 1 , dσ e (O 1 )) with O 1 a nilpotent orbit in l. Hence, only these nilpotent orbits admit a (G, σ)-equivariant real structure.

On the other hand, Case (A) will be handled by Proposition 3.5, but we first need to recall the partition type classification of nilpotent orbits in type D n . 1. G is of type D 4 , σ is an outer twist of a split real group structure on G (i.e. g dσe so 5,3 or so 7,1 ), and O belongs to one of the pairs of orbits swapped by the -action induced by σ (see Example 3.6 below). 2. G is of type D 2n with n ≥ 3, σ is an outer twist of a split real group structure on G (i.e. g dσe so p,q with p and q odd), and O ∈ {O Example 3.6. A partial order relation is defined on the set of nilpotent orbits of g by inclusion of closures. In type D 4 , the Hasse diagram representing this partial order is the following (see [CM93, Section 6.2]).

O [7,1] O [5,3] O I [4 2 ] z z $ $ o o / / O [5,1 3 ] o o / / O II [4 2 ] O [3 2 ,1 2 ] O [3,2 2 ,1] O I [2 4 ] z z $ $ o o / / O [3,1 5 ] o o / / O II [2 4 ] O [2 2 ,1 4 ] O [1 8 ]
Here the dotted arrows indicate which pairs of orbits can be swapped by the -action induced by an outer twist of a split real group structure on G.

Uniqueness of real structures on nilpotent orbits

We keep the notation of Section 3. In this section, we prove that all (G, σ)equivariant real structures on O are equivalent (Theorem 4.2).

We recall that, if X G/H is a homogeneous space, then there is an isomorphism of groups (see [Tim11, Proposition 1.8]) given by

N G (H)/H ∼ -→ Aut G C (X), nH → (ϕ nH : gH → gn -1 H), where N G (H) denotes the normalizer of H in G. Let O i∈I O i G/H be a nilpotent orbit in g i∈I g i , and let Q := N G (H)/H Aut G C (O).
The following result, which is essentially due to Brylinski and Kostant [START_REF] Brylinski | The variety of all invariant symplectic structures on a homogeneous space and normalizers of isotropy subgroups[END_REF], describes the structure of the algebraic group Q.

Theorem 4.1. The linear algebraic group Q is connected and solvable. Furthermore, the unipotent radical U of Q is of codimension

d = d(O) := Card({i ∈ I, O i = {0}}) in Q. In particular, we have T := Q/U G d m,C and Q U T. Proof. Let v ∈ O such that Stab G (v) = H. Then Q Q • v as algebraic varieties. Also, by [BK91, Theorem D], we have Q • v = O ∩ z(g v ), which is a dense open subset of z(g v ) by (N5). Hence, Q Q • v is irreducible, and so Q is connected.
Now the fact that Q is solvable and that its unipotent radical is of codimension Proof. If O admits no (G, σ)-equivariant real structure, then there is nothing to prove. Otherwise, according to Proposition 3.1, µ 0 := (dσ e ) |O is a (G, σ)-equivariant real structure on O. Thus, conjugation by µ 0 yields a Γ-action on the group Aut G C (O) Q, and according to Lemma 2.10, the first Galois cohomology set H 1 (Γ, Q) parametrizes the equivalence classes of (G, σ)-equivariant real structures on O. Therefore proving the theorem is equivalent to prove that this set is a singleton.

d in Q is precisely the content of [BK91,
Let K := G d m,C acting on g i∈I g i by scalar multiplication on each simple component g i such that O i = {0}. By (N3), the torus K acts faithfully on O, and so there is an inclusion of algebraic groups

K → Aut G C (O) Q.
Using the fact that µ 0 = dσ e is an antilinear Lie algebra involution that permutes the g i , we have the following two possibilities (with the notation of Lemma 2.2):

• Assume that O i = {0} and µ 0 (g i ) = g i . In this case, Γ acts on the corresponding G m,C through σ 0 . Indeed, denoting ϕ t :

g i → g i , v → t • v, we see that µ 0|gi • ϕ t • µ 0|gi = ϕ t .
• Assume that O i , O j = {0} and µ 0 (g i ⊕ g j ) = g j ⊕ g i . Then g i ⊕ g j l ⊕ l, for some simple Lie algebra l, and

µ 0|gi⊕gj identifies with l ⊕ l → l ⊕ l, (v 1 , v 2 ) → (ψ(v 2 ), ψ -1 (v 1 ))
, where ψ is an antilinear Lie algebra automorphism. In this case, Γ acts on the corresponding G 2 m,C through the real group structure σ 2 . Indeed, denoting ϕ (t1,t2) :

l ⊕ l → l ⊕ l, (v 1 , v 2 ) → (t 1 • v 1 , t 2 • v 2 ), we see that µ 0|gi⊕gj • ϕ (t1,t2) • µ 0|gi⊕gj = ϕ (t2,t1) .
Therefore, the Γ-action on Aut G C (O), given by the conjugation by µ 0 , stabilizes the subgroup K, and the restriction of the Γ-action on Aut G C (O) to K = G d m,C coincides with the Γ-action on K induced by a real group structure equivalent to σ ×n0 0 ×σ ×n2 2 for some n 0 , n 2 ∈ Z ≥0 such that d = n 0 + 2n 2 . But then [MJT21, Proposition 1.18] implies that H 1 (Γ, K) is trivial.

Moreover, the homomorphism Q → Q, q → γ • q is an antiregular algebraic group involution, and so the Γ-action on Q stabilizes the unipotent radical U of Q and induces a Γ-action on the quotient T = Q/U such that the quotient morphism π : Q → T is then Γ-equivariant. It follows that isomorphism of algebraic groups π |K : K T is Γ-equivariant. Consequently, it induces an isomorphism of abelian groups between H 1 (Γ, K) and H 1 (Γ, T), and so H 1 (Γ, T) is trivial.

Finally, the exact sequence of algebraic groups given by Theorem 4.1, namely

1 → U → Q → T → 1,
induces a long exact sequence of pointed sets (see [Ser02, Chp. I, Section 5.5]). In particular, we have

H 1 (Γ, U ) → H 1 (Γ, Q) → H 1 (Γ, T) = {e}.
But H 1 (Γ, U ) is a singleton by [Ser02, Chp. III, Section 2.1, Proposition 6], and so H 1 (Γ, Q) is also a singleton, which concludes the proof of the theorem.

Extension of real structures to nilpotent orbit closures

We maintain the notation of Section 3. In this section, we prove that every (G, σ)-equivariant real structure on O extends uniquely to O, and that all (G, σ)equivariant real structures on O are equivalent (Proposition 5.2). We then briefly review what is known about the (non-)normality of nilpotent orbit closures in semisimple Lie algebras, and finish this section by considering an example related to our extension problem (Example 5.3).

Lemma 5.1. Let X be a complex normal affine G-variety, and assume that there exists a G-stable dense open subset X 0 ⊆ X such that codim(X \ X 0 , X) ≥ 2. Then any G-equivariant automorphism of X 0 extends uniquely to a G-equivariant automorphism of X.

Proof. It is clear that if an automorphism extends from X 0 to X, then this extension is unique by [Har77, Chapter I, Lemma 4.1].

Since X is normal and codim(X \ X 0 , X) ≥ 2, the map

H 0 (X, O X ) → H 0 (X 0 , O X0 ), f → f |X0
is an isomorphism of C-algebras, and so Spec(H 0 (X 0 , O X0 )) Spec(H 0 (X, O X )) X. By [Sta22, Tag 01P9], the affinization morphism

X 0 → Spec(H 0 (X 0 , O X0 )) X
is an open immersion; it coincides with the inclusion morphism X 0 → X.

Let ϕ be an automorphism of X 0 . Then it follows that φ := Spec(ϕ * ) :

X Spec(H 0 (X 0 , O X0 )) → Spec(H 0 (X 0 , O X0 )) X
is an automorphism of X that extends (uniquely) ϕ. Furthermore, the set

g∈G {x ∈ X | φ(g • x) = g • φ(x)}
is a closed subset of X containing the dense open subset X 0 , and so it is equal to X. Therefore, φ is a G-equivariant automorphism of X. The question of which nilpotent orbits in a complex simple Lie algebra have normal closure has been studied by many authors. Let us note that, since a product variety is normal if and only if each factor is normal, the (non-)normality of O i∈I O i (see (N7)) is therefore easily deduced from the (non-)normality of the O i , which means that we can reduce to the case where g is simple. Here is a list of some known results on this topic, with references:

• The regular orbit, the subregular orbit, and the minimal orbit have normal closure [Kos63, VP72, Bro94]. We now give an example of a quasi-affine surface X 0 in A 4 C , endowed with a real structure µ that does not extend to the closure X of X 0 in A 4 C but does extend to the normalization X of X.

Example 5.3. Let X be the image of the morphism

f : A 2 C → A 4 C , (s, t) → (s, st, t 2 , t 3 ); it is a closed non-normal affine surface in A 4 C , defined by the prime ideal (u 2 w -v 2 , vw -ux, uw 2 -vx, w 3 -x 2 ) in C[u, v, w, x].
One can check that X has an isolated singularity at p := (0, 0, 0, 0), and that f restricts to an isomorphism A 2 C \ {(0, 0)} → X 0 := X \ p. Moreover, the normalization of X, which coincides with the affinization of X 0 , is X = A 2 C , and the normalization morphism corresponds to the inclusion of C-algebras C[y, yz, z 2 , z 3 ] ⊆ C[y, z].

Let μ be the real structure on X given by (s, t) → (t, s). It restricts to a real structure µ on X 0 (as (0, 0) is fixed by µ). But the corresponding comorphism

µ * : C[y, z] → C[y, z], Q(y, z) → Q(z, y)
does not preserve C[y, yz, z 2 , z 3 ] (as µ * exchanges y and z), and so µ does not extend to a real structure on X.

We do not know whether a similar phenomenon can occur when X 0 is a nilpotent orbit with non-normal closure. More precisely, by the results we have proven, we know that any (G, σ)-equivariant real structure on a nilpotent orbit is equivalent to one that extends (whether the closure of the orbit is normal or not), namely dσ e . However, in the case where the nilpotent orbit does not have a normal closure, it is a priori possible that we have two equivalent real structures, one that extends, and one that does not, or even two that extend to inequivalent real structures.

Back to Example 1.4

In this last section, we justify that the map µ, given in Example 1.4, is a (G, σ)equivariant real structure on O reg that does not extend to a (G, σ)-equivariant real structure on sl 3 . (Of course, it follows from our Main Theorem that µ is equivalent to a (G, σ)-equivariant real structure that does extend to g, namely (dσ e ) |Oreg , but we will see that this does not imply that µ itself extends to g.)

Recall that, in this example, G = SL 3 (C), and σ(g) = g for all g ∈ G. To check that µ is indeed a (G, σ)-equivariant real structure on O reg , it suffices to check that the element g 0 :=   1 i 0 0 1 0 0 0 1   satisfies the two conditions of Lemma 2.9 with Let f 1 , f 2 be two C-scheme endomorphisms of A 1 C \ {0} such that Φ(u a,1 ) = u f1(a),f2(a) . It is well-known that f i (a) = ν i a i with ν i ∈ C * and i ∈ Z. Let

  ν 1 a 1 ν 2 a 2 1 3 (ν 2 a 2 -1 -ν 1 a 1-2 ) 0 ν 1 a 1 ν 1 a 1 -1 0 0 -2ν 1 a 1   .
This implies readily that 1 = 1, 2 = 0, ν 1 = ν 2 = 1, but then ν 2 a 2-1 -ν 1 a 1-2 = 0, which is absurd. Hence, there is no Φ ∈ Aut G C (sl 3 ) such that Φ |Oreg = ϕ, and so the (G, σ)-equivariant real structure µ given in Example 1.4 is not obtained by restricting a (G, σ)-equivariant real structure from sl 3 to O reg .

  (a) O is dσ e -stable (i.e. dσ e (O) = O); (b) O admits a (G, σ)-equivariant real structure; (c) O admits a (G, σ)-equivariant real structure; and (d) σ D (w(O)) = w(O), where • σ D is the Dynkin diagram automorphism of Dyn(G) induced by σ (see Definition 2.5); and • w(O) is the weighted Dynkin diagram associated to O (see [CM93, Section 3.5]). Moreover, if these equivalent assertions hold, then all (G, σ)-equivariant real structures on O, resp. on O, are equivalent.

  is the first part of Proposition 5.2 (see also Section 2.2.2). • (a) ⇔ (d) is Corollary 3.3. • If the equivalent conditions (a)-(d) of the Main Theorem hold, then the fact that all (G, σ)-equivariant real structures on O are equivalent is Theorem 4.2, and the fact that all (G, σ)-equivariant real structures on O are equivalent is the second part of Proposition 5.2.

  Definition 2.4. 1. If there exists a Borel subgroup B ⊆ G such that σ(B) = B, then σ is called quasi-split. If furthermore B contains a maximal torus T such that σ(T ) = T and the restriction σ |T is equal to the product σ × dim(T ) 0

  (N1) The nilpotent cone N is a finite union of G-orbits (see [CM93, Corollary 3.2.15]). (N2) The dimension of a nilpotent orbit is always even (see [CM93, Corollary 1.4.8]). (N3) Each nilpotent orbit is stable for the G m,C -action induced by the scalar multiplication on g (see the proof in [CM93, Lemma 4.3.1]). (N4) To every nilpotent orbit O ⊆ g is associated a unique weighted Dynkin diagram w(O), that is, a labeling of the nodes of the Dynkin diagram Dyn(G) with labels in {0, 1, 2} (see [CM93, Section 3.5]). When g is a complex simple Lie algebra, a complete classification of the weighted Dynkin diagrams corresponding to nilpotent orbits can be found in [CM93, Section 5.3], for the classical cases, and in [CM93, Section 8.4], for the exceptional cases.

  If dσ e (O) = O, then (dσ e ) |O is a (G, σ)-equivariant real structure on O. (2) If dσ e (O) = O, then O does not admit a (G, σ)-equivariant real structure. Proof. (1): Follows from Example 2.8. (2): Let v ∈ O and v := dσ e (v) ∈ O := dσ e (O). Let H := Stab G (v) and H := Stab G (v ).

  H and H are conjugate if and only if O = O . The result follows by contraposition. Proposition 3.2. Let w(O) be the weighted Dynkin diagram associated to O. Then w(dσ e (O)) = σ D (w(O)), where σ D is the Dynkin diagram automorphism induced by σ (see Definition 2.5).

  Then w(O) is defined by associating to each node α ∈ S of Dyn(G) the label dα e (t) ∈ {0, 1, 2}. Let us now consider the weighted Dynkin diagram σ D (w(O)). By definition of σ D , it is the weighted Dynkin diagram defined by associating to each node α ∈ S of Dyn(G) the label ( ) d( γ α) e (t) = dα e (dθ -1 e (t)) ∈ {0, 1, 2}, where dθ -1 e is the antilinear Lie algebra automorphism obtained by differentiating θ -1 = σ • inn c -1 σ : G → G at the identity element. On the other hand, if (x, t, y) is an sl 2 -triple as above, then (x , t , y ) := (dθ -1 e (x), dθ -1 e (t), dθ -1 e (y)) is an sl 2 -triple with x ∈ dθ -1 e (O) = dσ e (c -1 σ • O) = dσ e (O), t ∈ t (since θ(T ) = T ), and such that for every simple root α ∈ S, we have dα e (t ) ∈ N by ( ). It follows that w(dσ e (O)) = σ D (w(O)).

Corollary 3. 3 .

 3 We have dσ e (O) = O if and only if σ D (w(O)) = w(O).

  Theorem 3.4. ([CM93, Theorem 5.1.4]) Nilpotent orbits in so 2n (type D n with n ≥ 4) are parametrized by partitions of 2n in which even parts occur with even multiplicity, except that "very even" partitions d = [d 1 , . . . , d r ] (those with only even parts, each having even multiplicity) correspond to two orbits, denoted by O I d and O II d . Proposition 3.5. Assume that G is simple. Then dσ e (O) = O if and only if one of the following two conditions holds:

  I d , O II d } with d a very even partition. Proof. • If σ is an inner twist of a split real group structure on G, then the corresponding Dynkin diagram automorphism σ D is trivial (see Remark 2.6). Hence, Corollary 3.3 yields dσ e (O) = O in this case. • In types A n (n ≥ 2), D 2n+1 (n ≥ 2) and E 6 , the weighted Dynkin diagram w(O) is invariant under the non-trivial Dynkin diagram automorphism; see [CM93, Lemma 3.6.5] for the type A n , [CM93, Lemma 5.3.4 and Remark 5.3.6] for the type D 2n+1 , and [CM93, Section 8.4] for the case E 6 . Hence, Corollary 3.3 yields again dσ e (O) = O in this case. • In type D 2n (n ≥ 2), if σ is an outer twist of a split real group structure on G, then σ D is a non-trivial involution of Dyn(G) (see Remark 2.6). In type D 4 , the group Aut(Dyn(G)) S 3 contains three (conjugate) involutions, and σ D can be each of them depending on the choice of σ in its equivalence class. In type D 2n (n ≥ 3), we have Aut(Dyn(G)) = σ D Z/2Z. The result follows then from the description of the weighted Dynkin diagrams given in [CM93, Example 5.3.7] in type D 4 and in [CM93, Lemmas 5.3.4 and 5.3.5, Remark 5.3.6] in type D 2n (n ≥ 3).

  Theorem 23 and Remark 23.1]. The last sentence follows from general results on the structure of connected solvable linear algebraic groups (see e.g. [Bor91, Section 10.6]). Theorem 4.2. All (G, σ)-equivariant real structures on O are equivalent.

Proposition 5. 2 .

 2 Every (G, σ)-equivariant real structure on O extends uniquely to the normalization O of O. Furthermore, all (G, σ)-equivariant real structures on O are equivalent. Proof. If µ is a (G, σ)-equivariant real structure on O, then by Theorem 4.2, there exists ϕ ∈ Aut G C (O) such that µ = ϕ • (dσ e ) |O • ϕ -1 . The equivariant real structure (dσ e ) |O is the restriction of dσ e : g → g, so it extends (uniquely by [Har77, Chapter I, Lemma 4.1]) to O, and also to O by the universal property of the normalization morphism (see e.g. [GW09, Section 12.11]). With a slight abuse of notation, we will write (dσ e ) | O to denote the extension of (dσ e ) |O to O. In addition, ϕ ∈ Aut G C (O) also extends (uniquely) to φ ∈ Aut G C ( O) by Lemma 5.1 applied to X = O and X 0 = O (here the fact that codim(X \ X 0 , X) ≥ 2 follows from (N2)). Hence, μ := φ • (dσ e ) | O • φ-1 is a (G, σ)-equivariant real structure on O extending µ. The fact that this extension is uniquely defined by µ follows again from [Har77, Chapter I, Lemma 4.1]. It remains to prove the second part of the proposition. Let µ be any (G, σ)equivariant real structure on O. Then its restriction to O, denoted by µ, is equivalent to (dσ e ) |O by Theorem 4.2. Hence, there exists ϕ ∈ Aut G C (O) such that µ = ϕ • (dσ e ) |O • ϕ -1 . By Lemma 5.1, ϕ extends to ϕ ∈ Aut G C ( O), and so we must have µ = ϕ • (dσ e ) | O • ϕ -1 . In particular, µ is equivalent to (dσ e ) | O . Therefore, all (G, σ)-equivariant real structures on O are equivalent.



  ; a, b, c ∈ C with a 3 = 1    , which is a straightforward computation left to the reader. Let µ 0 := (dσ e ) |Oreg , and let ϕ := µ 0 • µ ∈ Aut G C (O reg ). We check that ϕ : O reg → O reg , g• Since µ 0 is the restriction of dσ e : sl 3 → sl 3 , we see that µ extends to a (G, σ)equivariant real structure on sl 3 if and only if there exists Φ ∈ Aut G C (sl 3 ) such that Φ |Oreg = ϕ. Suppose that such a Φ exists. Then, for any u ∈ sl 3 , the isotropy subgroups of u and Φ(u) are the same. Let u a,b := subgroup of u a,b in G. Moreover, we check that {u a,b | a, b ∈ C * } are the only elements in sl 3 with this isotropy subgroup, and so this set must be stable by Φ.

  we check that w a = g a • u a,1 , and thus a ) = lim a→0 (g a • Φ (u a,1 )) = lim a→0 g a • u f1(a),f2(a) = lim a→0

  If n = 2, then σ 2 : (t 1 , t 2 ) → (t 2 , t 1 ) defines a real group structure on T . (iii) If n ≥ 2, then every real group structure on T is equivalent to exactly one real group structure of the form σ ×n0 Let σ be a real group structure on a complex simply-connected semisimple algebraic group G i∈I G i , where the G i are the simple factors of G. Then, for a given i ∈ I, we have the following possibilities:

	(ii) 0	× σ ×n1 1	× σ ×n2 2	, where n = n 0 + n 1 + 2n 2 .
	2.1.3. Let G be a complex semisimple algebraic group. There exists a central
	isogeny G → G, where G is a simply-connected semisimple algebraic group. Then
	G is isomorphic to a product of simply-connected simple algebraic groups (see
	[Con14, Exercise 1.6.13 and Section 6.4]). Moreover, every real group structure σ
	on G lifts uniquely to a real group structure σ on G (this follows for instance from
	[Pro07, Section 3.4, Theorem]).
	The next result reduces the classification of real group structures on simply-
	connected semisimple algebraic groups to the classification of real group structures
	on simply-connected simple algebraic groups.
	Lemma 2.3. ([MJT21, Lemma 1.7])
	two real forms of G are isomorphic (as real algebraic groups) if and only
	if the corresponding real group structures are equivalent.
	2.1.2. In this article, we are mostly interested in complex semisimple algebraic
	groups; nevertheless, the case of complex tori will play a role in the proof of Theo-
	rem 4.2.
	Lemma 2.2. (Real group structures on complex tori; see [MJT21, Lemma 1.5] and
	[Cas08, Theorem 2].) Let T G n m,C be an n-dimensional complex torus with n ≥ 1.
	(i) If n = 1, then T has exactly two inequivalent real group structures, defined
	by σ 0 : t → t and σ 1 : t → t	-1 .

  TY05, Propositions 35.3.3 and 35.3.4]). (N6) Let v 1 , v 2 ∈ N . Then Stab G (v 1 ) and Stab G (v 2 ) are conjugate in G if and only if v 1 and v 2 belong to the same nilpotent orbit (see [Bro98b, Theorem 3.7.1],

•

  Nilpotent orbits in sl n have normal closure [Hes79, KP79]. • Nilpotent orbits in so n and sp 2n having normal closure are determined in [KP82, Som05]. • Nilpotent orbits in g 2 , f 4 , and e 6 having normal closure are classified in [LS88, Kra89, Bro98c, Som03]. A list of nilpotent orbits in e 7 and e 8 whose closure is non-normal, which is expected to be the complete list, is given in [Bro98a, Section 7.8].

Acknowledgment. We are grateful to the anonymous referee for the careful reading of the former version of this paper and for his/her helpful comments.