Lucas Fresse
email: lucas.fresse@univ-lorraine.fr

Salah Mehdi
email: salah.mehdi@univ-lorraine.fr

APPROXIMATION OF NILPOTENT ORBITS FOR SIMPLE LIE GROUPS

Keywords: Lie groups, semisimple and nilpotent orbits, approximation, asymptotic cones. 2010 Mathematics Subject Classification. Primary: 17B08, Secondary: 22E15

We propose a systematic and topological study of limits lim ν→0 + G R ⋅(νx) of continuous families of adjoint orbits for a non-compact simple real Lie group G R . This limit is always a finite union of nilpotent orbits. We describe explicitly these nilpotent orbits in terms of Richardson orbits in the case of hyperbolic semisimple elements. We also show that one can approximate minimal nilpotent orbits or even nilpotent orbits by elliptic semisimple orbits. The special cases of SLn(R) and SU(p, q) are computed in detail.

1. Introduction 1.1. Continuous families of adjoint orbits and their limits. The structure of a connected real Lie group G R is closely related with the topology and the geometry of its adjoint orbits on its Lie algebra g R .

For instance, when G R is Abelian, its adjoint orbits are singletons, whereas, when G R is simple, adjoint orbits are symplectic manifolds. If G R is compact, its adjoint orbits are compact symplectic manifolds. For example, each adjoint orbit of the 3-dimensional special unitary group G R = SU(2) takes the form G R ⋅ (rx), where x = i 0 0 -i ∈ g R = su(2) and r ≥ 0, and it is diffeomorphic with the sphere of radius r ≥ 0.

Figure 1. Adjoint orbits for SU [START_REF] Barbasch | The local structure of characters[END_REF].

The picture in Figure 1 suggests that the continuous family {G R ⋅ (rx)} r>0 converges to the trivial orbit in the following sense:

(1.1)

lim r→0 + G R ⋅ (rx) ∶= ⋂ >0 ⋃ r∈(0, ) G R ⋅ (rx) = {0}
(here and throughout the paper, unless otherwise specified, we consider real analytic topology). It turns out that this is true for any x, whenever G R is compact and simple (see Remark 5.2).

The non-compact case is more subtle. For instance, the nontrivial adjoint orbits of the 3-dimensional special linear group G R = SL 2 (R) split into three families (see Example 3.3):

• G R ⋅ (λf ± ), λ > 0, where f ± = ± 0 1 -1 0 , which identifies with the upper sheet z ≥ 0, in the case of f + (resp., the lower sheet z ≤ 0, in the case of f -) of the two-sheeted hyperboloid z 2 -x 2 -y 2 = λ 2 ;

• G R ⋅(sϕ), s > 0, where ϕ = 1 0 0 -1 , which identifies with the one-sheeted hyperboloid z 2 -x 2 -y 2 = -s 2 ;

• G R ⋅ ψ ± , where ψ ± = 0

1 2 ± 1 2 1 2 ∓ 1 2 0
, which identifies with the connected component z > 0 (resp., z < 0) of the cone z 2 -x 2 -y 2 = 0. By the definition, these orbits are homogeneous spaces for G R = SL 2 (R). More precisely, if

P R = M R A R N R
is the Langlands decomposition of a minimal parabolic subgroup of G R , K R a maximal compact subgroup of G R and N - R the opposite of N R (i.e., defined by the opposite of the positive system of restricted roots), then one has the following diffeomorphisms:

G R ⋅ f ± ≃ G R K R , G R ⋅ ϕ ≃ G R M R A R , G R ⋅ ψ + ≃ G R M R N R and G R ⋅ ψ -≃ G R M R N - R .
In particular, besides of being symplectic homogeneous manifolds, the orbits G R ⋅ f ± and G R ⋅ ϕ are respectively Hermitian Riemannian and pseudo-Riemannian symmetric spaces, while the orbits G R ⋅ ψ ± are neither symmetric nor reductive. Moreover, using the limit defined as in (1.1), one checks that (see Example 3.3):

lim λ→0 + G R ⋅ (λf ± ) = G R ⋅ ψ ± = lim ν→0 + G R ⋅ (νψ ± ), lim s→0 + G R ⋅ (sϕ) = G R ⋅ ψ + ∪ G R ⋅ ψ -.
In other words, the continuous family {G R ⋅ (λf + )} λ>0 (resp., {G R ⋅ (λf -)} λ>0 ) of Riemannian manifolds converges towards the singular variety defined as the closure G R ⋅ ψ + (resp., G R ⋅ ψ -) of the nilpotent orbit

G R ⋅ ψ + (resp., G R ⋅ ψ -).
Nilpotent orbits of non-compact simple Lie groups G R play an important role in Mathematical Physics in the following sense. A general space time with symmetry group G R is a homogeneous space G R H R , where H R is a closed subgroup of G R . The cotangent bundle T ⋆ (G R H R ) is naturally equipped with a structure of a symplectic manifold on which the group G R acts in such a way that there is a G R -equivariant moment map

J ∶ T ⋆ (G R H R ) → g * R ≅ g R .
In particular, the image of J is a union of (co)adjoint orbits for G R . Let π ∶ T ⋆ (G R H R ) → G R H R be the natural projection. Then, following Souriau [21, §14-15], an orbit O = G R ⋅ x in the image of J is said to be the classical phase space of a free particle moving on G R H R if the projection π(J -1 (x)) of J -1 (x) is a timelike or a lightlike geodesic on G R H R (assuming that geodesics exist on G R H R ). In this picture, a semisimple orbit describes the classical dynamics of a massive particle, while a nilpotent orbit describes the classical dynamics of a massless particle. However, unlike massive particles, it is not known how to quantize canonically classical dynamics of massless particles. In this respect, a systematic approximation of nilpotent orbits by semisimple orbits could help to understand better the quantization of massless particles.

On the other hand, limits of semisimple (most often elliptic) orbits are used in Representation Theory to bridge objects of different nature such as associated varieties, wave front sets, and characters, see [START_REF] Barbasch | The local structure of characters[END_REF][START_REF] Barbasch | Weyl group representations and nilpotent orbits[END_REF][START_REF] Božičević | A limit formula for even nilpotent orbits[END_REF][START_REF] Harris | Tempered representations and nilpotent orbits[END_REF][START_REF] Harris | Wave front sets of reductive Lie group representations[END_REF][START_REF] Harris | Irreducible characters and semisimple coadjoint orbits[END_REF][START_REF] Kobayashi | Conformal geometry and branching laws for unitary representations attached to minimal nilpotent orbits[END_REF][START_REF] Kobayashi | Analysis on the minimal representation of O(p, q). I. Realization via conformal geometry[END_REF][START_REF] Rossmann | Limit characters of reductive Lie groups[END_REF][START_REF] Rossmann | Limit orbits in reductive Lie algebras[END_REF][START_REF] Schmid | Characteristic cycles and wave front cycles of representations of reductive Lie groups[END_REF][START_REF] Trapa | Annihilators and associated varieties of A q(λ) modules for U(p, q)[END_REF][START_REF] Trapa | Richardson orbits for real classical groups[END_REF] and references therein. However, these limits are used as a tool and, to our best knowledge, there is no systematic study of the limit lim ν→0 + G R ⋅ (νx) for an arbitrary non-compact simple Lie group G R and any x ∈ g R , which is based on topological arguments only.

1.2. Outline of the paper. This paper aims to provide a self-contained and systematic topological study of the limit of adjoint semisimple orbits for connected non-compact simple linear real Lie groups. We also address the reverse problem of realizing (the closure of) a prescribed nilpotent orbit as the limit of a specific family of semisimple orbits. In this respect, our main results appear in Sections 6-8 (Theorems 6.1, 7.4, 7.7, 8.2, 8.3, 8.6).

Let us describe in more detail the content of the paper. For the convenience of the reader, in Section 2, we collect basic definitions and properties about Jordan decompositions, elliptic and hyperbolic elements, sl 2 -triples and the classification of complex nilpotent orbits.

The definition and first properties of limits of orbits are given in Section 3. We first observe that lim ν→0 + G R ⋅ (νx) is a finite union of nilpotent orbits which contains the limit lim ν→0 + G R ⋅ (νx s ) (resp., the orbit closure G R ⋅ x n ) associated to the semisimple (resp., nilpotent) part of x in its Jordan decomposition (Proposition 3.2). In fact, every nilpotent orbit appears in the limit set lim ν→0 + G R ⋅ (νx) of a semisimple (elliptic or hyperbolic) element x ∈ g R (Proposition 3.7). This supports the idea that limits of orbits can be used to approximate nilpotent orbits by semisimple orbits. Our main concern will be then to study to what extent a nilpotent orbit O can be approached by a continuous family of semisimple orbits {G R ⋅ (νx)} ν>0 through its limit; the most desirable case is when lim ν→0 + G R ⋅ (νx) coincides with the closure of O. In Section 4, we relate the limit set lim ν→0 + G R ⋅ (νx) with various asymptotic cones in g R or in its complexification g. Namely, consider the sphere

S(g R ) ∶= (g R ∖ {0}) R ⋆ + and the embedding ι + ∶ g R ↪ S(g R ⊕ R), v ↦ [v ⊕ 1]. We define the real asymptotic cone AC R (G R ⋅ x) of G R ⋅ x as follows (Definition 4.2): AC R (G R ⋅ x) ∶= {0} ∪ (π + ) -1 (S(g R ) ∩ ι + (G R ⋅ x)), where π + ∶ g R ∖ {0} → (g R ∖ {0}) R ⋆
+ is the natural surjection. We observe in Proposition 4.4 that our asymptotic cone coincides with the cone introduced by Harris-He-Ólafsson in [START_REF] Harris | Wave front sets of reductive Lie group representations[END_REF]. Then we show that this cone coincides with the limit set lim

ν→0 + G R ⋅ (νx) (Theorem 4.7): lim ν→0 + G R ⋅ (νx) = AC R (G R ⋅ x).
In Section 5, we deduce that the limit lim ν→0 + G R ⋅ (νx) is always nontrivial whenever x = 0 (Theorem 5.1). Recall that G R is assumed to be simple and non-compact.

From the formulation in terms of the asymptotic cone, we also deduce that, when x is semisimple,

lim ν→0 + G R ⋅ (νx) is contained in the Zariski closure of the Richardson orbit of x (Theorem 4.7): lim ν→0 + G R ⋅ (νx) ⊂ O Rich (x) ∩ g R . Recall that O Rich (x) is the unique dense orbit in G ⋅ u(x)
, where G is the complexification of G R with Lie algebra g and u(x) is the nilradical of any parabolic subalgebra of g which contains the centralizer of x as a Levi factor. If, moreover, x is hyperbolic, i.e., ad(x) has real eigenvalues on g R , and thus on g, the above inclusion becomes an equality in the case of SL n (R) (Corollary 6.3). Writing u R (x) for the nilradical of the real parabolic subalgebra of g R defined by the positive eigenspaces of ad(x) we obtain that (Theorem 6.1):

lim ν→0 + G R ⋅ (νx) = G R ⋅ u R (x) = O 1 ∪ . . . ∪ O k ,
where O 1 , . . . , O k are nilpotent orbits which are all of the same dimension, for an arbitrary non-compact connected simple linear real Lie group G R and a hyperbolic semisimple element x ∈ g R .

Hyperbolic semisimple elements naturally arise in the study of sl 2 -triples. More precisely, by Jacobson-Morozov theorem, every nonzero nilpotent element e ∈ g R lies in an sl 2 -triple {h, e, f } for some semisimple element h and a nilpotent element f in g R which are unique up to conjugation under G R . We show that if the Richardson orbit O Rich (h) has real forms which all intersect u R (h), then (Proposition 7.6):

lim ν→0 + G R ⋅ (νh) = O Rich (h) ∩ g R = O 1 ∪ . . . ∪ O r ,
where O 1 , . . . , O r are the real forms of O Rich (h). (By "real forms" of a complex nilpotent orbit O ⊂ g, we mean G R -orbits (equivalently, connected components) of O ∩ g R ; see, e.g., [24, §6].) In particular, this equality holds for any e when G R = SL n (R) (see Theorem 7.7, which includes an explicit description of the limit set), and for an arbitrary non-compact connected simple linear real Lie group G R when e is even, i.e., when the eigenvalues of ad(h) are all even integers (Theorem 7.4).

Finally, we consider the complementary case where x ∈ g R is semisimple elliptic, i.e., the eigenvalues of ad(x) are all pure imaginary. Suppose that G R contains a compact Cartan subgroup T R . Let K R be a maximal compact subgroup of G R with complexified Lie algebra k and let g = k ⊕ p be the corresponding Cartan decomposition of g. Let u = u(ix) ⊂ g be the nilradical of the parabolic subalgebra associated to the semisimple element ix ∈ g. There is a unique K-orbit O + in p which intersects the subspace u ∩ p along a dense open subset. As a consequence of deep results in Representation Theory, it turns out that (see (8.4)):

lim ν→0 + G R ⋅ (νx) = KS -1 (O + ),
where KS is the Kostant-Sekiguchi bijection between nilpotent adjoint orbits of G R and nilpotent K-orbits of p. To our best knowledge, there is no direct proof of this equality. Based on this equality, we prove that every even nilpotent orbit of g R can be approximated by a continuous family of elliptic semisimple orbits. Specifically, if {h, e, f } is an even sl 2 -triple, then we have (Theorem 8.2):

lim ν→0 + G R ⋅ (ν(e -f )) = G R ⋅ e.
In the case where G R is classical, we provide continuous families of elliptic semisimple orbits whose limits attempt to approximate minimal nilpotent orbits (Theorem 8.3). Finally, for G R = SU(p, q), we show that (the closure of) every nilpotent orbit can be realized as the limit of a family of elliptic semisimple orbits (Theorem 8.6).

We thank the referees for their careful reading and their suggestions which helped to improve the paper.

Preliminaries

Let G R be a connected simple linear real Lie group with Lie algebra g R and let G be the complexification of G R whose Lie algebra g is the complexification of g R . Fix a maximal compact subgroup K R of G R with Lie algebra k R fixed pointwise by a Cartan involution θ. The Lie algebra k of the complexification

K of K R is the complexification of k R . Write (2.1) g R = k R ⊕ p R and g = k ⊕ p
for the corresponding Cartan decompositions of g R and g, respectively.

An element x of g R (resp., g) is said to be semisimple if the endomorphism ad(x) ∶ g R → g R (resp., ad(x) ∶ g → g) is semisimple. A semisimple element x of g R is said to be elliptic (resp., hyperbolic) if the eigenvalues of ad(x) are all pure imaginary (resp., real). An element x of g R (resp., g) is said to be nilpotent

if the endomorphism ad(x) ∶ g R → g R (resp., ad(x) ∶ g → g) is nilpotent
. By Jordan decomposition, any element x of g R (resp., g) can be written in a unique way as

x = x s + x n ,
where x s (resp., x n ) is semisimple (resp., nilpotent) in g R (resp., g) and [x s , x n ] = 0. Moreover, any element in g commuting with x commutes with x s and x n as well.

If x belongs to g R , then its semisimple part x s can be written in a unique way as

x s = x e + x h , that is, x = x e + x h + x n ,
where x e is elliptic, x h is hyperbolic, and x e , x h , x n commute with each other. Moreover, any G R -orbit in g R contains an element x such that x e belongs to k R and x h belongs to p R (see [START_REF] Vogan | The method of coadjoint orbits for real reductive groups[END_REF]Proposition 2.10]).

If e is a nonzero nilpotent element of g R (resp., g), by Jacobson-Morozov theorem there exist a semisimple element h and a nilpotent element f in g R (resp., g) such that ([6, Theorem 9.2.1]):

[h, e] = 2e, [h, f ] = -2f, and [e, f ] = h.
The triple {h, e, f } is said to be a standard triple (or sl 2 -triple), while h and e are respectively the neutral element and the nilpositive element of the triple. It is known that any standard triple {h, e, f } ⊂ g R is G Rconjugate to a Cayley triple {h ′ , e ′ , f ′ }, i.e., such that θ(e ′ ) = -f ′ , θ(f ′ ) = -e ′ , and θ(h ′ ) = -h ′ . Furthermore, we associate with a Cayley triple {h, e, f } ⊂ g R the standard triple {h ′ , e ′ , f ′ } ⊂ g given by

h ′ = i(e -f ), e ′ = 1 2 (e + f + ih), f ′ = 1 2 (e + f -ih).
The triple {h ′ , e ′ , f ′ } is the Cayley transform of {h, e, f }. Note that h ′ lies in k, while e ′ and f ′ lie in p. Such a standard triple is called normal. The Kostant-Sekiguchi correspondence is the bijection such that

(2.2) KS ∶ G R ⋅ e ↦ K ⋅ e ′
between nilpotent G R -orbits of g R and nilpotent K-orbits of p ([6, Chapter 9]). We will use the following result due to Mal'cev about conjugation of standard triples.

Theorem 2.1 (Mal'cev, [6, §3.4.12]). Any two standard triples of g with the same neutral element h are conjugate by an element of the connected component of the centralizer Z G (h) of h in G.

Fix a standard triple {h, e, f } in g. The eigenvalues of ad(h) are integers, therefore one gets the following grading of g:

g = ⊕ m∈Z g m , g m ∶= {X ∈ g [h, X] = mX}.
Set l = g 0 and u = ⊕ m>0 g m . The Lie subalgebra

(2.3) q = l ⊕ u
is a parabolic subalgebra associated with the nilpotent element e. The corresponding parabolic subgroup Q in G has Levi decomposition Q = LU , where L (resp., U ) has Lie algebra l (resp., u). In the case where g 2j+1 = {0} for all j ∈ Z, the nilpotent element e (or the standard triple {h, e, f }) is said to be even. In other words, g decomposes as a direct sum of irreducible representations of the subalgebra spanned by the sl 2 -triple {h, e, f } and all of the summands have even highest weight. In the special case where l is a maximal torus t in g, the nilpotent radical u coincides with the subspace n generated by positive t-roots in g, and the

parabolic subgroup Q is a Borel subgroup B = LN with Lie algebra b = t ⊕ n and N = exp(n).
By assumption, G is a closed subgroup of GL n (C) for some integer n ≥ 1. The nilpotent cone N (gl n (C))

is defined as the set of nilpotent elements in End(C n ). Similarly, the nilpotent cones N (g) and N (g R ) in g and g R are defined, respectively, as the sets of nilpotent elements in g and in g R . We have

N (g) = g ∩ N (gl n (C)).
Since N (gl n (C)) is Zariski closed in gl n (C), then the nilpotent cone N (g) is Zariski closed in g. On the other hand, the algebra S(g * ) G of G-invariant polynomial functions on g is graded by the degree:

S(g * ) G = ⊕ r≥0 S r (g * ) G ,
where S r (g * ) is the space of homogeneous polynomial functions of degree r on g.

If S + (g * ) G ∶= ⊕ r>0 S r (g * ) G , then N (g) = x ∈ g f (x) = 0 for all f ∈ S + (g * ) G = G ⋅ n.
In particular, the variety N (g) is irreducible and dim N (g) = 2 dim n = dim g -rank g.

Here rank g denotes the rank of g, i.e., the dimension of a Cartan subalgebra in g. Yet there is another particular type of nilpotent orbits in g, induced from any parabolic subalgebra q in g. Namely, the Richardson orbit O Rich (q) associated to q is the unique dense orbit in G ⋅ u, where u denotes the nilradical of q. Note that any semisimple element x ∈ g also gives rise to a Richardson orbit. Indeed, if x is semisimple, then its centralizer g x ∶= {z ∈ g [x, z] = 0} is a Levi subalgebra of g, which means that there is a parabolic subalgebra q such that g x is a Levi subalgebra of q. The Richardson orbit

(2.4) O Rich (x) ∶= O Rich (q)
is independent of the choice of q (see, e.g., [START_REF] Collingwood | Nilpotent orbits in semisimple Lie algebras[END_REF]Theorem 7.1.3]). Note also that

(2.5) dim O Rich (x) = 2 dim u = dim g -dim g x = dim G ⋅ x
(see [6, §7] for the first equality). In the case where x has real eigenvalues on g (which occurs for instance when x is a hyperbolic semisimple element in g R ), then g has an eigenspace decomposition indexed by real numbers

(2.6) g = ⊕ μ∈R g μ , g μ ∶= {z ∈ g [x, z] = μz},
and q ∶= ⊕ μ≥0 g μ is a parabolic subalgebra which contains g 0 = g x as a Levi factor; we have

(2.7) O Rich (x) = G ⋅ ⊕ μ>0 g μ .
If x = h belongs to an sl 2 -triple {h, e, f }, then the nilpotent orbit G ⋅ e is contained in the closure of O Rich (h).

The equality O Rich (h) = G ⋅ e holds if and only if e is even. Any even nilpotent orbit is therefore a Richardson orbit, but the converse is not true. Since g is simple, the subregular orbit O subreg is a Richardson orbit.

Note that every nilpotent orbit of sl n (C) is a Richardson orbit.

Recall that nilpotent G-orbits in the classical Lie algebras are parametrized by partitions (d 1 , . . . , d k ) with Chapter 5]):

d 1 ≥ d 2 ≥ . . . ≥ d k ≥ 1 such that ([6,
• d 1 + d 2 + . . . + d k = n, when g ≃ sl n (C); • d 1 + d 2 + . . . + d k = 2n + 1
and the even d j 's occur with even multiplicity, when g ≃ so 2n+1 (C);

• d 1 + d 2 + . . . + d k = 2n
and the odd d j 's occur with even multiplicity, when g ≃ sp 2n (C);

• d 1 + d 2 + . . . + d k = 2n
and the even d j 's occur with even multiplicity, when g ≃ so 2n (C).

In each case the parametrization is one-to-one, except in the case of so 2n (C) where the partitions having all the d j 's even (and occurring with even multiplicity) are each associated to two orbits.

Finally, the Zariski closure of nilpotent orbits can be described as follows. Given two partitions d

= (d 1 , . . . , d k ) and f = (f 1 , . . . , f ), d is said to dominate f , i.e., f ≤ d, if we have k ≤ and 1≤i≤j d i ≥ 1≤i≤j f i for 1 ≤ j ≤ k.

Limit of adjoint orbits: definition and basic properties

In this section we introduce a topological limit of adjoint orbits (Definition 3.1) and establish its basic properties (Propositions 3.2, 3.5, and 3.7). Recall that G R is a connected simple linear real Lie group with Lie algebra g R , G is the complexification of G R , and its Lie algebra g is the complexification of g R .

Definition 3.1. Given an element x ∈ g R , consider the family {G R ⋅ (νx)} ν>0 of G R -orbits (indexed by ν).

The limit of this family of orbits is defined as the topological set

lim ν→0 + G R ⋅ (νx) ∶= ⋂ >0 ⋃ ν∈(0, ) G R ⋅ (νx).
In other words, an element z belongs to the limit if and only if there are sequences

{ν k } k≥1 ⊂ R * + converging to 0 and {g k } k≥1 ⊂ G R such that z = lim k→+∞ ν k Ad(g k )(x). Proposition 3.2. Let x ∈ g R with Jordan decomposition x = x s + x n . (a) The limit of orbits lim ν→0 + G R ⋅ (νx) is nonempty, closed, G R -stable and contained in the nilpotent cone of g R : lim ν→0 + G R ⋅ (νx) ⊂ N (g R ).
In particular, the limit of orbits lim ν→0 + G R ⋅ (νx) is a finite union of nilpotent orbits. (b) The nilpotent part x n belongs to the limit of orbits lim

ν→0 + G R ⋅ (νx). In particular, lim ν→0 + G R ⋅ (νx) always contains the closure of G R ⋅ x n : G R ⋅ x n ⊂ lim ν→0 + G R ⋅ (νx). (c)
The limit of orbits of the semisimple part x s is contained in the limit of orbits of x:

lim ν→0 + G R ⋅ (νx s ) ⊂ lim ν→0 + G R ⋅ (νx). (d) If x = x n
is a nilpotent element then the limit of orbits lim ν→0 + G R ⋅ (νx) coincides with the closure of the G R -orbit of x:

lim ν→0 + G R ⋅ (νx) = G R ⋅ x. Proof. (a) By definition lim ν→0 + G R ⋅ (νx) is closed and G R -stable, and nonempty (since it contains 0). Let z ∈ lim ν→0 + G R ⋅ (νx) and let us show that z is nilpotent in g R . There are sequences {ν k } k≥1 ⊂ R * + converging to 0 and {g k } k≥1 ⊂ G R such that z = lim k→+∞ ν k Ad(g k )(x), hence ad(z) = lim k→+∞ ν k Ad(g k ) ○ ad(x) ○ Ad(g k ) -1 .
This yields the following relation between characteristic polynomials:

det(ad(z) -Xid) = lim k→+∞ det(ν k ad(x) -Xid),
from which we conclude that the endomorphism ad(z) is nilpotent, hence z is a nilpotent element. This shows (a).

To show (b), (c), and (d), it is useful to note that

(3.1) G R ⋅ (x s + tx n ) = G R ⋅ (x s + x n ) for all t ∈ R * + .
Indeed, the Jacobson-Morozov theorem implies that we can find an element x ′ in the semisimple part [l, l]

of the Levi subalgebra l = z g R (x s ) ∶= {z ∈ g R [x s , z] = 0} such that [x ′ , x n ] = 2x n .
Then, for every t > 0 we get

x s + tx n = Ad(exp( ln t 2 x ′ ))(x s + x n ) ∈ G R ⋅ (x s + x n ). (b) By (3.1), we have x n + νx s ∈ G R ⋅ (νx) for all ν > 0, hence x n ∈ lim ν→0 + G R ⋅ (νx)
, and the desired inclusion follows from the properties of the limit given in (a).

(c) By (3.1), for every ν > 0, we have

νx s + 1 k x n ∈ G R ⋅ (νx) ⊂ G R ⋅ (νx) for all k ≥ 1, hence G R ⋅ (νx s ) ⊂ G R ⋅ (νx),
so that the claimed inclusion follows from the definition of the limit.

(d) By (3.1), we have G R ⋅ x = G R ⋅ (νx) for all ν > 0, and the claimed equality immediately follows from the definition of the limit.

Example 3.3. The elements of the Lie algebra g R = sl 2 (R) can be written in the form

x = x 1 x 2 + x 3 x 2 -x 3 -x 1 with x 1 , x 2 , x 3 ∈ R.
The nilpotent cone is

N (sl 2 (R)) = {x ∈ sl 2 (R) det x = 0} = {x ∈ sl 2 (R) x 2 1 + x 2 2 = x 2 3 }
and it comprises two nontrivial nilpotent orbits,

O + ∶= {x ∈ sl 2 (R) x 2 1 + x 2 2 = x 2 3 , x 3 > 0} = SL 2 (R) ⋅ 0 1 0 0 , O -∶= {x ∈ sl 2 (R) x 2 1 + x 2 2 = x 2 3 , x 3 < 0} = SL 2 (R) ⋅ 0 0 1 0 .
We consider the elliptic semisimple element

η ∶= 0 1 -1 0 .
In fact, every elliptic semisimple element of sl 2 (R) is conjugate to νη for some ν ∈ R. For ν ∈ R * , we have

SL 2 (R) ⋅ (νη) = {x ∈ sl 2 (R) x 2 3 = x 2 1 + x 2 2 + ν 2 , sign(x 3 ) = sign(ν)}.
This yields

lim ν→0 + SL 2 (R) ⋅ (νη) = ⋂ >0 ⋃ 0<ν< SL 2 (R) ⋅ (νη) = ⋂ >0 {x ∈ sl 2 (R) 0 ≤ x 2 3 -x 2 1 -x 2 2 ≤ 2 , x 3 ≥ 0} = {x ∈ sl 2 (R) x 2 3 = x 2 1 + x 2 2 , x 3 ≥ 0} = {0} ∪ O + = O + .
A similar calculation shows that

lim ν→0 + SL 2 (R) ⋅ (ν(-η)) = {0} ∪ O -= O -.
We now consider the hyperbolic semisimple element

h ∶= 1 0 0 -1
and let us note that every hyperbolic semisimple element of sl 2 (R) is conjugate to νh for some ν ≥ 0; in particular h and -h are conjugate under SL 2 (R). For ν > 0, we have

SL 2 (R) ⋅ (νh) = {x ∈ sl 2 (R) x 2 3 = x 2 1 + x 2 2 -ν 2 }, so that lim ν→0 + SL 2 (R) ⋅ (νh) = {0} ∪ O + ∪ O -= N (sl 2 (R))
. This example is illustrated in Figure 3. Remark 3.4. Given x, z ∈ g R , it is readily seen that we have

z ∈ lim ν→0 + G R ⋅ (νx) ⇔ -z ∈ lim ν→0 + G R ⋅ (ν(-x)).
For instance, in Example 3.3 above, we have ν(-η)) can also be deduced from the formula giving lim

O -= SL 2 (R) ⋅ 0 -1 0 0 , hence the formula for lim ν→0 + SL 2 (R) ⋅ (
ν→0 + SL 2 (R) ⋅ (νη).
The following is a characterization of the limit of orbits in terms of the Slodowy slice of a nilpotent element.

This characterization appears in [8, Corollary 2.2] (see also the remark below), where it is attributed to Barbasch and Vogan. For the sake of completeness, we give a proof.

Proposition 3.5. Let x ∈ g R . Let e ∈ g R ∖ {0} be a nilpotent element and let {h, e, f } ⊂ g R be a standard triple. Let z g R (f ) = {z ∈ g R [z, f ] = 0}.
The affine space e + z g R (f ) is called a Slodowy slice. The following conditions are equivalent:

(i) e belongs to the limit of orbits lim

ν→0 + G R ⋅ (νx); (ii) G R ⋅ x intersects the Slodowy slice e + z g R (f ).
Proof. (ii)⇒(i): We may assume that x = e + z with z ∈ z g R (f ). It follows from sl 2 -theory that z can be written as

z = m j=0 z j with [h, z j ] = -jz j . Then Ad(exp(th))(x) = k≥0 t k k! ad(h) k (e + z) = k≥0 (2t) k k! e + m j=0 k≥0 (-jt) k k! z j = exp(2t)e + m j=0
exp(-jt)z j hence, letting t = -ln(ν) 2 , we get

Ad(exp(th))(νx) = e + m j=0 ν 1+ j 2 z j ν→0 + → e.
Therefore, e ∈ lim

ν→0 + G R ⋅ (νx). (i)⇒(ii): Assume that e ∈ lim ν→0 + G R ⋅ (νx). This implies that e ∈ ⋃ ν∈(0,1) G R ⋅ (νx). Note that G R ⋅ (e + z g R (f ))
is an open neighborhood of e in g R , hence there are g ∈ G R , ν ∈ (0, 1), and z ∈ z g R (f ) such that

x = Ad(g) 1 ν e + z .
Letting t = ln ν 2 , we have

Ad(exp(th)) 1 ν e = 1 ν k≥0 t k k! ad(h) k (e) = 1 ν k≥0 (2t) k k! e = e while z ′ ∶= Ad(exp(th))(z) ∈ z g R (f ), since z g R (f ) is stable by ad(h). Therefore, e + z ′ = (Ad(exp(th)) ○ Ad(g -1 ))(x) ∈ (G R ⋅ x) ∩ (e + z g R (f )).
This shows (ii).

Remark 3.6. In [8, Corollary 2.2], the set

N x ∶= N (g R ) ∩ ⋃ t>0 G R ⋅ (tx)
(or, in fact, its analogue in the dual Lie algebra g * R ) is considered. Note that this set coincides with the limit of orbits:

(3.2) N x = lim ν→0 + G R ⋅ (νx). Indeed, the inclusion lim ν→0 + G R ⋅ (νx) = ⋂ >0 ⋃ ν∈(0, ) G R ⋅ (νx) ⊂ ⋃ ν>0 G R ⋅ (νx)
is clear, and since we know that lim

ν→0 + G R ⋅ (νx) ⊂ N (g R ) (see Proposition 3.2 (a)
), we get the inclusion ⊃ in (3.2). For the other inclusion, first note that this inclusion holds in the case where x is nilpotent, because

we then have G R ⋅ (tx) = G R ⋅ x for all t > 0, hence N x = G R ⋅ x = lim ν→0 + G R ⋅ (νx) (see Proposition 3.2 (d))
. We now assume that x is not nilpotent. Let z ∈ N x . Then z is nilpotent and there are sequences {t k } k≥1 ⊂ (0, +∞) and {g k } k≥1 ⊂ G R such that z = lim k→+∞ t k Ad(g k )(x). We claim that the sequence {t k } k≥1 converges to 0; in which case we are able to conclude that z belongs to the limit of orbits as explained below Definition 3.1. Arguing by contradiction, assume that along a relabeled subsequence

{t k } k≥1 , we have t k → t 0 > 0 or t k → +∞, then this yields 1 t 0 z ∈ G R ⋅ x or 0 ∈ G R ⋅ x,
and in both cases we get that x must be nilpotent since 1 t0 z and 0 are nilpotent, a contradiction.

We conclude this section with the observation that every nilpotent orbit fits into a limit of semisimple orbits. The statement is actually more precise.

Proposition 3.7. Let e ∈ g R be a nilpotent element. Then there is x h ∈ g R semisimple, hyperbolic and there

is x e ∈ g R semisimple, elliptic such that G R ⋅ e ⊂ lim ν→0 + G R ⋅ (νx h ) and G R ⋅ e ⊂ lim ν→0 + G R ⋅ (νx e ).
Proof. We can assume that e = 0 (if e = 0, we can take x e = x h = 0). Let {h, e, f } ⊂ g R be an sl 2 -triple which contains e. Since ad(h) has integer eigenvalues in g R , the element x h ∶= h is hyperbolic. We have Ad(exp(-1 2ν e))(νx h ) = νx h + e → e as ν → 0, hence e belongs to the limit lim

ν→0 + G R ⋅ (νx), whence G R ⋅ e ⊂ lim ν→0 + G R ⋅ (νx) (see Proposition 3.2 (a)
). Up to G R -conjugation, we may assume that {h, e, f } is a Cayley triple. Then, the element x e ∶= e -f belongs to k R , so it is semisimple and elliptic, and we have

Ad(exp(th))(νx e ) = ν(exp(2t)e -exp(-2t)f ) for all t ∈ R, hence if we take t = -ln ν 2 we conclude that e ∈ lim ν→0 + G R ⋅ (νx e ), and therefore G R ⋅ e ⊂ lim ν→0 + G R ⋅ (νx e ).

Asymptotic cones

In this section, we define and relate various (complex or real) asymptotic cones that can be attached to a subset X of a (complex or real) vector space V (Section 4.1). In the case where X is the adjoint orbit of a semisimple element in the complex Lie algebra g, we recall a result due to Borho and Kraft, which describes the asymptotic cone as the closure of the Richardson orbit (Theorem 4.6). In the case where X is an adjoint orbit in the real Lie algebra g R , the limit of orbits of Definition 3.1 can be described as an asymptotic cone, and useful information can be deduced from the result of Borho and Kraft (Section 4.2).

4.1. General definitions of asymptotic cones. In this section, V denotes a finite-dimensional complex vector space and V R is a real vector space realized as a real form of V .

We consider the extension V ⊕ C of V . Both projective spaces P(V ) and P(V ⊕ C) are equipped with the Zariski topology. Consider the maps

ι ∶ V → P(V ⊕ C), v ↦ [v ⊕ 1], κ ∶ P(V ) → P(V ⊕ C), [v] ↦ [v ⊕ 0], π ∶ V ∖ {0} → P(V ), v ↦ [v].
Definition 4.1. For every subset X ⊂ V , the projective asymptotic cone for X is defined as

AC p (X) = ι(X) ∩ κ(P(V ))
and the (affine) asymptotic cone is then the following closed cone in V :

AC(X) = {0} ∪ π -1 (κ -1 (AC p (X))).
We adapt this construction to the real setting in the following way. We consider the sphere

S(V R ) ∶= (V R ∖ {0}) R * + ,
the standard embedding

ι + ∶ V R ↪ S(V R ⊕ R), v ↦ [v ⊕ 1],
and the surjection

π + ∶ V R ∖ {0} → S(V R ).
Definition 4.2. For a subset X ⊂ V R , we define the real asymptotic cone to be

AC R (X) ∶= {0} ∪ (π + ) -1 (S(V R ) ∩ ι + (X)).
Here we consider the analytic topology on V R ⊕ R and the quotient topology on S(V R ⊕ R).

Remark 4.3. The construction of the cone AC R (X) can also be described in the following explicit way.

• Consider V R and V R ⊕ R endowed with Euclidean norms, so that S(V R ) and S(V R ⊕ R) coincide with the usual spheres.

• Note that V R and V R ⊕ 1 ∶= {v ⊕ 1 v ∈ V R } are two parallel hyperplanes in V R ⊕ R. Project the subset X ⊕ 1 ⊂ V R ⊕ 1 onto the sphere S(V R ⊕ R)
and consider the closure of this projection.

• Intersect this closure with the hyperplane V R , and take the (nonnegative) cone of V R determined by this intersection.

Figure 4 illustrates the construction in the case where X is a conic. We point out a relation between the real and complex asymptotic cones, as well as additional characterizations of the real cone. In particular, we shall see that AC R (X) coincides with the asymptotic cone AC(X)

introduced by Harris-He-Ólafsson in [START_REF] Harris | Wave front sets of reductive Lie group representations[END_REF]Introduction]:

AC(X) ∶= {z ∈ V R Γ an open cone containing z ⇒ Γ ∩ X is unbounded} ∪ {0}. Proposition 4.4. (a) For every subset X ⊂ V R , we have AC R (X) ⊂ AC(X). (b) Let X ⊂ V R .
For every z ∈ V R ∖ {0}, the following conditions are equivalent:

(i) z ∈ AC R (X);
(ii) there are sequences {ν n } n≥1 ⊂ R * + converging to 0 and {x n } n≥1 ⊂ X such that z = lim n→+∞ ν n x n ;

(iii) for every open (positive) cone Γ ⊂ V R which contains z, the intersection Γ ∩ X is unbounded.

In particular, AC R (X) and AC(X) coincide:

AC(X) = AC R (X).
We also point out the following simple criterion for the asymptotic cone to be nontrivial: Proposition 4.5. For every subset X ⊂ V R , the cone AC R (X) is nontrivial (i.e., = {0}) if and only if the set X is unbounded.

Proof of Proposition 4.4. (a) The inclusion V R ⊂ V yields a continuous mapping φ ∶ S(V R ⊕ R) → P(V ⊕ C)
with φ(S(V R )) ⊂ P(V ). Moreover, we have a commutative diagram

V R ∖ {0} π + G G S(V R ) 1 G G φ S(V R ⊕ R) φ V R ι + o o V ∖ {0} π G G P(V ) 1 G G P(V ⊕ C) V ι o o
For a subset X ⊂ V R , we get

(π + ) -1 (S(V R ) ∩ ι + (X)) ⊂ (π + ) -1 (φ -1 (P(V )) ∩ φ -1 (ι(X))) = (V R ∖ {0}) ∩ π -1 (P(V ) ∩ ι(X)).
This yields the desired inclusion.

(b) Let π + also denote the canonical map

(V R ⊕ R) ∖ {0} → S(V R ⊕ R).
By virtue of the continuity of π + , for every neighborhood

W of π + (z) in S(V R ⊕ R), there are > 0 and a neighborhood V of z in V R such that π + (V ⊕ (-, )) ⊂ W. Conversely, if > 0 and V is a neighborhood of z in V R , then π + (V ⊕ (-, )) is a neighborhood of π + (z) in S(V R ⊕ R).
Condition (ii) is equivalent to

∀ > 0, ∀V ⊂ V R neighborhood of z, ∃ν ∈ (0, ), ∃x ∈ X, such that νx ∈ V. (4.1)
By definition of the cone AC R (X), we have

z ∈ AC R (X) ⇐⇒ π + (z) ∈ ι + (X) ⇐⇒ ∀ > 0, ∀V ⊂ V R neighborhood of z, π + (V ⊕ (-, )) ∩ ι + (X) = ∅. The intersection π + (V ⊕ (-, )) ∩ ι + (X) is nonempty if and only if there is x ∈ X such that ι + (x) = [x ⊕ 1]
belongs to π + (V ⊕ (-, )), that is, if and only if there are x ∈ X and ν ∈ R * + such that

ν(x ⊕ 1) ∈ V ⊕ (-, ).
This is equivalent to saying that there are x ∈ X and ν ∈ (0, ) such that νx ∈ V. This establishes the equivalence between (i) and (ii).

(ii)⇒(iii): Let {ν n } n≥1 ⊂ R * + and {x n } n≥1 ⊂ X be as in (ii). Note that, since the sequence {ν n x n } n≥1 ⊂ V R has a nonzero limit while {ν n } n≥1 converges to 0, the sequence {x n } n≥1 is necessarily unbounded. Now, let Γ ⊂ V R be an open cone which contains z. Since Γ is open, there is n 0 ≥ 1 such that, for all n ≥ n 0 , we have ν n x n ∈ Γ. Since Γ is a cone, we get x n ∈ Γ ∩ X for all n ≥ n 0 . Therefore, the intersection Γ ∩ X is unbounded.

Therefore, (ii) implies (iii).

(iii)⇒(ii): Let > 0 and let V ⊂ V R be an open, bounded neighborhood of z. Let us consider the open cone Γ ∶= {tv t ∈ R * + , v ∈ V}. Since z belongs to that cone, condition (iii) implies that there is an unbounded sequence {x n } n≥1 contained in Γ ∩ X. The definition of Γ yields a sequence {ν n } n≥1 ⊂ R * + such that ν n x n ∈ V for all n ≥ 1. Since V is bounded while {x n } n≥1 is unbounded, the sequence {ν n } n≥1 has to converge to 0, hence we can find n ≥ 1 such that ν n ∈ (0, ). We have shown the condition stated in (4.1), and this implies that (ii) holds whenever (iii) is satisfied. The proof of the proposition is complete.

Proof of Proposition 4.5. Let ⋅ be any norm on the finite-dimensional space V R .

Assume first that there exists z ∈ AC R (X), z = 0. By Proposition 4.4 (b), there is a sequence {ν n } n≥1 ⊂ R * + converging to 0 and a sequence {x n } n≥1 ⊂ X such that

lim n→+∞ ν n x n = z, hence lim n→+∞ ν n x n = z = 0.
This implies that x n → +∞ as n → +∞. Therefore, X contains an unbounded sequence.

Next assume that X contains a sequence {x n } n≥1 such that x n → +∞ as n → +∞. Up to considering a subsequence, we may assume that the bounded sequence { xn xn } n≥1 converges to some y ∈ V R such that y = 1. This yields

ι + (x n ) = [x n ⊕ 1] = x n x n ⊕ 1 x n → [y ⊕ 0] as n → +∞, hence S(V R ) ∩ ι + (X) is nonempty as it contains [y ⊕ 0]
. This implies that the cone AC R (X) is nontrivial as it contains the half line {ty t ≥ 0}.

4.2. Asymptotic cones of adjoint orbits. In this section, we take V = g and V R = g R . The asymptotic cone of the orbit of a semisimple element x ∈ g can be described in the following way.

AC(G ⋅ x) = O Rich (x).
In the real setting, Proposition 4.4 yields the following result which relates the asymptotic cone with the limit of orbits.

Theorem 4.7.

(a) For every x ∈ g R , we have

lim ν→0 + G R ⋅ (νx) = AC R (G R ⋅ x) ⊂ AC(G R ⋅ x) ⊂ AC(G ⋅ x). (b)
In particular, for every x ∈ g R semisimple, we have

lim ν→0 + G R ⋅ (νx) ⊂ O Rich (x) ∩ g R .

Nontriviality of the limit of orbits

Relying on the identification of the limit of orbits with an asymptotic cone (Theorem 4.7) and on the criterion stated in Proposition 4.5 to guarantee the nontriviality of an asymptotic cone, we are in position to state the following result regarding the nontriviality of the limit of orbits.

Theorem 5.1. Recall that the group G R is connected simple linear and non-compact. Then, for every nonzero x ∈ g R , we have lim

ν→0 + G R ⋅ (νx) = {0}.
Proof. Since ad(x) = 0, knowing that k R = [p R , p R ] (because g R is simple and non-compact; see [11, VI.12.24]), we can find z ∈ p R such that [x, z] = 0. Then z is semisimple and hyperbolic (see, e.g., [START_REF] Knapp | Lie groups beyond an introduction[END_REF]Theorem 6.51]), so that the Lie algebra g R decomposes as a sum of eigenspaces

g R = ⊕ k=1 (g R ) k where (g R ) k ∶= {y ∈ g R [z, y] = μ k y}
for a sequence {μ 1 , . . . , μ } of real numbers. In particular we can write

x = x 1 + . . . + x with x k ∈ (g R ) k .
Moreover, since [z, x] = 0, there must be k 0 ∈ {1, . . . , } such that x k 0 = 0 and μ k 0 = 0. Then

Ad(exp(tz))(x) = k=1 exp(tμ k )x k (for t ∈ R)
determines an unbounded curve contained in the orbit G R ⋅ x. The claim now follows from Proposition 4.5

and Theorem 4.7.

Remark 5.2. (a) In the case where the Lie algebra g R is compact, its nilpotent cone is reduced to {0}, hence every limit of orbits is trivial.

(b) Theorem 5.1 extends to the case where g R is semisimple, with decomposition as a sum of simple ideals

g R = g 1 R ⊕ . . . ⊕ g R
, not all of them being compact, and x = x 1 + . . . + x has at least one nonzero summand x i in a non-compact simple factor g i R of g R .

Note that one can bound from above the dimension of the limit of orbits as

(5.1) dim lim ν→0 + G R ⋅ (νx) ≤ dim G R ⋅ x.
Moreover, the equality holds if and only if the Richardson orbit O Rich (x) intersects the limit lim ν→0 + G R ⋅ (νx). This follows from (2.5), Theorem 4.7, and the fact that dim

C G ⋅ x = dim R G R ⋅ x.
It is however more difficult to bound from below the dimension of the limit. Take for example G R = SU(n -1, 1) with n ≥ 4. On one hand, if x ∈ su(n -1, 1) is regular, then dim G R ⋅ x = n 2 -n. On the other hand, every nilpotent element of su(n -1, 1) is of nilpotency order ≤ 3 (see [START_REF] Collingwood | Nilpotent orbits in semisimple Lie algebras[END_REF]Theorem 9.3.3]) and its nilpotent orbit has dimension ≤ 4n -6 (see [START_REF] Collingwood | Nilpotent orbits in semisimple Lie algebras[END_REF]Corollary 6.1.4]). Thus, in this case, the inequality in (5.1) is strict.

Even in cases in which every nilpotent orbit of the complexified Lie algebra g admits real forms, equality does not necessarily hold in (5.1). Take for instance G R = SU(2, 1) and

x = ⎛ ⎜ ⎝ -2i 0 0 0 -i 0 0 0 3i ⎞ ⎟ ⎠ ∈ su(2, 1).
The element x is regular, semisimple (and elliptic), but the limit lim ν→0 + G R ⋅ (νx) does not contain the sole principal nilpotent orbit of su(2, 1) as it only consists of elements of nilpotency order ≤ 2; this follows from Section 8.3 below.

Limit of hyperbolic orbits

In this section, we consider the limit of orbits associated with a nonzero hyperbolic semisimple element x ∈ g R . This means that ad(x) has real eigenvalues on g R , thus also on g, and it determines a grading of the Lie algebra g as in (2.6). Note that the eigenspace g μ of the grading is given by g μ = (g R ) μ + i(g R ) μ , where (g R ) μ stands for the eigenspace corresponding to the same eigenvalue in g R . Let

u(x) ∶= ⊕ μ>0 g μ and u R (x) ∶= ⊕ μ>0 (g R ) μ
denote the nilradical of the parabolic subalgebra defined by x and the corresponding real form. By (2.7), the Richardson nilpotent orbit O Rich (x) corresponding to x is such that

O Rich (x) = G ⋅ u(x).
The following result provides a characterization of the limit of the orbits G R ⋅ (νx). The proof relies on a standard argument in Lie theory involving polarization of nilpotent orbits (see, e.g., [START_REF] Borho | Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen[END_REF], [START_REF] Rossmann | Limit orbits in reductive Lie algebras[END_REF]). Theorem 6.1. Let x ∈ g R be a hyperbolic semisimple element.

(a) The subset G R ⋅ u R (x) ⊂ g R is closed and equidimensional. It is of the form G R ⋅ u R (x) = O 1 ∪ . . . ∪ O k
where O 1 , . . . , O k ⊂ g R are nilpotent G R -orbits which are all of the same dimension.

(b) We have

lim ν→0 + G R ⋅ (νx) = G R ⋅ u R (x) = O 1 ∪ . . . ∪ O k ⊂ O Rich (x) ∩ g R . Proof. (a) The fact that G R ⋅ u R (x) is closed follows from Lemma 6.2 below. Let d = max{dim G R ⋅ z z ∈ u R (x)}.
We claim:

(6.1) the subset Z d ∶= {z ∈ u R (x) dim G R ⋅ z = d} is open and dense in u R (x).
Indeed, we first note that

dim G R ⋅ z = dim g R -dim{y ∈ g R [z, y] = 0} = rank(ad(z))
hence we have rank(ad(z)) ≤ d for all z ∈ u R (x) and

Z d = {z ∈ u R (x) rank(ad(z)) ≥ d}.
(z) + M (z 0 ) is a submatrix of Mat B (ad(z + z 0 )) of size d × d which is invertible, ensuring that z + z 0 ∈ Z d , whenever > 0 is small enough. This shows that Z d is dense in u R (x).
The claim made in (6.1) is justified.

We infer that G R ⋅ Z d is dense in G R ⋅ u R (x). By definition, G R ⋅ Z d is G R -stable, nilpotent, hence it is a
union of finitely many nilpotent orbits:

G R ⋅ Z d = O 1 ∪ . . . ∪ O k , whence G R ⋅ u R (x) = G R ⋅ Z d = O 1 ∪ . . . ∪ O k . Moreover, since O j ⊂ G R ⋅ Z d ,
lim ν→0 + G R ⋅ (νx) ⊂ G R ⋅ u R (x). Let Q R ⊂ G R be the parabolic subgroup of Lie algebra q R = ⊕ μ≥0 (g R ) μ . Hence u R (x) is the nilradical of q R .
First we note that

lim ν→0 + G R ⋅ (νx) = ⋂ >0 ⋃ ν∈(0, ) G R ⋅ (νx) ⊂ ⋃ ν∈(0,1) G R ⋅ (νx) = G R ⋅ {νx 0 < ν < 1}. Since the subset Q R ⋅ {νx 0 < ν < 1} ⊂ g R is closed and Q R -stable, its image by G R is closed (see Lemma 6.
2). This yields

lim ν→0 + G R ⋅ (νx) ⊂ G R ⋅ Q R ⋅ {νx 0 < ν < 1}. Moreover, we have Q R = U R Z G R (x), where U R = exp(u R (x)) and Z G R (x) is the stabilizer of x in G R . This implies that Q R ⋅ {νx 0 < ν < 1} = U R ⋅ {νx 0 < ν < 1}.
Every u ∈ U R can be written as u = exp(z) with z ∈ u R (x), and we have

Ad(u)(x) = k≥0 1 k! ad(z) k (x) = x + z ′ with z ′ ∈ u R (x) since u R (x) is both ad(x)-stable and ad(u R (x))-stable. Whence Q R ⋅ {νx 0 < ν < 1} ⊂ {νx + z ′ 0 < ν < 1, z ′ ∈ u R (x)}.
Altogether, we get the inclusion

lim ν→0 + G R ⋅ (νx) ⊂ G R ⋅ {νx + z ′ 0 ≤ ν ≤ 1, z ′ ∈ u R (x)}. We have also that lim ν→0 + G R ⋅ (νx) is contained in the nilpotent cone N (g R ) (see Proposition 3.2)
, and an element of the form νx + z ′ can be nilpotent only if ν = 0. Therefore, (6.2) is established.

It remains to show the inclusion

(6.3) u R (x) ⊂ lim ν→0 + G R ⋅ (νx). We claim that (6.4) {x + z z ∈ u R (x)} ⊂ G R ⋅ x.
Once we have shown (6.4), we can deduce that, for all z ∈ u R (x), all ν > 0, the element x + 1 ν z belongs to G R ⋅ x, so that νx + z belongs to G R ⋅ (νx). This yields z ∈ lim ν→0 + G R ⋅ (νx). Since z ∈ u R (x) is arbitrary, this establishes (6.3). Therefore, it remains to show (6.4).

For proving (6.4), let z ∈ u R (x). Let (0 <)μ 1 < . . . < μ N be the list of all positive eigenvalues of ad(x). We write z = z 1 +z 2 +. . .+z N with z j ∈ (g R ) μj . We construct an element g ∈ G R such that Ad(g)(x) = x+z. Arguing by induction, we show that, for all k = 0, . . . , N , there is

g (k) ∈ G R such that Ad(g (k) )(x) = x + z (k) 1 + . . . + z (k) N with z (k) j ∈ (g R ) μ j for all j and z (k) j = z j whenever j ≤ k. If k = 0, then g (0)
= id fulfills the required property.

Assume the construction done until rank

k < N . Let y = z k+1 -z (k) k+1 ∈ (g R ) μ k+1 . Thus [x, y] = μ k+1 y and [y, z (k) j ] ∈ ⊕ >k+1 (g R ) μ for all j ∈ {1, . . . , N }. Letting t = -1 μ k+1 , this implies that Ad(exp(ty)g (k) )(x) = ≥0 t ! ad(y) (x + z (k) 1 + . . . + z (k) N ) = x + z (k) 1 + . . . + z (k) N -t[x, y] + z ′ = x + z 1 + . . . + z k + z k+1 + z ′′ for some z ′ , z ′′ ∈ ⊕ >k+1 (g R ) μ
. This establishes the property at the rank k + 1, and the proof of the theorem is complete.

The above proof uses the following well-known fact. We give a proof for the sake of completeness.

Lemma 6.2. Let Q R ⊂ G R be a parabolic subgroup. If M ⊂ g R is a closed and Q R -stable subset, then G R ⋅ M is closed. Proof. Let {Ad(g k )(m k )} k≥1 be a sequence of elements of G R ⋅M , converging to some limit m 0 . Since G R Q R is compact, the sequence {g k Q R } k≥1 has a convergent subsequence {g k Q R } ≥1 with limit g 0 Q R ∈ G R Q R . Hence there is a sequence {q } ≥1 ⊂ Q R such that g 0 = lim →+∞ g k q . Let m ′ = Ad(q -1 )(m k ) ∈ M . Then m ′ = Ad(q -1 )(m k ) = Ad(q -1 g -1 k )(Ad(g k )(m k )) →+∞ → Ad(g -1 0 )(m 0 ). Since M is closed, we deduce that m 0 ∈ Ad(g 0 )(M ), hence m 0 ∈ G R ⋅ M .
In the case of SL n (R), we obtain the following refined description of the limit. Note that the inclusion in Theorem 6.1 becomes an equality in this case.

Corollary 6.3. Assume that G R = SL n (R).
Then, for all hyperbolic semisimple element x ∈ g R , we have

lim ν→0 + G R ⋅ (νx) = (G ⋅ u(x)) ∩ g R = O Rich (x) ∩ g R .
Proof. Up to conjugation, we may assume that x ∈ sl n (R) is a diagonal matrix.

Recall that for every nilpotent G-orbit O of g = sl n (C), the intersection O ∩ sl n (R) is nonempty and consists of exactly one GL n (R)-orbit, which splits into at most two SL n (R)-orbits. In particular, letting

σ = ⎛ ⎜ ⎜ ⎜ ⎝ 1 0 ⋯ 0 0 ⋱ ⋱ ⋮ ⋮ ⋱ 1 0 0 ⋯ 0 -1 ⎞ ⎟ ⎟ ⎟ ⎠
, we get that Ad(σ) induces an involution on the set of nilpotent SL n (R)-orbits of sl n (R), which switches the two real forms of O whenever O has two real forms, and which stabilizes every orbit which is the sole real form of a complex nilpotent orbit. This justifies the following definition.

L(e) = lim ν→0 + G R ⋅ (νh).
The set L(e) is a closed subset of the nilpotent cone N (g R ), which is independent of the choice of h. Since L(e) is in fact independent of the choice of e ∈ O, we set

L(O) ∶= L(e).

We also define L({0}) = L(0) = {0}.

Note that L(O) always contains O (see the proof of Proposition 3.7). Hence the definition of the mapping L is motivated by the fact that it provides us with a limit of semisimple orbits which contains a specified nilpotent orbit O. Note however that this mapping does not separate the orbit O = G R ⋅ e from the other real forms of the complex orbit G ⋅ e:

Proposition 7.2. For every nilpotent element e ∈ g R , we have (G ⋅ e) ∩ g R ⊂ L(e).

Proof. We can assume that e = 0, otherwise the inclusion is clear. Let e ′ ∈ (G ⋅ e) ∩ g R , and let {h, e, f } and {h ′ , e ′ , f ′ } be sl 2 -triples in g R containing e and e ′ . Since G ⋅ e = G ⋅ e ′ , these sl 2 -triples are G-conjugate, hence so are in particular the elements h and h ′ . These elements being semisimple and hyperbolic, they are in fact G R -conjugate; see, e.g., [11, §6] or [START_REF] Rothschild | Orbits in a real reductive Lie algebra[END_REF]Theorem 2.1]. This yields the equality G R ⋅ (νh) = G R ⋅ (νh ′ ) for all ν > 0, hence L(e) = L(e ′ ), and therefore e ′ ∈ L(e). In this case, we have

L(e) = L(f ) = N (sl 2 (R)).
In Definition 7.1, the semisimple element h is in particular hyperbolic. Hence Theorem 6.1 gives information on the nilpotent set L(e). As we show in the next subsection, this information is more precise in the case where the nilpotent element e is even.

7.1. The mapping L on even nilpotent orbits. Recall from Section 2 that an sl 2 -triple {h, e, f } is said to be even if the eigenvalues of ad(h) are all even integers; in this case, we also say that e is an even nilpotent element and that G R ⋅ e is an even nilpotent orbit.

Theorem 7.4. Let e ∈ g R be an even nilpotent element. Then, we have

L(e) = (G ⋅ e) ∩ g R .
In other words, if O 1 , . . . , O r are the real forms of G ⋅ e, then

L(e) = O 1 ∪ . . . ∪ O r .
Proof. The inclusion ⊃ follows from Proposition 7.2. For the second inclusion, let {h, e, f } ⊂ g R be an even sl 2 -triple containing e. We have

g = ⊕ j∈Z g(2j) ⊃ g R = ⊕ j∈Z g R (2j) where g(2j) = {z ∈ g [h, z] = 2jz} ⊃ g R (2j) = {z ∈ g R [h, z] = 2jz}.
As in Section 6, we consider

u R (h) = ⊕ j>0 g R (2j) and u(h) = ⊕ j>0 g(2j).
The fact that e is even implies that G ⋅ u(h) = G ⋅ e. The semisimple element h is in particular hyperbolic, and Theorem 6.1, applied with x = h, yields the equalities

L(e) = G R ⋅ u R (h) = O ′ 1 ∪ . . . ∪ O ′ k ,
where O ′ 1 , . . . , O ′ k ⊂ g R are nilpotent G R -orbits of the same dimension. We thus obtain:

(7.1) (G ⋅ e) ∩ g R ⊂ L(e) = G R ⋅ u R (h) ⊂ (G ⋅ u(h)) ∩ g R = G ⋅ e ∩ g R .
We claim that O ′ 1 , . . . , O ′ k are the real forms of G ⋅ e. Indeed, if G ⋅ O ′ j denotes the complex nilpotent orbit that contains O ′ j , then (7.1) implies that we have G ⋅ O ′ j ⊂ G ⋅ e for all j ∈ {1, . . . , k} on one hand and e ∈ G ⋅ O ′ j 0 so G ⋅ e ⊂ G ⋅ O ′ j 0 for some j 0 ∈ {1, . . . , k} on the other hand.

Since O ′ 1 , . . . , O ′ k are of the same dimension, we deduce that dim G ⋅ O ′ j = dim G ⋅ O ′ j 0 = dim G ⋅ e and therefore G ⋅ O ′ j =
G ⋅ e for all j ∈ {1, . . . , k}. This implies that every O ′ j is a real form of G ⋅ e. Conversely, (7.1) now implies that

(G ⋅ e) ∩ g R ⊂ (O ′ 1 ∪ . . . ∪ O ′ k ) ∩ (G ⋅ e) = O ′ 1 ∪ . . . ∪ O ′ k ,
and so every real form of G ⋅ e is of the form O ′ j for some j. This shows the claim. The stated equality L(e) = (G ⋅ e) ∩ g R ensues. ; see, e.g., [START_REF] Djoković | Closures of conjugacy classes in classical real linear Lie groups[END_REF].

In the case of G R = SL n (R), however, the equality (G ⋅ e) ∩ g R = G ⋅ e ∩ g R is true for all nilpotent element e, in view of the parametrization of real forms and the description of their closures given in [START_REF] Djoković | Closures of conjugacy classes in classical real linear Lie groups[END_REF]; see also Theorem 7.7 below.

(b) Let us consider a symmetric subgroup K ⊂ G compatible with the real form G R , and the corresponding decomposition g = k ⊕ p. Nishiyama describes the (complex) asymptotic cone AC(K ⋅ (i(e ′ -f ′ ))) where {h ′ , e ′ , f ′ } is a normal triple with e ′ , f ′ even nilpotent [START_REF] Nishiyama | Asymptotic cone of semisimple orbits for symmetric pairs[END_REF]. Thus Theorem 7.4 above appears to be a real counterpart of [14, Theorem 0.2]. Specifically, in [START_REF] Nishiyama | Asymptotic cone of semisimple orbits for symmetric pairs[END_REF], it is shown that the above asymptotic cone coincides with the closure of (G ⋅ e) ∩ p, and is in general strictly contained in G ⋅ e ∩ p. The equidimensionality of the complex cone (which is the counterpart of Theorem 6.1 (a)) is used as a key ingredient in the proof of [14, Theorem 0.2].

7.2. The mapping L on arbitrary nilpotent orbits. Theorem 6.1 also gives information on L(e) in the case where e is not even. The key fact in the proof of Theorem 7.4 is that all the real forms of G ⋅ e have a representative in the space u R (h) and, when e is even, G ⋅ e = O Rich (h) is in addition the largest nilpotent orbit that intersects u R (h).

If e ∈ g R is an arbitrary nonzero nilpotent element, then Theorem 6.1 and Proposition 7.2 imply that

(7.2) (G ⋅ e) ∩ g R ⊂ L(e) = G R ⋅ u R (h) ⊂ O Rich (h) ∩ g R ,
where L(e) is equidimensional. Here the difficulty is that, if e is not even, then G ⋅ e does not coincide with the Richardson orbit O Rich (h), and we cannot guarantee that O Rich (h) has real forms or that these real forms have representatives in u R (h). We can however formulate the following statement.

Proposition 7.6. Let e ∈ g R be an arbitrary nonzero nilpotent element. Assume that the nilpotent G-orbit O Rich (h) has real forms which all intersect u R (h). Then,

L(e) = O Rich (h) ∩ g R .
Proof. The assumption made in the proposition combined with (7.2) implies that

O Rich (h) ∩ g R ⊂ L(e) = G R ⋅ u R (h) ⊂ O Rich (h) ∩ g R .
Then, by arguing as in the proof of Theorem 7.4, we can deduce the claimed equality by invoking the

equidimensionality of G R ⋅ u R (h) (Theorem 6.1).
The condition in the proposition is fulfilled if e is even (by Proposition 7.2), in which case we simply retrieve the conclusion of Theorem 7.4. The condition in the proposition is fulfilled for all e in the case where G R = SL n (R), as illustrated in the next subsection.

7.3. Complete description of the mapping L in the case of SL n (R). In the case where

G R = SL n (R),
we can describe explicitly the mapping L. Recall from Section 2 that the nilpotent orbits

O Λ of g = sl n (C)
are parametrized by partitions Λ = (Λ 1 , . . . , Λ m ). Let Λ 0 (resp., Λ 1 ) be the subsequence of even (resp., odd) parts of Λ. The nilpotent orbit O Λ is even if and only if Λ 0 = ∅ or Λ 1 = ∅. We also recall that a (complex) (b) Assume that e is not even, i.e., Λ 0 = ∅ and Λ 1 = ∅. Let Λ 0 + Λ 1 ∶= (μ 0 k + μ 1 k ) k=1 be the partition of n obtained by summing the sequences Λ 0 = (μ 0 k ) k=1 and Λ 1 = (μ 1 k ) k=1 term by term (adding 0's if necessary, we may assume that both sequences have the same length ). Then one has:

nilpotent orbit O Λ such that Λ 1 = ∅ has only one real form O R Λ .
L(O R Λ ) = O R Λ 0 +Λ 1 .
Proof. Part (a) follows from Corollary 6.

O R (2,1,1,1) ⊂ O R (2,2,1) ⊂ O R (3,1,1) ⊂ O R (5)
while we have

L(O R (2,1,1,1) ) = L(O R (3,1,1) ) = O R (3,1,1) ⊂ L(O R (2,2,1) ) = O R (3,2) ⊂ L(O R (5) ) = O R (5)
.

Approximation of nilpotent orbits by elliptic orbits

In this section, we consider the limit of orbits associated to a nonzero elliptic semisimple element x ∈ g R .

Throughout this section, we assume that we are in the equal-rank situation (a priori needed for (8.4) below):

(8.1) rank(g R ) = rank(k R ).
Since x is elliptic, ix has real eigenvalues on g. Then we can consider the decomposition g = ⊕ μ∈R g μ associated to ix as in (2.6), and this gives rise to the parabolic subalgebra q ∶= ⊕ μ≥0 g μ , corresponding to a parabolic subgroup Q ⊂ G, and to the nilpotent radical

(8.2) u ∶= u(ix) = ⊕ μ>0 g μ .
Recall the Cartan decomposition g = k ⊕ p of (2.1). Up to conjugation, we may assume that x ∈ k R . Then

K ⋅ (u ∩ p) is a closed, irreducible, K-stable subvariety of p (it is closed since u ∩ p is K ∩ Q-stable, and K ∩ Q is a parabolic subgroup of K; it is irreducible since K is connected). It follows that (8.3) there is a unique nilpotent K-orbit O(ix) ⊂ p such that K ⋅ (u ∩ p) = O(ix).
The limit of orbits of x can now be characterized as the closure of the nilpotent G R -orbit of g R corresponding to O(ix) through the Kostant-Sekiguchi bijection (see (2.2)):

(8.4) lim ν→0 + G R ⋅ (νx) = KS -1 (O(ix)
). This result can be obtained by combining several results from Representation Theory, that relate the limit of orbits to the asymptotic support and the associated variety of (g, K)-modules defined by parabolic induction: [2, Proposition 3.7], [START_REF] Barbasch | Weyl group representations and nilpotent orbits[END_REF]Proposition 3.4], and [START_REF] Trapa | Annihilators and associated varieties of A q(λ) modules for U(p, q)[END_REF]Proposition 5.4] (which relies mainly on [START_REF] Schmid | Characteristic cycles and wave front cycles of representations of reductive Lie groups[END_REF]Theorem 1.4]).

Other relevant references on this topic are [START_REF] Harris | Irreducible characters and semisimple coadjoint orbits[END_REF] and [START_REF] Trapa | Richardson orbits for real classical groups[END_REF]. To our knowledge, there is no direct proof of the result stated in (8.4), that relies on purely topological arguments. Also we do not know whether the equalrank condition (8.1) (which is assumed in some of the aforementioned referenced results) is indeed necessary for guaranteeing (8.4).

In this section, we use the formula stated in (8.4) for studying approximations of certain nilpotent orbits by elliptic semisimple orbits. In Section 8.1, we consider even nilpotent orbits. In Section 8.2, we study approximation of minimal nilpotent orbits. In Section 8.3, we focus on the case where G R = SU(p, q). 8.1. Approximation of even nilpotent orbits. We give an "elliptic analogue" of the mapping e ↦ L(e) constructed in Section 7.

Definition 8.1. Let e ∈ g R be a nonzero nilpotent element and let O = G R ⋅ e be its nilpotent orbit. Given an sl 2 -triple {h, e, f } which contains e, we let

L ′ (e) = lim ν→0 + G R ⋅ (ν(e -f )
). The set L ′ (e) is a closed subset of the nilpotent cone N (g R ). It is independent of the choice of the triple {h, e, f }, and in fact it is independent of e up to its nilpotent orbit. We can therefore set

L ′ (O) ∶= L ′ (e).
We also define L ′ ({0}) = L ′ (0) = {0}.

As for L(O) in Section 7, the set L ′ (O) always contains O (see the proof of Proposition 3.7). Hence the mapping L ′ also provides us with a limit of orbits which contains the specified orbit O. Unlike L(O) in general, the set L ′ (O) is irreducible: this follows from (8.4), taking into account that the element e -f is elliptic. In the case of even nilpotent elements, the following result (which is the "elliptic analogue" of Theorem 7.4) implies that any even nilpotent orbit closure can be obtained as a limit of elliptic orbits. Theorem 8.2. Assume that condition (8.1) holds and that the nilpotent element e ∈ g R is even. Then,

L ′ (e) = G R ⋅ e.
Proof. Let {h, e, f } ⊂ g R be an sl 2 -triple containing e. Up to dealing with G R -conjugates, we may assume that {h, e, f } is a Cayley triple, and we denote by {h ′ , e ′ , f ′ } ⊂ g its Cayley transform. In particular, we have h ′ = i(e -f ), and K ⋅ e ′ is the image of G R ⋅ e by the Kostant-Sekiguchi correspondence (see (2.2)).

Let g = ⊕ j∈Z g j be the decomposition of g into eigenspaces for ad(h ′ ) and let u = ⊕ j>0 g j . Since the triple {h, e, f } is even, then so is {h ′ , e ′ , f ′ }, and we have

O Rich (h ′ ) = G ⋅ e ′ = G ⋅ e and G ⋅ u = O Rich (h ′ ) = G ⋅ e ′ = G ⋅ e.
Let O ⊂ p be the unique nilpotent K-orbit which is dense in K ⋅ (u ∩ p) (see (8.3)). By (8.4), we have

L ′ (e) = KS -1 (O).
We claim that O = K ⋅ e ′ . Once we show this, we deduce that L ′ (e) = KS -1 (K ⋅ e ′ ) = G R ⋅ e, which will complete the proof of the theorem. For this, we have

O = K ⋅ (u ∩ p) ⊂ G ⋅ u = G ⋅ e ′ .
On the other hand, since e ∈ L ′ (e), and since the Kostant-Sekiguchi correspondence preserves the closure relations [START_REF] Barbasch | Closure ordering and the Kostant-Sekiguchi correspondence[END_REF], we must have e ′ ∈ O, hence

K ⋅ e ′ ⊂ O ⊂ G ⋅ e ′ .
This implies that O is contained in G⋅e ′ , and in fact O = K ⋅e ′ . The proof of the theorem is now complete. 8.2. Approximation of minimal nilpotent orbits. We still assume equal-rank situation (8.1), so that the result stated in (8.4) applies. The assumption also implies that we can take a Cartan subalgebra of g of the form h = h R ⊕ ih R ⊂ k, where h R ⊂ k R . Then the root system Φ = Φ(g, h) decomposes as Φ = Φ c ⊔ Φ nc so that we have the root space decompositions

k = h ⊕ ⊕ α∈Φ c g α and p = ⊕ α∈Φ nc g α .
Moreover, since each root has pure imaginary values on h R , we have g α = g -α for all root α. We consider a set of positive roots Φ + and the corresponding set of simple roots Δ, which decomposes as Δ = Δ c ⊔ Δ nc in the same way as Φ.

We focus on the case where g is classical.

k O min ∩ g R G R AIII sl n (C), n ≥ 2 su(p, q), 1 ≤ p ≤ q sl p (C) × sl q (C) ⊕ C 2 BI so 2n+1 (C), n ≥ 2 so(2p, 2q + 1), p, q ≥ 1 so 2p (C) × so 2q+1 (C) 2 if p = 1, 1 if p ≥ 2 CI sp 2n (C), n ≥ 2 sp 2n (R) gl n (C) 2 DI so 2n (C), n ≥ 4 so(2p, 2q), 1 ≤ p ≤ q so 2p (C) × so 2q (C) 2 if p = 1, 1 if p ≥ 2 DIII so 2n (C), n ≥ 4 so * 2n gl n (C) 2
In each case, we point out that the set Δ nc consists of a single element α 0 , which is always a long root. This is shown in the following table .  type

Φ + Φ + ∩ Φ nc α 0 AIII {ε i -ε j } 1≤i<j≤n {ε i -ε j } i≤p<j ε p -ε p+1 BI {ε i ± ε j } 1≤i<j≤n ∪ {ε i } 1≤i≤n {ε i ± ε j } i≤p<j ∪ {ε i } i≤p ε p -ε p+1 CI {ε i ± ε j } 1≤i<j≤n ∪ {2ε i } 1≤i≤n {ε i + ε j } i≤j 2ε n DI {ε i ± ε j } 1≤i<j≤n {ε i ± ε j } i≤p<j ε p -ε p+1 DIII {ε i ± ε j } 1≤i<j≤n {ε i + ε j } i<j ε n-1 + ε n We then focus on the real form O + min ∶= KS -1 (K ⋅ e 0 ) ⊂ O min where e 0 ∈ g α0 ∖ {0}.
In each case, we consider the usual numbering of the simple roots, so that the first root is α

1 = ε 1 -ε 2 .
Let h 1 ∶= ∨ 1 ∈ h be the first fundamental coweight, characterized by

α 1 ( ∨ 1 ) = 1 and α( ∨ 1 ) = 0 for all α ∈ Δ ∖ {α 1 }. Since α( ∨ 1
) is a real number for all root α, we must have ∨ 1 ∈ ih R , and therefore x 1 ∶= -i ∨ 1 (= -ih 1 ) belongs to h R and is elliptic. The following statement describes the limit of orbits lim ν→0 + G R ⋅ (νx 1 ) and shows in particular that it is close to the real nilpotent orbit O + min . In the special case of type DI, we retrieve a result stated in [START_REF] Kobayashi | Conformal geometry and branching laws for unitary representations attached to minimal nilpotent orbits[END_REF]Theorem 4.3]. The statement uses the parametrization of complex nilpotent orbits of classical simple Lie algebras by admissible partitions d = (d 1 , . . . , d k ), which is recalled in Section 2. Note also that, in types A and C, the minimal nilpotent orbit is parametrized by the partition (2, 1, . . . , 1), while in types B and D, it is parametrized by (2, 2, 1, . . . , 1).

lim ν→0 + G R ⋅ (νx 1 ) = {0} ∪ O + min ∪ O 0 ,
where O 0 is a real form of the complex nilpotent orbit O d ⊂ g corresponding to the partition

d = (3, 1, . . . , 1)
in types BI and DI, (2, 2, 1, . . . , 1) in type CI.

Proof. From (8.4), we know that the limit lim ν→0 + G R ⋅ (νx 1 ) has a dense nilpotent G R -orbit O 0 ; moreover, O 0 is characterized by the fact that the corresponding K-orbit KS(O 0 ) intersects the space u ∩ p along a dense open subset, where u = u( ∨ 1 ) is given by (8.2). We have

u ∩ p = ⊕ β∈Ψ 1 g β where Ψ 1 ∶= {β ∈ Φ + ∩ Φ nc β( ∨ 1 ) > 0}.
In the different cases, the set Ψ 1 can be described as follows.

type AIII BI CI DI DIII Ψ 1 {ε 1 -ε j } j>p {ε 1 ± ε j } j>p ∪ {ε 1 } {ε 1 + ε j } j≥1 {ε 1 ± ε j } j>p {ε 1 + ε j } j>1 Let α ′ 1 ∶= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ε 1 -ε p+1 in
) = α ′ 1 . Whence Ad(g w )(g α0 ) = g α ′ 1 ⊂ u ∩ p. • In the other situations, α ′ ∶= α ′ 1 -α 0 is a compact positive root such that α ′ 1 + kα ′ , α 0 -kα ′ ∉ Φ ∪ {0} for all k ≥ 1.
Denoting by {u ±α ′ (t)} t∈C ⊂ K the one-parameter subgroup attached to ±α ′ , for every

e 0 ∈ g α0 ∖ {0}, we can find t, s ∈ C such that Ad(u -α ′ (t)u α ′ (s))(e 0 ) ∈ g α ′ 1 ⊂ u ∩ p.
In all the cases, this implies that (8.5)

O + min ⊂ lim ν→0 + G R ⋅ (νx 1 ). In each type, the description of the set Ψ 1 allows us to determine the Jordan normal form of the elements of the space u ∩ p, viewed as matrices. For each root β, let E β be a nonzero element of the root space g β .

• In type AIII, any element of u ∩ p is a matrix of rank ≤ 1. The element z 0 ∶= E ε 1 -ε p+1 is of rank 1.

• In types CI and DIII, any element of u ∩ p is a matrix of rank ≤ 2 and nilpotency order ≤ 2. The element z 0 ∶= E ε1+εn is of rank 2 and nilpotency order 2.

• In types BI and DI, any element of u ∩ p is a matrix of rank ≤ 2 and nilpotency order ≤ 3. The element z 0 ∶= E ε 1 -ε p+1 + E ε 1 +ε p+1 is a matrix of rank 2 and nilpotency order 3.

In each case, we obtain that the G-orbit, and so the K-orbit, of the considered element z 0 intersects the space u ∩ p along a dense open subset. In view of (8.4), this yields lim ν→0 + G R ⋅ (νx 1 ) = KS -1 (K ⋅ z 0 ). In types AIII and DIII, the element z 0 actually belongs to O min , hence we must have O + min = KS -1 (K ⋅ z 0 ). In types BI, CI, and DI, the Jordan normal form of z 0 (viewed as a matrix) coincides with the partition d indicated in the theorem. Moreover, it follows from [START_REF] Djoković | Closures of conjugacy classes in classical real linear Lie groups[END_REF][START_REF] Ohta | The closures of nilpotent orbits in the classical symmetric pairs and their singularities[END_REF] that the closure of O 0 ∶= KS -1 (K ⋅ z 0 ) is the union of O 0 , {0}, and only one real form of O min , which is precisely O + min (see (8.5)). The proof is complete.

8.3. Limit of elliptic orbits in the case of G R = SU(p, q). In this section, we assume that G R = SU(p, q)

and g R = su(p, q), with p, q ≥ 1, p + q = n. In this case, the complexification g = sl Each elliptic orbit has a representative x in k R = k ∩ g R , which is diagonalizable with pure imaginary eigenvalues. The spectra of the diagonal blocks ia, ib of ix can be written as a pair of nonincreasing sequences of real numbers (8.6) Λ = (λ 1 ≥ λ 2 ≥ . . . ≥ λ p ), (μ 1 ≥ μ 2 ≥ . . . ≥ μ q ) ∈ R p × R q such that p k=1

λ k + q =1 μ = 0.
Conversely, if Λ is a pair of sequences as in (8.6), then the diagonal matrix (8.7)

x ∶= diag(-iλ 1 , . . . , -iλ p , -iμ 1 , . . . , -iμ q )

is an elliptic element of g R corresponding to Λ.

Let us recall from [15, §1.4] that the nilpotent K-orbits of p can be parametrized by the set of signed Let V + = C p × {0} q and V -= {0} p × C q , so that p can be viewed as the space of endomorphisms z of V = C n such that z(V + ) ⊂ V -and z(V -) ⊂ V + . Given a signed Young diagram σ with signature (p, q), let O σ denote the set of nilpotent endomorphisms z ∈ p which have a Jordan basis (ε c ) parametrized by the boxes c of σ, in such a way that Proof. (a) Let (u 1 , . . . , u p ), resp. (v 1 , . . . , v q ), be the standard basis of V + , resp. V -. Hence each vector u k , resp. v , is an eigenvector of x 0 ∶= ix corresponding to the eigenvalue λ k , resp. μ . Viewing p as above as the space of endomorphisms z ∶ V → V such that z(V + ) ⊂ V -and z(V -) ⊂ V + , the intersection u ∩ p coincides with the subspace of such endomorphisms such that z(u k ) ∈ ⟨v j μ j > λ k ⟩, z(v ) ∈ ⟨u j λ j > μ ⟩ for all k, .

In particular, for r, s as in Notation 8.4 (b), we have that u 1 , . . . , u r , v 1 , . . . , v s ∈ ker z. Hence z has the following matrix form:

z = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 γ 0 ⋮ 0 c ′ 0 δ ⋮ 0 0 d ′ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ with ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
c ′ ∈ M p-r,q-s (C), d ′ ∈ M q-s,p-r (C), γ ∈ M r,q-s (C), δ ∈ M s,p-r (C).

Arguing by induction, we obtain that whenever z is generic in u ∩ p, the submatrix The proof of the theorem is complete.

Remark 8.7. The calculation of K ⋅ (u ∩ p) made in the above proof was done previously by Trapa [START_REF] Trapa | Annihilators and associated varieties of A q(λ) modules for U(p, q)[END_REF],

though our combinatorial algorithm is somewhat different. We also mention that a representation theoretic analogue of Theorem 8.6 is given in [START_REF] Barbasch | Weyl group representations and nilpotent orbits[END_REF]Theorem 4.2].

Example 8.8. (a) The proof of Theorem 8.6 also indicates an explicit way to determine an elliptic element

x whose associated limit lim ν→0 + G R ⋅ (νx) coincides the closure of a specified nilpotent orbit. For instance the

Figure 2 .

 2 Figure 2. Adjoint orbits for SL 2 (R).

If

  O and O ′ are nilpotent orbits associated with distinct partitions d and d ′ , then O is contained in the Zariski closure of O ′ if and only if d ≤ d ′ .

Figure 3 .

 3 Figure 3. The nilpotent cone N (sl 2 (R)) and the orbits SL 2 (R) ⋅ (νη) and SL 2 (R) ⋅ (νh) for ν = √ 3.

Figure 4 .

 4 Figure 4. The asymptotic cone of a hyperbola and of a parabola.

Theorem 4 . 6 (

 46 ). The asymptotic cone AC(G ⋅ x) of the semisimple orbit G ⋅ x coincides with the Zariski closure of the Richardson orbit O Rich (x) defined in (2.4):

Proof.

  The equality in (a) is implied by Proposition 4.4 (b) (i)⇔(ii). The first inclusion is implied by Proposition 4.4 (a). The second inclusion is obvious. (b) follows from (a) and Theorem 4.6.

Fix a basis B

  of g R and consider the matrix Mat B (ad(z)). The last formula shows that Z d is the subset of elements z ∈ u R (x) such that at least one of the d × d-minors of Mat B (ad(z)) is nonzero. This implies that Z d is open in u R (x). Moreover, let us fix z 0 ∈ Z d (by assumption, Z d is nonempty) and take any z ∈ u R (x). There is a submatrix M (z 0 ) of Mat B (ad(z 0 )) of size d × d which is invertible, and let M (z) be the corresponding submatrix of Mat B (ad(z)). Then M

  we have dim O j = d for all j ∈ {1, . . . , k}. The proof of part (a) is complete. (b) The last inclusion is clear. First we show that (6.2)

Definition 7 . 1 .

 71 Let e ∈ g R be a nonzero nilpotent element and let O = G R ⋅ e be its nilpotent orbit. Given an sl 2 -triple {h, e, f } which contains e, we let

Example 7 . 3 .

 73 In Example 3.3 we have seen that the nilpotent cone N (sl 2 (R)) consists of three nilpotent orbits: {0}, O + = SL 2 (R) ⋅ e, and O -= SL 2 (R) ⋅ f , where

Remark 7 .

 7 5. (a) In general L(e) = (G ⋅ e) ∩ g R can be strictly contained in G ⋅ e ∩ g R . Take for instance G R = SU(2, 2). The nilpotent SL 4 (C)-orbit O d parametrized by the partition d = (3, 1) has two real forms, parametrized by the signed Young diagrams (see [6, Theorem 9.3.3] or Section 8.3 below): d is an even orbit. The nilpotent SU(2, 2)-orbits parametrized by the signed Young diagrams in SL 4 (C) ⋅ e ∩ su(2, 2), but not in (SL 4 (C) ⋅ e) ∩ su(2, 2)

Theorem 7 . 7 .

 77 Let e, h, f ∈ sl n (R) form a standard triple. Let O Λ = SL n (C) ⋅ e. (a) Assume that e is even. Then L(e) = O Λ ∩ sl n (R) = O Λ ∩ sl n (R).

.

  3 and Theorem 7.4. Let us show part (b). We construct a semisimple element h ∈ sl n (R) which belongs to an sl 2 -triple together with an element of O R Λ . For an integer d ≥ 1, let D(d) denote the diagonal matrix of size d whose coefficients are d -1, d -3, . . . , -(d -1), and let J(d) denote the Jordan matrix of size d whose coefficients are 1 just above the diagonal and 0 elsewhere. We define h, e as the blockwise diagonal matrices Then e is an element in O R Λ and we have [h, e] = 2e. We claim that O Rich (h) = O Λ 0 +Λ 1 . Once we have shown this equality, we deduce in particular that O Rich (h) has only one real form (because the first part μ 0 1 + μ 1 1 of the partition Λ 0 + Λ 1 is odd), and the result claimed in Theorem 7.7 (b) finally follows from Corollary 6.3. Recall that the Richardson nilpotent orbit O Rich (h) can be computed as follows. For d ∈ Z, let N d be the number of diagonal coefficients of h which are equal to d. We obtain a partition μ = (μ 1 , μ 2 , . . .) ⊢ n by letting μ j ∶= {d ∈ Z N d ≥ j} for all j ≥ 1. Then, by [6, Theorem 7.2.3], we have O Rich (h) = O μ . Hence, we have to check that μ = Λ 0 + Λ 1 . For d even, we have N d = {j d ≤ μ 1 j -1} . For d odd, we have N d = {j d ≤ μ 0 j -1} . Hence, for d even we have N d ≥ j if and only if d ≤ μ 1 j -1, whereas for d odd we have N d ≥ j if and only if d ≤ μ 0 j -1. This yields μ j = {d even d ≤ μ 1 j -1} + {d odd d ≤ μ 0 j -1} = μ 1 j + μ 0 j for all j, whence μ = Λ 0 + Λ 1 . Example 7.8. (a) If Λ = (8, 7, 6, 4, 4, 3, 2, 1), then we get Λ 0 = (8, 6, 4, 4, 2), Λ 1 = (7, 3, 1) and Λ 0 + Λ 1 = (15, 9, 5, 4, 2). Hence, in sl 35 (R), Theorem 7.7 yields L(O R (8,7,6,4,4,3,2,1) ) = O R (15,9,5,4,2) . (b) It is worth noting that the map L is not monotone nor injective. Indeed, in the Lie algebra g R = sl 5 (R), we have the following inclusions of orbit closures:

  n (C) has Cartan decomposition sl n (C) = k ⊕ p with k = a 0 0 b a ∈ gl p (C), b ∈ gl q (C), Tr(a) + Tr(b) = 0 and p = 0 c d 0 c ∈ M p,q (C), d ∈ M q,p (C) .

  Young diagrams with signature (p, q): A signed Young diagram σ is obtained by filling in the boxes of a Young diagram with signs + and -in such a way that the signs + and -alternate in each row (but not necessarily in each column). Moreover, two signed Young diagrams are considered equal if they coincide up to permutation of the rows. The signature of σ is the pair (n + , n -), where n ± indicates the number of ±'s in σ. For example, the set of signed Young diagrams of signature (3, 2) consists of the following list:

  z ′ ∶= 0 c ′ d ′ 0 belongs to the nilpotent orbit parametrized by the signed Young diagram σ Λ ′ for Λ ′ as in Notation 8.4 (b). This means that z ′ has a Jordan basis parametrized by the boxes of σ Λ ′ , which satisfies the above conditions (C1)-(C2). The fact that z generically belongs to the orbit O σ Λ now follows straightforward from the definition of σ Λ , in view of the form of the matrix z given above. (b) In view of part (a), the claim made in part (b) becomes equivalent to the claim that, for every signed Young diagram σ of signature (p, q), we can find Λ as in (8.6) such that σ = σ Λ . Let be the number of columns in σ and, for k ∈ {1, . . . , }, let r k , resp. s k , denote the number of signs +, resp. -, contained in the k-th column of σ. Thus p = ∑ k=1 r k and q = ∑ k=1 s k . Define the pair of sequences Λ = (N -1) r 1 , . . . , (N -) r ), ((N -1) s 1 , . . . , (N -) s ) ,where the exponents r k and s k indicate the multiplicity of the coefficient N -k in each sequence, and N is a number chosen so that the total sum of the coefficients in Λ is equal to 0, namely,N = 1 p + q k=1 k(r k + s k ).It is easy to check (by induction) that, for every k ∈ {0, . . . , }, the signed diagram L r -k+1 ,s -k+1 * ⋯ * L r ,s * ∅ coincides with the subdiagram formed by the last k columns of σ. Whence σ Λ = L r1,s1 * ⋯ * L r ,s * ∅ = σ.

  It is known as the regular or principal nilpotent orbit, and it is the largest nilpotent G-orbit in N (g). Elements in O reg will be called regular nilpotent. Second, since g is simple, there exists a unique nilpotent G-orbitO subreg of dimension dim O reg -2 = dim g -rank g -2 in N (g).The orbit O subreg , known as the subregular nilpotent orbit, is open and dense in N (g) O reg . Third, there exists a nonzero nilpotent G-orbit O min in N (g) which is contained in the Zariski closure of any nonzero nilpotent orbit. The orbit O min has minimal dimension and is known as the minimal nilpotent orbit.

	The nilpotent cone N (g)
	is a finite union of nilpotent G-orbits. There are several nilpotent orbits of particular interest in N (g) [6,
	Chapters 4 and 7]. First, the nilpotent cone coincides with the Zariski closure of a unique nilpotent G-orbit
	O reg which is open and dense in N (g):
	N (g) = O reg .

  Recall that, by Jacobson-Morozov theorem, every nonzero nilpotent element e ∈ g R is part of an sl 2 -triple {h, e, f } ⊂ g R . Moreover, two such triples {h, e, f } and {h ′ , e, f ′ } that contain e are conjugate under G R .

It is well known that the Richardson orbit O Rich (x) has a representative in u R (x); see, for instance, the explicit construction made in [6, §7.2]. Since u R (x) is stable by Ad(σ), we deduce that each real form of

O Rich (x) has a representative in u R (x). Moreover, if O is a nilpotent orbit of sl n (C)

contained in the closure of O Rich (x), then each real form of O is contained in the closure of a real form of O Rich (x); this follows from [7, Theorem 3]. The claimed equality ensues. 7. A mapping on the set of nilpotent orbits

  Moreover, we assume that the minimal nilpotent orbit O min ⊂ g has real forms, or equivalently (by virtue of the Kostant-Sekiguchi correspondence) that the intersection O

min ∩ p is nonempty. Every root vector e ∈ g α associated to a long root α is an element of O min , and O min ∩ p is nonempty if and only if Φ nc contains a long root. In this case, O min ∩ p is the union of one or two K-orbits (equivalently, O min has one or two real forms) depending on whether the symmetric pair (G, K) is non-Hermitian or Hermitian; see

[START_REF] Okuda | Smallest complex nilpotent orbits with real points[END_REF]

. The following table lists the cases that we are left to consider. In each case, we indicate the number of real forms in O min . See [11, §VII.9] and

[START_REF] Okuda | Smallest complex nilpotent orbits with real points[END_REF]

for more details. type g g R

  Theorem 8.3. With the above notation, the limit of orbits lim ν→0 + G R ⋅ (νx 1 ) always contains the real nilpotent orbit O + min . Moreover: (a) In types AIII and DIII, the limit lim ν→0 + G R ⋅ (νx 1 ) is the closure of O + min . (b) In types BI, CI, and DI, we have

  In types AIII, BI, and DI, for p = 1, we have α ′ 1 = α 0 .• In type CI, the product of transpositions w = (1, n)(n + 1, 2n) is a Weyl group element which has a representant g w in K, and such that w(α 0

		types AIII, BI, and DI,
	2ε 1	in type CI,
	ε 1 + ε n	in type DIII.
	In each case, α ′ 1 is a long root which belongs to Ψ 1 .
	•

(C1) the vector ε c belongs to V + if the box c contains a sign + and to V -if c contains a -;

(C2) if c lies in the first column of σ, then z(ε c ) = 0; otherwise, z(ε c ) = ε c ′ where c ′ is the box on the left of c.

The so-obtained subset O σ is a K-orbit of p, and every orbit is of this form for a unique signed Young diagram of signature (p, q). Next, we define a procedure to associate a signed Young diagram of signature (p, q) to each Λ as in (8.6): Notation 8.4. (a) If σ is a signed Young diagram of signature (p, q), then let L r,s * σ denote the signed Young diagram of signature (p + r, q + s) obtained by adding a new box with sign + on the left of each one of the first r rows of σ which start with a -, and a new box with sign -on the left of each one of the first s rows of σ which start with a + (understanding that these new boxes can be inserted in empty rows if σ has less than r rows starting with a -or less than s rows starting with a +). For example,

a pair of sequences of real numbers, we define a signed Young diagram σ Λ of signature (p, q) by induction:

• If p = q = 0 (Λ is an empty sequence), then let σ Λ be the empty signed Young diagram;

are the coefficients of Λ equal to max Λ (the maximal value in Λ) and Λ ′ ∶= ((λ r+1 , . . . , λ p ), (μ s+1 , . . . , μ q )) is the pair of subsequences formed by the coefficients < max Λ.

Example 8.5. For Λ = (6, 3, 2, 2, -1, -3, -4), (3, 2, -3, -3, -4) we get

Theorem 8.6. Assume that G R = SU(p, q) with p, q ≥ 1.

(a) Let x ∈ g R be the elliptic semisimple element corresponding to a pair of sequences