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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Properties of Nilpotent Orbit Complexification

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.

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Abstract

We consider aspects of the relationship between nilpotent orbits in a semisim-ple real Lie algebra g and those in
its complexification g�. 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 g�. Finally, for g quasi-split, we characterize those complex nilpotent orbits containing real ones.

Keywords

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

Introduction

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 image lies in a unique complex nilpotent orbit image , the complexification of image 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 image is realizable as the complexification of a real nilpotent orbit if and only if image is invariant under conjugation with respect to the real form image

• If image are real nilpotent orbits satisfying image then either image 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 image 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 image 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 image 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 image.

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 Orbit Generalities

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 gc One has the adjoint representations

image

of G and image respectively. Differentiation then gives the adjoint representations of g and gc, namely

image

Recall that an element ξ∈g (resp. ξcimage is called nilpotent if ad(ξ):g→g (resp.image is a nilpotent vector space endomorphism. The nilpotent cone N(g) (resp.image is then the subvariety of nilpotent elements of g (resp. gc). A real (resp. complex) nilpotent orbit is an orbit of a nilpotent element in g (resp. gc) under the adjoint representation of G (resp. gc). 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. Gc-orbit) in image By virtue of being an orbit of a smooth G -action, each real nilpotent orbit is an immersed submanifold of g. However, as gc is a complex linear algebraic group, a complex nilpotent orbit is a smooth locally closed complex subvariety of gc.

The closure orders

The sets image and image 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

image if and only if image (1)

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

Example 1: Suppose that image whose adjoint group is image The nilpotent elements of image are precisely the nilpotent nn matrices, so that the nilpotent image -orbits are exactly the image conjugacy classes of nilpotent matrices. The latter are indexed by the partitions of n via Jordan canonical forms. Given a partition λ=(λ12,…,λk) of n, let Qλ be the image orbit of the nilpotent matrix with Jordan blocks of sizes λ12,…,λk, read from top-to-bottom. It is a classical result of Gerstenhaber that image if and only if ≤ μ in the dominance order [10,11].

The poset image has a unique maximal elementimage called the regular nilpotent orbit. It is the collection of all elements of gc which are simultaneously regular and nilpotent. In the framework of Example image 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 image,where V is a finite-dimensional vector space over image or image The choice of V determines an assignment of partitions to nilpotent orbits in both g and image. To this end, fix a real nilpotent orbit image. and choose a point ξ∈ïimage. We may include ξ as the nilpositive We may include ξ as the nilpositive element of an image –triple (ξ,h,n), so that

image

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

image

where image denotes the irreducible λj -dimensional representation of image over image. Let us require that λ1 ≥ λ2 ≥ … ≥ λk, so that (1,λ2,…,λk) is a partition of dim F(V). Accordingly, we define the partition of image to be to be

image

It can be established that image depends only on image.

The faithful representation V of g canonically gives a faithful epresentation image of gc Indeed, if V is over image,then one has an inclusion image If V is over image then the inclusionimage complexifies to give a faithful representation image 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 image.The only notable difference with the real case is that image is replaced withimage

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

Example 3: The nilpotent orbits in image 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 image 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 image with n ≥ 4. Taking our faithful representation to beimage nilpotent orbits in image 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 image with 1≤q≤p and p+q=n. Note that so(p,q) is a real form of image 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.

Nilpotent Orbit Complexification

The complexification map

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

image

image

Concretely, image is just the unique complex nilpotent orbit containing image,and we shall call it the complexification of image Let us then call image 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 image, If image is a real nilpotent orbit, then image

Proof: Choose a point imageand include it in an so image-tripleimage as in Section 2.3. Note that image is then additionally an image -triple in gc Hence, we will prove that the faithful representationimage of gc decomposes into irreducible image-representations according to the partition image

Let us write λ(ï)=(λ1,…,λk), so that

image (2)

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

image

is the decomposition of image into irreducible representations of sl2(c). In each of these two cases, we haveimage

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

image

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

.image

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

image

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

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

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

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 image noting that image then has signature (n,n). Hence,image corresponds to a nilpotent orbit in su(n,n) andimage It follows that ψg is surjective Sinceimage and φg= ψg, we have shown φg 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 image 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 image 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 image has signature (2n+2,2n). It follows that image represents a nilpotent orbit in so(2n+2,2n) andimage.λ g . Furthermore, image andimage so that φg 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 image 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 image 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 φg are not surjective.

Surjectivity

We now address the matter of classifying those semisimple real Lie algebras g for which φg is surjective. To proceed, we will require some additional machinery. Let image 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 image Given a complex nilpotent orbitimage ,there exists an sl2(r)-triple (ξ,h,) in gc 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 φg 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 bcsub> is a Borel subalgebra of gc. 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 Qreg(gc) is in the image of φg. In particular, g being quasi-split is a necessary condition for φg 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 gc. Since Qreg(gc) consists of all such elements, this is equivalent to having Qreg(gc)∩g ≠∅ hold. This latter condition holds precisely when Qreg(gc) is in the image of φg.

Lemmas 1 and 2 establish that φg 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 image,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 image 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

image

Where g1,...,gk are simple real Lie algebras. Let G1,…,Gk denote the respective adjoint groups..

• The map image is surjective if and only each orbit complexification mapimage 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

image

defines an isomorphism of real varieties. Note that image 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

image

Analogous considerations give a second bijection

image

Furthermore, we have the commutative diagram

image

image

Hence, φg is surjective if and only if image is so, proving (i).

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

image

To see this, note that image is theimage orbit of maximal dimension inimage This orbit is therefore the image ofimage under the Gc-equivariant variety isomorphism image,implying that (4) holds.

Theorem 2: If g is a semisimple real Lie algebra, then φg 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 φg 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 φg is surjective.

The Image of φg

Having investigated the surjectivity of φg, let us consider the more subtle matter of characterizing its image. Accordingly, let image denote complex conjugation with respect to the real formimage The

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

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

image

where image is the connected, simply-connected Lie group with Lie algebraimage and ξ∈gc, then

image

Hence, σg sends the (G)SC -orbit of ξ to theimage -orbit of σg(ξ). To complete the proof, we need only observe that (Gc)SC -orbits coincide with Gc-orbits in gc, and that σg(ξ) is nilpotent whenever ξ is nilpotent.

We may now use σg to explicitly describe the image of φg 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 φg. If g is quasi-split, then this condition is also sufficient.

Proof: Assume that Q belongs to the image of φg, 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 image with respect to the real structure on gc 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 φg 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 image. If Q is a complex nilpotent orbit, then

image

Proof: Choose an image in g with ξ∈Q. Since σg preserves Lie brackets, it follows thatimage is also an sl2(image)-triple. The exercise is then to show that our two image-triples give isomorphic representations of image on image . For this, it will suffice to prove that h and σg(h) act on Vimage with the same eigenvalues, and that their respective eigenspaces for a given eigenvalue are equi dimensional. To this end, letimage be complex conjugation with respect to image. Note that

image.

for all x∈Vimage, where . is used to denote the action of gc on Vimage. Hence, if x is an eigenvector of h with eigenvalue λ∈image, 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 overimage. If Q is the unique complex nilpotent orbit with partition λ(), then Q is in the image of φg.

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 φg.

A few remarks are in order.

Remark 1: One can use Proposition 3 to investigate whether φg is surjective without appealing to the partition-type description of φg discussed in Section 3.1. For instance, suppose that g=so(2n+2,2n), a quasi-split real form of image 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 φg is surjective.

Remark 2: The converse of Proposition 3 does not hold. Indeed, suppose that image the split real form ofimage. 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 φg is surjective, so that these orbits are in the image of φg.

Fibres

In this section, we investigate the fibres of the orbit complexification map image 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

image

This gives a second decomposition

image

where image and image are the complexifications of k and p, respectively. Letimage be the connected closed subgroups with respective Lie algebras k and kc. The Kostant-Sekiguchi Correspondence is one between the nilpotent orbits in g and the Kc-orbits in the (Kc-invariant) subvarietyimage

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

image

image

with the following properties.

• It is an isomorphism of posets, where image

• If image is a real nilpotent orbit, then image and imageÚ 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 image is a real nilpotent orbit, then image is the unique G-orbit of maximal dimension in image .

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

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

Lemma 6: If ξ∈g, then Gξ is a real form of (Gc)ξ.

Proof: We are claiming that the Lie algebra of image is the complexification of the Lie algebra of Gξ. The former is (gc) ξ={η∈gc:[η,ξ]=0}, while the Lie algebra of Gξ is gξ={η∈gc:[η,ξ]=0}. If η=η1+iη2∈gc with η1,2∈g, then [η,ξ]=[η1, ξ]+i[η2, ξ]. So, η∈(gc)ξ if and only if η12∈gξ. This is equivalent to the condition that η(gξ)c ⊆ gc, so that (gc)ξ=(gξ)c.

We may now prove the main result of this section.

Theorem 5: If imageand imageare real nilpotent orbits with the property thatimage then eitherimage are incomparable in the closure order. In other words, each fibre of φg consists of pairwise incomparable nilpotent orbits..

Proof: Assume that imageand image are comparable. Without the loss of generality,imageWe will prove that image which by Lemma 5 will amount to showing that the dimensions of image and image agree. we haveimage Using Lemma 6, this becomesimage ξ . Hence, the (real) dimensions of image and image 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.

Acknowledgements

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.

References

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