Medical, Pharma, Engineering, Science, Technology and Business

**Peter Crooks ^{*}**

Department of Mathematics, University of Toronto, Canada

- *Corresponding Author:
- Peter Crooks

Department of Mathematics

University of Toronto, Canada

**E-mail:**[email protected]

**Received Date**: April 24, 2016; **Accepted Date:** June 09, 2016; **Published Date**: June 30, 2016

**Citation: **Crooks P (2016) Properties of Nilpotent Orbit Complexification. J Generalized Lie Theory Appl S2:012. doi:10.4172/1736-4337.S2-012

**Copyright:** © 2016 Crooks P. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

We consider aspects of the relationship between nilpotent orbits in a semisim-ple real Lie algebra g and those in

its complexification gï��. In particular, we prove that two distinct real nilpotent orbits lying in the same complex orbit are

incomparable in the closure order. Secondly, we characterize those g having non-empty intersections with all nilpotent

orbits in gï��. Finally, for g quasi-split, we characterize those complex nilpotent orbits containing real ones.

Nilpotent orbit; Quasi-split Lie algebra; Kostant- Sekiguchi correspondence

**Background and statement of results**

Real and complex nilpotent orbits have received considerable attention in the literature. The former have been studied in a variety of contexts, including differential geometry, symplectic geometry, and Hodge theory [1]. Also, there has been some interest in concrete descriptions of the poset structure on real nilpotent orbits in specific cases [2,3]. By contrast, complex nilpotent orbits are studied in algebraic geometry and representation theory — in particular, Springer Theory [4-7].

Attention has also been given to the interplay between real and complex nilpotent orbits, with the Kostant-Sekiguchi Correspondence being perhaps the most famous instance [8]. Accordingly, the present article provides additional points of comparison between real and complex nilpotent orbits. Specifically, let g be a finite-dimensional semisimple real Lie algebra with complexification *g* Each real nilpotent orbit lies in a unique complex nilpotent orbit , the complexification of The following is our main result.

**Theorem 1: **The process of nilpotent orbit complexification has the following properties.

• Every complex nilpotent orbit is realizable as the complexification of a real nilpotent orbit if and only if g is quasi-split and has no simple summand of the form so (2n+1, 2n −1).

• If g is quasi-split, then a complex nilpotent orbit is realizable as the complexification of a real nilpotent orbit if and only if is invariant under conjugation with respect to the real form

• If are real nilpotent orbits satisfying then either or these two orbits are incomparable in the closure order.

**Structure of the article**

We begin with an overview of nilpotent orbits in semisimple real and complex Lie algebras. In recognition of Theorem 1 (iii), and of the role played by the unique maximal complex nilpotent orbit throughout the article, Section 2.2 reviews the closure orders on the sets of real and complex nilpotent orbits. In Section 2.3, we recall some of the details underlying the use of decorated partitions to index nilpotent orbits.

Section 3 is devoted to the proof of Theorem 1. In Section 3.1, we represent nilpotent orbit complexification as a poset map between the collections of real and complex nilpotent orbits. Next, we show this map to have a convenient description in terms of decorated partitions. Section 3.2 then directly addresses the proof of Theorem 1 (i), formulated as a characterization of when is surjective. Using Proposition 2, we reduce this exercise to one of characterizing surjectivity for g simple. Together with the observation that surjectivity implies g is quasi-split and is implied by g being split, Proposition 2 allows us to complete the proof of Theorem 1 (i).

We proceed to Section 3.3, which provides the proof of Theorem 1 (ii). The essential ingredient is Kottwitz’s work [9]. We also include Proposition 3, which gives an interesting sufficient condition for a complex nilpotent orbit to be in the image of .

In Section 3.4, we give a proof of Theorem 1 (iii). Our proof makes extensive use of the Kostant-Sekiguchi Correspondence, the relevant parts of which are mentioned.

**Nilpotent orbits**

We begin by fixing some of the objects that will persist throughout this article. Let g be a finite-dimensional semisimple real Lie algebra g_{c} One has the adjoint representations

of G and respectively. Differentiation then gives the adjoint representations of g and g_{c}, namely

Recall that an element ξ∈g (resp. ξc is called nilpotent if ad(ξ):g→g (resp. is a nilpotent vector space endomorphism. The nilpotent cone *N(g)* (resp. is then the subvariety of nilpotent elements of g (resp. g_{c}). A real (resp. complex) nilpotent orbit is an orbit of a nilpotent element in g (resp. g_{c}) under the adjoint representation of G (resp. g_{c}). Since the adjoint representation occurs by means of Lie algebra automorphisms, a real (resp. complex) nilpotent orbit is equivalently defined to be a G -orbit (resp. G_{c}-orbit) in By virtue of being an orbit of a smooth G -action, each real nilpotent orbit is an immersed submanifold of g. However, as g_{c} is a complex linear algebraic group, a complex nilpotent orbit is a smooth locally closed complex subvariety of g_{c}.

**The closure orders**

The sets and of real and complex nilpotent orbits are finite and carry the so-called closure order. In both cases, this is a partial order defined by

if and only if (1)

In the real case, one takes closures in the classical topology on g. For the complex case, note that a complex nilpotent orbit is a constructible subset of g_{c},so that its Zariski and classical closures agree. Accordingly, shall denote this common closure.

**Example 1: **Suppose that whose adjoint group is The nilpotent elements of are precisely the nilpotent nn matrices, so that the nilpotent -orbits are exactly the conjugacy classes of nilpotent matrices. The latter are indexed by the partitions of n via Jordan canonical forms. Given a partition λ=(λ_{1},λ_{2},…,λ_{k}) of n, let Qλ be the orbit of the nilpotent matrix with Jordan blocks of sizes λ_{1},λ_{2},…,λ_{k}, read from top-to-bottom. It is a classical result of Gerstenhaber that if and only if ≤ μ in the dominance order [10,11].

The poset has a unique maximal element called the regular nilpotent orbit. It is the collection of all elements of g_{c} which are simultaneously regular and nilpotent. In the framework of Example corresponds to the partition(n).

**Partitions of nilpotent orbits**

Generalizing Example 1, it is often natural to associate a partition to each real and complex nilpotent orbit. One sometimes endows these partitions with certain decorations and then uses decorated partitions to enumerate nilpotent orbits. It will be advantageous for us to recall the construction of the underlying (undecorated) partitions. Our exposition will be largely based on Chapters 5 and 9 of [12].

Suppose that g comes equipped with a faithful representation *g* ,where V is a finite-dimensional vector space over or The choice of V determines an assignment of partitions to nilpotent orbits in both g and . To this end, fix a real nilpotent orbit . and choose a point ξ∈ï. We may include ξ as the nilpositive We may include ξ as the nilpositive element of an –triple (ξ,h,n), so that

Regarding V as an -module, one has a decomposition into irreducibles,

where denotes the irreducible λj -dimensional representation of over . Let us require that λ_{1} ≥ λ_{2} ≥ … ≥ λ_{k}, so that (1,λ_{2},…,λ_{k}) is a partition of dim _{F}(V). Accordingly, we define the partition of to be to be

It can be established that depends only on .

The faithful representation V of g canonically gives a faithful epresentation of g_{c} Indeed, if V is over ,then one has an inclusion If V is over then the inclusion complexifies to give a faithful representation In either case, one proceeds in analogy with the real nilpotent case, using the faithful representation to yield a partition λ(Q) of a complex nilpotent orbit .The only notable difference with the real case is that is replaced with

**Example 2:** One can use the framework developed above to index the nilpotent orbits in using the partitions of n. This coincides with the indexing given in Example 1.

**Example 3:** The nilpotent orbits in are indexed by the partitions of n, after one replaces certain partitions with decorated counterparts. Indeed, if λ is a partition of n having only even parts, we replace λ with the decorated partitions λ+ and λ−. Otherwise, we leave λ undecorated.

**Example 4: **Suppose that n ≥ 3 and consider g=su(p,q) with 1≤qp and p+q=n. This Lie algebra is a real form of Now, let us regard a partition of n as a Young diagram with n boxes. Furthermore, recall that a signed Young diagram is a Young diagram whose boxes are marked with + or −, such that the signs alternate across each row [10]. We restrict our attention to the signed Young diagrams of signature (p,q), namely those for which + and − appear with respective multiplicities p and q. It turns out that the nilpotent orbits in su(p,q) are indexed by the signed Young diagrams of signature (p,q).

**Example 5: **Suppose that with n ≥ 4. Taking our faithful representation to be nilpotent orbits in are assigned partitions of 2n. The partitions realized in this way are those in which each even part appears with even multiplicity. One extends these partitions to an indexing set by replacing each λ having only even parts with the decorated partitions λ+ and λ−.

**Example 6:** Suppose that n ≥ 3 and consider with 1≤q≤p and p+q=n. Note that so(p,q) is a real form of As with Example 4, we will identify partitions of n with Young diagrams having n boxes. We begin with the signed Young diagrams of signature (p,q) such that each even-length row appears with even multiplicity and has its leftmost box marked with +. To obtain an indexing set for the nilpotent orbits in so(p,q), we decorate two classes of these signed Young diagrams Y. Accordingly, if Y has only even-length rows, then remove Y and add the four decorated diagrams Y+,+,Y+,−,Y−,+ and Y−,−. Secondly, suppose that Y has at least one odd-length row, and that each such row has an even number of boxes marked +, or that each such row has an even number of boxes marked −. In this case, we remove Y and add the decorated diagrams Y+ and Y.

**The complexification map**

There is a natural way in which a real nilpotent orbit determines a complex one. Indeed, the inclusion gives rise to a map.

Concretely, is just the unique complex nilpotent orbit containing ,and we shall call it the complexification of Let us then call the complexification map for g.

It will be prudent to note that the process of nilpotent orbit complexification is well-behaved with respect to taking partitions. More explicitly, we have the following proposition.

**Proposition 1: **Suppose that g is endowed with a faithful representation , If is a real nilpotent orbit, then

**Proof: **Choose a point and include it in an so -triple as in Section 2.3. Note that is then additionally an -triple in g_{c} Hence, we will prove that the faithful representation of g_{c} decomposes into irreducible -representations according to the partition

Let us write λ(ï)=(λ_{1},…,λ_{k}), so that

(2)

is the decomposition of V into irreducible representations. If V is over then and (2) is a decomposition of into irreducible - representations. If v is over then and

is the decomposition of into irreducible representations of sl_{2}(c). In each of these two cases, we have

Proposition 1 allows us to describe in more combinatorial terms. To this end, fix a faithful representation As in Examples 2-6, we obtain index sets I(g) and I(g_{c}) of decorated partitions for the real and complex nilpotent orbits, respectively. We may therefore regard as a map

Now, let be the set of all partitions of the form λ(Q), wit Q ⊆ g_{c} a complex nilpotent orbit. One has the map.

.

sending a decorated partition to its underlying partition. Proposition 1 is then the statement that the composite map

sends an index in to its underlying partition. Let us denote this composite map by

We will later give a characterization of those semisimple real Lie algebras g for which is surjective. To help motivate this, we investigate the matter of surjectivity in some concrete examples.

**Example 7:** Recall the parametrizations of nilpotent orbits in and outlined in Examples 3 and 2, respectively. We see that and The surjectivity of φg then follows immediately from that of

**Example 8:** Let the nilpotent orbits in g=su(n,n) be parametrized as in Example 4. We then have , whose nilpotent orbits are indexed by the partitions of 2n. Given such a partition λ, let Y denote the corresponding Young diagram. Since Y has an even number of boxes, it has an even number, 2k, of odd-length rows. Label the leftmost box in k of these rows with +, and label the leftmost box in each of the remaining k rows with −. Now, complete this labelling to obtain a signed Young diagram noting that then has signature (n,n). Hence, corresponds to a nilpotent orbit in su(n,n) and It follows that ψg is surjective Since and φ_{g}= ψg, we have shown φ_{g} to be surjective. A similar argument establishes surjectivity when g=su(n+1,n).

**Example 9: **Let us consider g=so(2n+2,2n), with nilpotent orbits indexed as in Example 6. Noting Example 5, a partition λ of 4n+2 represents a nilpotent orbit in if and only if each even part of λ occurs with even multiplicity. Since 4n+2 is even and not divisible by 4, it follows that any such λ has exactly 2k odd parts for some k ≥ 1 . Let Y be the Young diagram corresponding to λ, and label the leftmost box in k−1 of the odd-length rows with +. Next, label the leftmost box in each of k−1 different odd-length rows with −. Finally, use + to label the leftmost box in each of the two remaining odd-length rows. Let be any completion of our labelling to a signed Young diagram, such that the leftmost box in each even-length row is marked with +. Note that has signature (2n+2,2n). It follows that represents a nilpotent orbit in so(2n+2,2n) and.λ g . Furthermore, and so that φ_{g} is surjective

**Example 10: **Suppose that g=so(2n+2,2n−1), whose nilpotent orbits are parametrized in Example 6. Let the nilpotent orbits in be indexed as in Example 5. There exist partitions of 4n having only even parts, with each part appearing an even number of times. Let λ be one such partition, which by Example 6 represents a nilpotent orbit in Note that every signed Young diagram with underlying partition λ must have signature (2n,2n). In particular, λ cannot be realized as the image under ψg of a signed Young diagram indexing a nilpotent orbit in so(2n+2,2n−1). It follows that ψg and φ_{g} are not surjective.

**Surjectivity**

We now address the matter of classifying those semisimple real Lie algebras g for which φ_{g} is surjective. To proceed, we will require some additional machinery. Let be the (−1)-eigenspace of a Cartan involution, and let a be a maximal abelian subspace of p. Also, let h be a Cartan subalgebra of g containing a, and choose a fundamental Weyl chamber Given a complex nilpotent orbit ,there exists an sl2(r)-triple (ξ,h,) in g_{c} with the property that ξ∈Q and h∈C. The element hC is uniquely determined by this property, and is called the characteristic of Q. Theorem 1 of [5] then states that Qg ≠∅ if and only if h∈a. If g is split, then a=h, and the following lemma is immediate.

**Lemma 1: **If g is split, then φ_{g} is surjective

Let us now consider necessary conditions for surjectivity. To this end, recall that g is called quasi-split if there exists a subalgebra b ⊆ g such that b_{c}sub> is a Borel subalgebra of g_{c}. However, the following characterization of being quasi-split will be more suitable for our purposes.

**Lemma 2:** The Lie algebra g is quasi-split if and only if Q_{reg}(g_{c}) is in the image of φ_{g}. In particular, g being quasi-split is a necessary condition for φ_{g} to be surjective.

**Proof: **Proposition 5.1 of [13] states that g is quasi-split if and only if g contains a regular nilpotent element of g_{c}. Since Q_{reg}(g_{c}) consists of all such elements, this is equivalent to having Q_{reg}(g_{c})∩g ≠∅ hold. This latter condition holds precisely when Q_{reg}(g_{c}) is in the image of φ_{g}.

Lemmas 1 and 2 establish that φ_{g} being surjective is a weaker condition than having g be split, but stronger than having g be quasisplit. Furthermore, since su(n,n) is not a split real form of ,Example 8 establishes that surjectivity is strictly weaker than g being spilt,Yet, as so(2n+2,2n−1) is a quasi-split real form of Example 10 demonstrates that surjectivity is strictly stronger than having g be quasi-split. To obtain a more precise measure of the strength of the surjectivity condition, we will require the following proposition.

**Proposition 2:** Suppose that g decomposes as a Lie algebra into

Where g_{1},...,g_{k} are simple real Lie algebras. Let G_{1},…,G_{k} denote the respective adjoint groups..

• The map is surjective if and only each orbit complexification map is surjective.

Proof: For each j∈{1,…,k}, let πj:g→gj be the projection map. Note that ξ ∈g is nilpotent if and only if πj(ξ) is nilpotent in gj for each j. It follows that

defines an isomorphism of real varieties. Note that with the former group acting on *N*(g) and the latter group acting on the product of nilpotent cones.

One then sees that π is G-equivariant, so that it descends to a bijection

Analogous considerations give a second bijection

Furthermore, we have the commutative diagram

Hence, φ_{g} is surjective if and only if is so, proving (i).

By Lemma 2, proving (ii) will be equivalent to proving that Q_{reg}(g_{c}) is in the image of φ_{g} if and only if Q_{reg}((gj)) is in the image of j φ_{g} for all j. Using the diagram (3), this will follow from our proving that the image of Q_{reg}(g_{c}) under π c is the k-tuple of the regular nilpotent orbits in the , namely that

To see this, note that is the orbit of maximal dimension in This orbit is therefore the image of under the G_{c}-equivariant variety isomorphism ,implying that (4) holds.

Theorem 2: If g is a semisimple real Lie algebra, then φ_{g} is surjective if and only if g is quasi-split and has no simple summand of the form so(2n+1,2n−1).

Proof: If φ_{g} is surjective, then Lemma 2 implies that g is quasisplit. Also, Proposition 2 implies that each simple summand of g has a surjective orbit complexification map, and the above discussion then establishes that g has no simple summand of the form so(2n+1,2n−1). Conversely, assume that g is quasi-split and has no simple summand of the form so(2n+1,2n−1). By Proposition 2 (ii), each simple summand of g is quasi-split. Furthermore, the above discussion implies that the only quasi-split simple real Lie algebras with non-surjective orbit complexification maps are those of the form so(2n+1,2n−1). Hence, each simple summand of g has a surjective orbit complexification map, and Proposition 2 (i) implies that φ_{g} is surjective.

**The Image of φg**

Having investigated the surjectivity of φ_{g}, let us consider the more subtle matter of characterizing its image. Accordingly, let denote complex conjugation with respect to the real form The

**Lemma 3:** If is a complex nilpotent orbit, then so is σg(Q).

**Proof:** Note that σg integrates to a real Lie group automorphism

where is the connected, simply-connected Lie group with Lie algebra and ξ∈g_{c}, then

Hence, σg sends the (G)SC -orbit of ξ to the -orbit of σg(ξ). To complete the proof, we need only observe that (G_{c})SC -orbits coincide with G_{c}-orbits in g_{c}, and that σg(ξ) is nilpotent whenever ξ is nilpotent.

We may now use σg to explicitly describe the image of φ_{g} when g is quasi-split.

**Theorem 3:** If Q is a complex nilpotent orbit, the condition σg(Q)=Q is necessary for Q to be in the image of φ_{g}. If g is quasi-split, then this condition is also sufficient.

Proof: Assume that Q belongs to the image of φ_{g}, so that there exists ξ∈Q ∩g. Note that σg(Q) is then the complex nilpotent orbit containing σg(ξ)=ξ, meaning that σg(Q)=Q. Conversely, assume that g is quasi-split and that σg(Q)=Q. The latter means precisely that Q is defined over with respect to the real structure on g_{c} induced by the inclusion g ⊆ g. Theorem 4.2 of [9] then implies that Q∩g≠∅.

Using Theorem 3, we will give an interesting sufficient condition for a complex nilpotent orbit to be in the image of φ_{g} when g is quasi-split. In order to proceed, however, we will need a better understanding of the way in which σg permutes complex nilpotent orbits. To this end, we have the following lemma.

**Lemma 4: **Suppose that g comes with the faithful representation g ⊆ gl(V), where V is over . If Q is a complex nilpotent orbit, then

**Proof: **Choose an in g with ξ∈Q. Since σg preserves Lie brackets, it follows that is also an sl_{2}()-triple. The exercise is then to show that our two -triples give isomorphic representations of on . For this, it will suffice to prove that h and σg(h) act on V with the same eigenvalues, and that their respective eigenspaces for a given eigenvalue are equi dimensional. To this end, let be complex conjugation with respect to . Note that

.

for all x∈V, where . is used to denote the action of g_{c} on V. Hence, if x is an eigenvector of h with eigenvalue λ∈, then σV(x) is an eigenvector of σg(h) with eigenvalue λ. We conclude that h and σg(h) have the same eigenvalues. Furthermore, their respective eigenspaces for a fixed eigenvalue are related by σV, and so are equi-dimensional.

We now have the following

**Proposition 3:** Let g be a quasi-split semisimple real Lie algebra endowed with a faithful representation g ⊆ gl(V), where V is over. If Q is the unique complex nilpotent orbit with partition λ(), then Q is in the image of φ_{g}.

**Proof:** By Lemma 4, σg(Q) is a complex nilpotent orbit with partition λ(Q), and our hypothesis on Q gives σg(Q)=Q. Theorem 3 then implies that Q is in the image of φ_{g}.

A few remarks are in order.

**Remark 1: **One can use Proposition 3 to investigate whether φ_{g} is surjective without appealing to the partition-type description of φ_{g} discussed in Section 3.1. For instance, suppose that g=so(2n+2,2n), a quasi-split real form of We refer the reader to Example 5 for the precise assignment of partitions to nilpotent orbits in so4n+2(c). In particular, note that a complex nilpotent orbit is the unique one with its partition if and only if the partition does not have all even parts. Furthermore, as discussed in Example 9, there do not exist partitions of 4n+2 having only even parts such that each part appears with even multiplicity. Hence, each complex nilpotent orbit is specified by its partition, so Proposition 3 implies that φ_{g} is surjective.

**Remark 2:** The converse of Proposition 3 does not hold. Indeed, suppose that the split real form of. Recalling Example 5, every partition of 4n with only even parts, each appearing with even multiplicity, is the partition of two distinct complex nilpotent orbits. Yet, Lemma 1 implies that φ_{g} is surjective, so that these orbits are in the image of φ_{g}.

**Fibres**

In this section, we investigate the fibres of the orbit complexification map In order to proceed, it will be necessary to recall some aspects of the Kostant-Sekiguchi Correspondence. To this end, fix a Cartan involution θ:g→g. Letting k and p denote the 1 and (−1)-eigenspaces of θ, respectively, we obtain the internal direct sum decomposition

This gives a second decomposition

where and are the complexifications of k and p, respectively. Let be the connected closed subgroups with respective Lie algebras k and k_{c}. The Kostant-Sekiguchi Correspondence is one between the nilpotent orbits in g and the K_{c}-orbits in the (K_{c}-invariant) subvariety

**Theorem 4:** (The Kostant-Sekiguchi Correspondence) There is a bijective correspondence.

with the following properties.

• It is an isomorphism of posets, where

• If is a real nilpotent orbit, then and Ú are K-equivariantly

The first property was established by Barbasch and Sepanski in, while the second was proved by Vergne in [15,16]. Each paper makes extensive use of Kronheimer’s desciption of nilpotent orbits from [17].

**Lemma 5: **If is a real nilpotent orbit, then is the unique G-orbit of maximal dimension in .

Proof: Suppose that is another G -orbit lying in Property (i) in Theorem 4, it follows that is an orbit in different from However, is an orbit of the complex algebraic group K_{c} under an algebraic action, and therefore is the unique orbit of maximal dimension in its closure. Hence, Property (ii) of Theorem 4 implies that the Kostant-Sekiguchi Correspondence preserves real dimensions, so that

We will also require some understanding of the relationship between the G-centralizer of ξ∈g and the G_{c}-centralizer of ξ, viewed as an element of g_{c}. Denoting these centralizers by Gξ and (G_{c})ξ respectively, we have the following lemma.

**Lemma 6:** If ξ∈g, then Gξ is a real form of (G_{c})ξ.

**Proof: **We are claiming that the Lie algebra of is the complexification of the Lie algebra of Gξ. The former is (g_{c}) ξ={η∈g_{c}:[η,ξ]=0}, while the Lie algebra of Gξ is gξ={η∈g_{c}:[η,ξ]=0}. If η=η_{1}+iη_{2}∈g_{c} with η_{1},2∈g, then [η,ξ]=[η_{1}, ξ]+i[η_{2}, ξ]. So, η∈(g_{c})ξ if and only if η_{1},η_{2}∈gξ. This is equivalent to the condition that η(gξ)_{c} ⊆ g_{c}, so that (g_{c})ξ=(gξ)_{c}.

We may now prove the main result of this section.

**Theorem 5: **If and are real nilpotent orbits with the property that then either are incomparable in the closure order. In other words, each fibre of φ_{g} consists of pairwise incomparable nilpotent orbits..

**Proof: **Assume that and are comparable. Without the loss of generality,We will prove that which by Lemma 5 will amount to showing that the dimensions of and agree. we have Using Lemma 6, this becomes ξ . Hence, the (real) dimensions of and coincide.

**Proof: **If is surjective, and then Lemma 2 implies that is quasi-split. Also, Proposition 2 implies that each simple summand of has a surjective orbit complexification map, and the above discussion then establishes that has no simple summand of the form. Conversely, assume that is quasi-split and has no simple summand of the form. By Proposition 2 (ii), each simple summand of is quasi-split. Furthermore, the above discussion implies that the only quasi-split simple real Lie algebras with non-surjective orbit complexification maps are those of the form. Hence, each simple summand of has a surjective orbit complexification map, and Proposition 2 (i) implies that is surjective.

**The Image of φC**

Having investigated the surjectivity of φC, let us consider the more subtle matter of characterizing its image. Accordingly, let denote complex conjugation with respect to the real form. The following lemma will be useful.

Lemma 3 If is a complex nilpotent orbit, then so is.

**Proof: **Note that integrates to a real Lie group automorphism

Where is the connected, simply-connected Lie group with Lie algebra? If and, then

Hence, sends the -orbit of to the -orbit of. To complete the proof, we need only observe that -orbits coincide with -orbits in, and that is nilpotent whenever is nilpotent.

We may now use to explicitly describe the image of when is quasisplit.

**Theorem 3: **If is a complex nilpotent orbit, the condition is necessary for to be in the image of. If is quasi-split, then this condition is also sufficient.

Proof: Assume that belongs to the image of, so that there exists. Note that is then the complex nilpotent orbit containing, meaning that, Conversely, assume that is quasi-split and that. The latter means precisely that is defined over with respect to the real structure on

Using Theorem 3, we will give an interesting sufficient condition for a complex nilpotent orbit to be in the image of when is quasi-split. In order to proceed, however, we will need a better understanding of the way in which permutes complex nilpotent orbits. To this end, we have the following lemma. induced by the inclusion. Theorem 4.2 of [9] then implies that.

Lemma 4 Suppose that comes with the faithful representation, where is over. If is a complex nilpotent orbit then.

**Proof:** Choose an -triple in with. Since preserves Lie brackets, it follows that is also an -triple. The exercise is then to show that our two -triples give isomorphic representations of on. For this, it will suffice to prove that and act on with the same eigenvalues, and that their respective eigenspaces for a given eigenvalue are equi-dimensional. To this end, let be complex conjugation with respect to. Note that for all , where is used to denote the action of on . Hence, if is an eigenvector of with eigenvalue, then is an eigenvector of with eigenvalue. We conclude that and have the same eigenvalues. Furthermore, their respective eigenspaces for a fixed eigenvalue are related by, and so are equidimensional.

We now have the following

Proposition 3 Let be a quasi-split semisimple real Lie algebra endowed with a faithful representation, where is over. If is the unique complex nilpotent orbit with partition, then is in the image of.

**Proof: **By Lemma 4, is a complex nilpotent orbit with partition, and our hypothesis on gives. Theorem 3 then implies that is in the image of.

A few remarks are in order.

**Remark 1:** One can use Proposition 3 to investigate whether is surjective without appealing to the partition-type description of discussed in Section 3.1. For instance, suppose that, a quasi-split real form of. We refer the reader to Example 5 for the precise assignment of partitions to nilpotent orbits in. In particular, note that a complex nilpotent orbit is the unique one with its partition if and only if the partition does not have all even parts. Furthermore, as discussed in Example 9, there do not exist partitions of having only even parts such that each part appears with even multiplicity. Hence, each complex nilpotent orbit is specified by its partition, so Proposition 3 implies that is surjective.

**Remark 2: **The converse of Proposition 3 does not hold. Indeed, suppose that, the split real form of. Recalling Example 5, every partition of with only even parts, each appearing with even multiplicity, is the partition of two distinct complex nilpotent orbits. Yet, Lemma 1 implies that is surjective, so that these orbits are in the image of.

**Fibres**

In this section, we investigate the fibres of the orbit complexification map. In order to proceed, it will be necessary to recall some aspects of the Kostant-Sekiguchi Correspondence. To this end, fix a Cartan involution. Letting and denote the and Eigen spaces of, respectively, we obtain the internal direct sum decomposition

This gives a second decomposition

Where and are the complexifications of and, respectively. Let and be the connected closed subgroups with respective Lie algebras and. The Kostant-Sekiguchi Correspondence is one between the nilpotent orbits in and the - orbits in the (-invariant) subvariety of.

**Theorem 4:** (The Kostant-Sekiguchi Correspondence) There is a bijective correspondence with the following properties.

• It is an isomorphism of posets, where is endowed with the closure order (??).

• If is a real nilpotent orbit, then and are equivariantly diffeomorphic.

The first property was established by Barbasch and Sepanski in, while the second was proved by Vergne in [15,16]. Each paper makes extensive use of Kronheimer’s description of nilpotent orbits from [17].

We now prove two preliminary results, the first of which is a direct consequence of the Kostant-Sekiguchi Correspondence.

Lemma 5 If is a real nilpotent orbit, then is the unique -orbit of maximal dimension in.

**Proof: **Suppose that is another -orbit lying in. By Property (i) in Theorem 4, it follows that is an orbit in different from. However, is an orbit of the complex algebraic group under an algebraic action, and therefore is the unique orbit of maximal dimension in its closure. Hence, Property (ii) of Theorem 4 implies that the Kostant-Sekiguchi Correspondence preserves real dimensions, so that.

We will also require some understanding of the relationship between the -centralizer of and the -centralizer of, viewed as an element of. Denoting these centralizers by and, respectively, we have the following lemma.

Lemma 6 If, then is a real form of.

**Proof: **We are claiming that the Lie algebra of is the complexification of the Lie algebra of. The former is, while the Lie algebra of is. So, if and only if, this is equivalent to the condition that, so that.

We may now prove the main result of this section.

**Theorem 5:** If and are real nilpotent orbits with the property that, then either or and are incomparable in the closure order. In other words, each fiber of consists of pairwise incomparable nilpotent orbits.

**Proof: **Assume that and are comparable. Without the loss of generality. We will prove that, which by Lemma 5 will amount to showing that the dimensions of and agree. To this end, choose points and since, we have. Using Lemma 6, this becomes. Hence, the (real) dimensions of and coincide.

The author is grateful to John Scherk for discussions that prompted much of this work. The author also acknowledges Lisa Jeffrey and Steven Rayan for their considerable support. This work was partially funded by NSERC CGS and OGS awards.

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