Properties of Nilpotent Orbit Complexification Peter Crooks* Department of Mathematics, University of Toronto, Canada

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Research Article Open Access Properties of Nilpotent Orbit Complexification Peter Crooks* Department of Mathematics, University of Toronto, Canada

Abstract We consider aspects of the relationship between nilpotent orbits in a semisimple real Lie algebra g and those in its complexificationg . 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- map to have a convenient description in terms of decorated partitions. Sekiguchi correspondence Section 3.2 then directly addresses the proof of Theorem 1 (i), formulated as a characterization of when ϕg is surjective. Using Proposition 2, we 1. Introduction reduce this exercise to one of characterizing surjectivity for g simple. Together with the observation that surjectivity implies g is quasi-split 1.1 Background and statement of results and is implied by g being split, Proposition 2 allows us to complete the Real and complex nilpotent orbits have received considerable proof of Theorem 1 (i). attention in the literature. The former have been studied in a variety We proceed to Section 3.3, which provides the proof of Theorem of contexts, including differential geometry, symplectic geometry, 1 (ii). The essential ingredient is Kottwitz’s work [9]. We also include and Hodge theory [1]. Also, there has been some interest in concrete Proposition 3, which gives an interesting sufficient condition for a descriptions of the poset structure on real nilpotent orbits in specific complex nilpotent orbit to be in the image of ϕ . cases [2,3]. By contrast, complex nilpotent orbits are studied in algebraic g geometry [4,5,6] and representation theory — in particular, Springer In Section 3.4, we give a proof of Theorem 1 (iii). Our proof makes Theory [7]. extensive use of the Kostant-Sekiguchi Correspondence, the relevant parts of which are mentioned. Attention has also been given to the interplay between real and complex nilpotent orbits, with the Kostant-Sekiguchi Correspondence 2. Nilpotent Orbit Generalities being perhaps the most famous instance [8]. Accordingly, the present article provides additional points of comparison between real and 2.1 Nilpotent orbits g complex nilpotent orbits. Specifically, let be a finite-dimensional We begin by fixing some of the objects that will persist throughout semisimple real Lie algebra with complexification g . Each real  this article. Let g be a finite-dimensional semisimple real Lie algebra nilpotent orbit  ⊆ g lies in a unique complex nilpotent orbit  ⊆ g ,   with adjoint group G. Also, let g :=g⊗ be the complexification of g, the complexification of . The following is our main result.   whose adjoint group is the complexification G. One has the adjoint Theorem 1: The process of nilpotent orbit complexification has the representations following properties. g g Ad:G→GL( ) and Ad:G→ GL( ) (i) Every complex nilpotent orbit is realizable as the complexification of G and G , respectively. Differentiation then gives the adjoint g  of a real nilpotent orbit if and only if is quasi-split and has no simple representations of g and g , namely summand of the form so (2n+1, 2n −1).  ad:g→gl(g) and ad :g →gl(g ). (ii) If g is quasi-split, then a complex nilpotent orbit Q ⊆ g is     ∈g ∈g realizable as the complexification of a real nilpotent orbit if and only if Recall that an element ξ (resp. ξ ) is called nilpotent Q g ⊆ g if ad(ξ):g→g (resp. ad (ξ):g →g ) is a nilpotent vector space is invariant under conjugation with respect to the real form .    endomorphism. The nilpotent cone (g) (resp. (g )) is then the (iii) If  , ⊆ g are real nilpotent orbits satisfying ( ) =( ) ,  1 2 1  2  subvariety of nilpotent elements of g (resp. g ). A real (resp. complex)    then either 1= 2 or these two orbits are incomparable in the closure order. g g nilpotent orbit is an orbit of a nilpotent element in (resp. ) under the adjoint representation of G (resp. G ). Since the adjoint representation 1.2 Structure of the article  occurs by means of Lie algebra automorphisms, a real (resp. complex) We begin with an overview of nilpotent orbits in semisimple real and complex Lie algebras. In recognition of Theorem 1 (iii), and of the Q g role played by the unique maximal complex nilpotent orbit reg( ) *Corresponding author: Peter Crooks, Department of Mathematics, University of throughout the article, Section 2.2 reviews the closure orders on the sets Toronto, Canada; E-mail: [email protected] of real and complex nilpotent orbits. In Section 2.3, we recall some of Received April 24, 2016; Accepted June 09, 2016; Published June 30, 2016 the details underlying the use of decorated partitions to index nilpotent Citation: Crooks P (2016) Properties of Nilpotent Orbit Complexification. J orbits. Generalized Lie Theory Appl S2: 012. doi:10.4172/1736-4337.S2-012 Section 3 is devoted to the proof of Theorem 1. In Section 3.1, we Copyright: © 2016 Crooks P. This is an open-access article distributed under the ϕ terms of the Creative Commons Attribution License, which permits unrestricted represent nilpotent orbit complexification as a poset map g between use, distribution, and reproduction in any medium, provided the original author and the collections of real and complex nilpotent orbits. Next, we show this source are credited.

Recent Advances of Lie Theory in J Generalized Lie Theory Appl differential Geometry, in memory of ISSN: 1736-4337 GLTA, an open access journal John Nash Citation: Crooks P (2016) Properties of Nilpotent Orbit Complexification. J Generalized Lie Theory Appl S2: 012. doi:10.4172/1736-4337.S2-012

Page 2 of 6 nilpotent orbit is equivalently defined to be a G -orbit (resp. G -orbit) g ⊆ gl VV =  complexifies to give a faithful representation  (V) (ie.  ).  g  g in ( ) (resp. ( )). By virtue of being an orbit of a smooth G -action, In either case, one proceeds in analogy with the real nilpotent case, each real nilpotent orbit is an immersed submanifold of g. However, as using the faithful representation to yield a partition λ(Q) of a complex ⊆ G is a complex linear algebraic group, a complex nilpotent orbit is a nilpotent orbit Q (g). The only notable difference with the real case g smooth locally closed complex subvariety of . is that sl2() is replaced with sl2(). 2.2 The closure orders Example 2: One can use the framework developed above to index the nilpotent orbits in sl ( ) using the partitions of n. This coincides The sets (g)/G and (g )/G of real and complex nilpotent 2    with the indexing given in Example 1. orbits are finite and carry the so-called closure order. In both cases, this is a partial order defined by Example 3: The nilpotent orbits in sl2() are indexed by the partitions of n, after one replaces certain partitions with decorated  ≤ if and only if ⊆ . (1) 1 2 12 counterparts. Indeed, if λ is a partition of n having only even parts, we

In the real case, one takes closures in the classical topology on replace λ with the decorated partitions λ+ and λ−. Otherwise, we leave g. For the complex case, note that a complex nilpotent orbit Q is a λ undecorated. constructible subset of g , so that its Zariski and classical closures agree.  Example 4: Suppose that n ≥ 3 and consider g=su(p,q) with 1≤q≤p Accordingly, Q shall denote this common closure. and p+q=n. This Lie algebra is a real form of sln(). Now, let us regard a g sl partition of n as a Young diagram with n boxes. Furthermore, recall that Example 1: Suppose that = n(), whose adjoint group is sl a signed Young diagram is a Young diagram whose boxes are marked G=PSLn(). The nilpotent elements of n() are precisely the × nilpotent n n matrices, so that the nilpotent PSLn() -orbits are with + or −, such that the signs alternate across each row [12]. We exactly the (GLn()-) conjugacy classes of nilpotent matrices. The latter restrict our attention to the signed Young diagrams of signature (p,q), are indexed by the partitions of n via Jordan canonical forms. Given a namely those for which + and − appear with respective multiplicities p su partition λ=(λ1,λ2,…,λk) of n, let Qλ be the PSLn()-orbit of the nilpotent and q. It turns out that the nilpotent orbits in (p,q) are indexed by the matrix with Jordan blocks of sizes λ1,λ2,…,λk, read from top-to-bottom. signed Young diagrams of signature (p,q). It is a classical result of Gerstenhaber [10] that Q ≤Q if and only if λ≤μ λ µ g so Example 5: Suppose that = 2n() with n ≥ 4. Taking our faithful in the dominance order [11]. 2n representation to be  , nilpotent orbits in so2n() are assigned  g Q g partitions of 2n. The partitions realized in this way are those in which The poset ( )/G has a unique maximal element reg( ), called g each even part appears with even multiplicity. One extends these the regular nilpotent orbit. It is the collection of all elements of  which are simultaneously regular and nilpotent. In the framework of Example partitions to an indexing set by replacing each λ having only even parts

1, Qreg (sln()) corresponds to the partition(n). with the decorated partitions λ+ and λ−. g so 2.3 Partitions of nilpotent orbits Example 6: Suppose that n ≥ 3 and consider = (p,q) with so g so 1≤q≤p and p+q=n. Note that (p,q) is a real form of = n(). As Generalizing Example 1, it is often natural to associate a partition with Example 4, we will identify partitions of n with Young diagrams to each real and complex nilpotent orbit. One sometimes endows these having n boxes. We begin with the signed Young diagrams of signature partitions with certain decorations and then uses decorated partitions (p,q) such that each even-length row appears with even multiplicity to enumerate nilpotent orbits. It will be advantageous for us to recall and has its leftmost box marked with +. To obtain an indexing set for the construction of the underlying (undecorated) partitions. Our the nilpotent orbits in so(p,q), we decorate two classes of these signed exposition will be largely based on Chapters 5 and 9 of [12]. Young diagrams Y. Accordingly, if Y has only even-length rows, then

Suppose that g comes equipped with a faithful representation g remove Y and add the four decorated diagrams Y+,+,Y+,−,Y−,+ and Y−,−. ⊆ gl(V)=End (V), where V is a finite-dimensional vector space over 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 = or . The choice of V determines an assignment of partitions to nilpotent orbits in both g and g . To this end, fix a real nilpotent orbit has an even number of boxes marked −. In this case, we remove Y and  add the decorated diagrams Y and Y.  ⊆ (g) and choose a point ξ∈ . We may include ξ as the nilpositive + element of an sl ( ) –triple (ξ,h,n), so that 2  3. Nilpotent Orbit Complexification [ξ,n]=h,[h,ξ]=2ξ,[h,n]=− 2n 3.1 The complexification map Regarding V as an sl ( )-module, one has a decomposition into 2  There is a natural way in which a real nilpotent orbit determines a irreducibles,  g ⊆  g k complex one. Indeed, the inclusion ( ) ( ) gives rise to a map. VV=,λ ⊕ j j=1 ϕ  g  g g: ( )/G→ ( )/G where Vλ denotes the irreducible λ -dimensional representation of j j sl ( ) over . Let us require that λ ≥ λ ≥ … ≥ λ , so that (λ ,λ ,…,λ ) is    2   1 2 k 1 2 k . a partition of dim (V). Accordingly, we define the partition of  to be   Concretely,  is just the unique complex nilpotent orbit containing λ():=(λ ,λ ,…,λ ). 1 2 k , and we shall call it the complexification of . Let us then call ϕg the It can be established that λ() depends only on . complexification map for g. The faithful representation V of g canonically gives a faithful It will be prudent to note that the process of nilpotent orbit V g complexification is well-behaved with respect to taking partitions. representation of . Indeed, if V is over , then one has an inclusion g ⊆ gl VV = g ⊆ gl More explicitly, we have the following proposition.  (V) (so ). If V is over , then the inclusion (V)

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Proposition 1: Suppose that g is endowed with a faithful of λ occurs with even multiplicity. Since 4n+2 is even and not divisible g ⊆ gl    by 4, it follows that any such λ has exactly 2k odd parts for some k ≥ 1 . representation (V). If is a real nilpotent orbit, then λ( )=λ( ). Let Y be the Young diagram corresponding to λ, and label the leftmost Proof: Choose a point ξ∈ and include it in an sl ( )-triple (ξ,h,η) 2  box in k−1 of the odd-length rows with +. Next, label the leftmost box as in Section 2.3. Note that (ξ,h,η) is then additionally an sl ( )-triple 2  in each of k−1 different odd-length rows with −. Finally, use + to label in g . Hence, we will prove that the faithful representation V of g   the leftmost box in each of the two remaining odd-length rows. Let Y decomposes into irreducible sl ( )-representations according to the 2  be any completion of our labelling to a signed Young diagram, such that partition λ(). the leftmost box in each even-length row is marked with +. Note that Y Let us write λ()=(λ ,…,λ ), so that has signature (2n+2,2n). It follows that Y represents a nilpotent orbit 1 k  so ψλg (Y )= g g ϕ k in (2n+2,2n) and . Furthermore, I( )=P( ) and g= ψg, VV= λ ϕ ⊕ j (2) so that g is surjective. j=1 Example 10: Suppose that g=so(2n+2,2n−1), whose nilpotent is the decomposition of V into irreducible sl ()-representations. If V 2 orbits are parametrized in Example 6. Let the nilpotent orbits in is over , then VV = and (2) is a decomposition of V into irreducible g so sl  = 4n() be indexed as in Example 5. There exist partitions of n4 2()-representations. If V is over , then VV=  and k having only even parts, with each part appearing an even number of VV=()λ ⊕ j times. Let λ be one such partition, which by Example 6 represents a j=1 so  nilpotent orbit in 4n(). Note that every signed Young diagram with is the decomposition of V into irreducible representations of sl2(). In   underlying partition λ must have signature (2n,2n). In particular, λ each of these two cases, we have λ( )=λ( ). cannot be realized as the image under ψg of a signed Young diagram ϕ Proposition 1 allows us to describe g in more combinatorial terms. indexing a nilpotent orbit in so(2n+2,2n−1). It follows that ψg and ϕg To this end, fix a faithful representation g ⊆ gl(V). As in Examples 2-6, are not surjective. we obtain index sets I(g) and I(g ) of decorated partitions for the real  3.2 Surjectivity and complex nilpotent orbits, respectively. We may therefore regard ϕg as a map We now address the matter of classifying those semisimple real Lie

algebras g for which ϕg is surjective. To proceed, we will require some ϕg: I(g) → I(g ).  additional machinery. Let p ⊆ g be the (−1)-eigenspace of a Cartan g Q Q ⊆ Now, let P( ) be the set of all partitions of the form λ( ), with involution, and let a be a maximal abelian subspace of p. Also, let h be g  a complex nilpotent orbit. One has the map a Cartan subalgebra of g containing a, and choose a fundamental Weyl chamber C ⊆ h. Given a complex nilpotent orbit Q ⊆ g , there exists I(g) → P(g ),   sl η g ∈Q ∈ an 2()-triple (ξ,h, ) in  with the property that ξ and h C. The sending a decorated partition to its underlying partition. Proposition 1 element h C is uniquely determined by this property, and is called the is then the statement that the composite map characteristic of Q. ϕ ∈ g →g gg → Qg ∅ ∈a g I() IP () () Theorem 1 of [13] then states that ≠ if and only if h . If is a h sends an index in I(g) to its underlying partition. Let us denote this split, then = , and the following lemma is immediate. composite map by ψ :I(g)→P(g ). g  Lemma 1: If g is split, then ϕg is surjective. We will later give a characterization of those semisimple real Lie Let us now consider necessary conditions for surjectivity. To this algebras g for which ϕg is surjective. To help motivate this, we investigate end, recall that g is called quasi-split if there exists a subalgebra b the matter of surjectivity in some concrete examples. ⊆ g b g such that  is a Borel subalgebra of . However, the following Example 7: Recall the parametrizations of nilpotent orbits in characterization of being quasi-split will be more suitable for our g sl g sl purposes. = 2() and = 2() outlined in Examples 3 and 2, respectively. We see that I(g)=P(g ) and ϕg= ψg. The surjectivity of ϕg then follows g Q g  Lemma 2: The Lie algebra is quasi-split if and only if reg( ) immediately from that of ψ . g is in the image of ϕg. In particular, g being quasi-split is a necessary condition for ϕ to be surjective. Example 8: Let the nilpotent orbits in g=su(n,n) be parametrized g g sl g as in Example 4. We then have = 2n(), whose nilpotent orbits are Proof: Proposition 5.1 of [14], states that is quasi-split if and only indexed by the partitions of 2n. Given such a partition λ, let Y denote g g Q g if contains a regular nilpotent element of . Since reg( ) consists Q g ∩g ∅ the corresponding Young diagram. Since Y has an even number of of all such elements, this is equivalent to having reg( ) ≠ hold. Q g ϕ boxes, it has an even number, 2k, of odd-length rows. Label the leftmost This latter condition holds precisely when reg( ) is in the image of g. box in k of these rows with +, and label the leftmost box in each of Lemmas 1 and 2 establish that ϕg being surjective is a weaker the remaining k rows with −. Now, complete this labelling to obtain condition than having g be split, but stronger than having g be quasi- a signed Young diagram  , noting that Y then has signature (n,n). Y split. Furthermore, since su(n,n) is not a split real form of sl (), Y ψλ(Y )= 2n Hence, corresponds to a nilpotent orbit in su(n,n) and g . Example 8 establishes that surjectivity is strictly weaker than g being It follows that ψ is surjective. Since I(g)=P(g ) and ϕ = ψ , we have g  g g split. Yet, as so(2n+2,2n−1) is a quasi-split real form of so (), Example ϕ 4n shown g to be surjective. A similar argument establishes surjectivity 10 demonstrates that surjectivity is strictly stronger than having g be when g=su(n+1,n). quasi-split. To obtain a more precise measure of the strength of the Example 9: Let us consider g=so(2n+2,2n), with nilpotent orbits surjectivity condition, we will require the following proposition. indexed as in Example 6. Noting Example 5, a partition λ of 4n+2 Proposition 2: Suppose that g decomposes as a Lie algebra into g so represents a nilpotent orbit in = 4n+2() if and only if each even part

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k that ϕg is surjective for g=su(n,n), g=su(n+1,n), and g=so(2n+2,2n), gg=,⊕ j j=1 while Example 10 demonstrates that surjectivity does not hold for g=so(2n+1,2n−1). Also, a brief examination of the computations in Where g1,...,gk are simple real Lie algebras. Let G1,…,Gk denote the respective adjoint groups. [3] reveals that ϕg is surjective for g=EII. We then have the following characterization of the surjectivity condition. ϕ   g (i) The map g: (g)/G→ ( )/G is surjective if and only each

ϕg : (gg )/GG→ (( ) )/( ) Theorem 2: If g is a semisimple real Lie algebra, then ϕ is surjective orbit complexification map j jj j j is g surjective. if and only if g is quasi-split and has no simple summand of the form so(2n+1,2n−1). (ii) The Lie algebra g is quasi-split if and only if each summand gj is quasi-split. Proof: If ϕg is surjective, then Lemma 2 implies that g is quasi- split. Also, Proposition 2 implies that each simple summand of g has Proof: For each j∈{1,…,k}, let π :g→g be the projection map. Note j j a surjective orbit complexification map, and the above discussion then that ξ ∈g is nilpotent if and only if π (ξ) is nilpotent in g for each j. It j j establishes that g has no simple summand of the form so(2n+1,2n−1). follows that k Conversely, assume that g is quasi-split and has no simple summand of π → : ()gg∏ (j ) the form so(2n+1,2n−1). By Proposition 2 (ii), each simple summand j=1 of g is quasi-split. Furthermore, the above discussion implies that ξ πξk  (jj ( )) =1 the only quasi-split simple real Lie algebras with non-surjective orbit k complexification maps are those of the form so(2n+1,2n−1). Hence, defines an isomorphism of real varieties. Note that GG= ∏ j, with the j=1 g former group acting on (g) and the latter group acting on the product each simple summand of has a surjective orbit complexification map, and Proposition 2 (i) implies that ϕ is surjective. of nilpotent cones. g

One then sees that π is G-equivariant, so that it descends to a 3.3 The Image of φg bijection k Having investigated the surjectivity of ϕg, let us consider the more π gg→ subtle matter of characterizing its image. Accordingly, let σ :g →g : ( )/GG∏ (jj )/ . g   j=1 g ⊆ g denote complex conjugation with respect to the real form . The Analogous considerations give a second bijection following lemma will be useful. k

π  : (gg )/GG→ ((jj ) )/( ) . ∏   Lemma 3: If Q ⊆ g is a complex nilpotent orbit, then so is σg(Q). j=1  Furthermore, we have the commutative diagram Proof: Note that σg integrates to a real Lie group automorphism π k τ:(G ) →(G ) , (gg )/GG→ ( )/  SC  SC ∏ j=1 jj where (G ) is the connected, simply-connected Lie group with Lie ↓↓ϕϕk  SC g ∏ j=1 (3) algebra g . If g∈ (G ) and ξ∈g , then g j   SC  π  k σg(Ad(g)(ξ )=Ad(τ(g))(σg(ξ)). (gg )/GG→ (( ) )/( ) ∏ j=1 jj  Hence, σg sends the (G)SC -orbit of ξ to the (G)SC -orbit of σg(ξ). To k ϕ complete the proof, we need only observe that (G ) -orbits coincide Hence, ϕ is surjective if and only if ∏ j=1 g is so, proving (i).  SC g j g with G-orbits in , and that σg(ξ) is nilpotent whenever ξ is nilpotent. By Lemma 2, proving (ii) will be equivalent to proving that Q (g ) reg  We may now use σ to explicitly describe the image of ϕ when g is ϕ Q g ϕg g g is in the image of g if and only if reg(( j)) is in the image of j for all j. Using the diagram (3), this will follow from our proving that the quasi-split. Q g π image of reg( ) under  is the k-tuple of the regular nilpotent orbits Theorem 3: If Q is a complex nilpotent orbit, the condition in the (g ) , namely that j  σg(Q)=Q is necessary for Q to be in the image of ϕg. If g is quasi-split, π QQggk (4) then this condition is also sufficient.  (reg ( )) = (reg ((jj ) ))=1 . k Proof: Assume that Q belongs to the image of ϕ , so that there exists Q ((g ) ) k g To see this, note that ∏ j=1 reg j  is the GG=∏ ()j -orbit j=1 ξ∈Q ∩g. Note that σg(Q) is then the complex nilpotent orbit containing k  ((g ) ) Q Q of maximal dimension in ∏ j=1 j  . This orbit is therefore the σg(ξ)=ξ, meaning that σg( )= . Conversely, assume that g is quasi-split Q Q g and that σg(Q)=Q. The latter means precisely that is defined over  image of reg( ) under the G-equivariant variety isomorphism k with respect to the real structure on g induced by the inclusion g ⊆ g . (gg )≅ (( ) ) , implying that (4) holds.   ∏ j=1 j Theorem 4.2 of [9] then implies that Q∩g≠∅. In light of Proposition 2, we address ourselves to classifying Using Theorem 3, we will give an interesting sufficient condition for g the simple real Lie algebras with surjective orbit complexification a complex nilpotent orbit to be in the image of ϕg when g is quasi-split. maps ϕg. Noting Lemma 2, we may assume g to be quasi-split. Since In order to proceed, however, we will need a better understanding of g being split is a sufficient condition for surjectivity, we are further the way in which σg permutes complex nilpotent orbits. To this end, we reduced to finding those quasi-split simple g which are non-split but have the following lemma. have surjective ϕ . It follows that g belongs to one of the four families g Lemma 4: Suppose that g comes with the faithful representation su(n,n), su(n+1,n), so(2n+2,2n), and so(2n+1,2n−1), or that g=EII, g ⊆ gl(V), where V is over . If Q is a complex nilpotent orbit, then the non-split, quasi-split real form of E6 [15]. Our examples establish λ(σg(Q))=λ(Q).

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sl g ∈Q Proof: Choose an 2()-triple (ξ,h,η) in with ξ . Since σg between the nilpotent orbits in g and the K-orbits in the (K-invariant) p ∩ g g preserves Lie brackets, it follows that (σg(ξ),σg(h),σg(η)) is also an subvariety  ( ) of . sl ( )-triple. The exercise is then to show that our two sl ( )-triples 2  2  Theorem 4: (The Kostant-Sekiguchi Correspondence) There is a give isomorphic representations of sl ( ) on VV = . For this, it will 2   bijective correspondence suffice to prove that h and σ (h) act on V with the same eigenvalues, g  (g)/G→(p∩(g ))/K and that their respective eigenspaces for a given eigenvalue are equi-     Ú dimensional. To this end, let σV:V→V be complex conjugation with respect to V ⊆ V . Note that  with the following properties. σ (h).(σ (x))=σ (h.x) g V V p ∩ g (i) It is an isomorphism of posets, where (  ( ))/K is endowed ∈ g for all x V, where . is used to denote the action of  on V. Hence, if x with the closure order (??). is an eigenvector of h with eigenvalue λ∈ , then σ (x) is an eigenvector  V (ii) If  is a real nilpotent orbit, then  and Ú are K-equivariantly of σ (h) with eigenvalue λ. We conclude that h and σ (h) have the same g g diffeomorphic. eigenvalues. Furthermore, their respective eigenspaces for a fixed eigenvalue are related by σV, and so are equi-dimensional. The first property was established by Barbasch and Sepanski in [16], while the second was proved by Vergne in [17]. Each paper makes We now have the following extensive use of Kronheimer’s desciption of nilpotent orbits from [18]. Proposition 3: Let g be a quasi-split semisimple real Lie algebra We now prove two preliminary results, the first of which is a direct endowed with a faithful representation g ⊆ gl(V), where V is over . If  consequence of the Kostant-Sekiguchi Correspondence. Q is the unique complex nilpotent orbit with partition λ(Q), then Q is in the image of ϕg. Lemma 5: If  is a real nilpotent orbit, then  is the unique G-orbit of maximal dimension in  . 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 Proof: Suppose that ′≠ is another G -orbit lying in  . By Q ϕ Ú ∨ then implies that is in the image of g. Property (i) in Theorem 4, it follows that (′) is an orbit in () Ú Ú A few remarks are in order. different from . However,  is an orbit of the complex algebraic group K under an algebraic action, and therefore is the unique orbit ϕ Ú Ú Remark 1: One can use Proposition 3 to investigate whether g of maximal dimension in its closure. Hence, dim ((′) )

Theorem 5: If 1 and 2 are real nilpotent orbits with the property 3.4 Fibres       that ( 1)=( 2) , then either 1= 2 or 1 and 2 are incomparable in In this section, we investigate the fibres of the orbit complexification the closure order. In other words, each fibre ofϕ g consists of pairwise ϕ  g  g incomparable nilpotent orbits. map g: ( )/G→ ( )/G. In order to proceed, it will be necessary to recall some aspects of the Kostant-Sekiguchi Correspondence. To this Proof: Assume that  and  are comparable. Without the loss end, fix a Cartan involution θ :g→g. Letting k and p denote the 1 and 1 2 of generality, ⊆ . We will prove that  = , which by Lemma (−1)-eigenspaces of θ, respectively, we obtain the internal direct sum 12 1 2 5 will amount to showing that the dimensions of 1 and 2 agree. decomposition ∈ ∈   To this end, choose points ξ1 1 and ξ2 2. Since ( 1)=( 2), g k ⊕p ((GG )ξξ ) = (( ) ) = . we have dim12 dim . Using Lemma 6, this becomes

dim(GGξξ )= dim ( ). Hence, the (real) dimensions of  and  This gives a second decomposition 12 1 2 coincide. g =k ⊕p ,    Proof: If is surjective, and then Lemma 2 implies that is quasi-split. k p k p where  and  are the complexifications of and , respectively. Let Also, Proposition 2 implies that each simple summand of has a surjective ⊆ ⊆ K G and K G be the connected closed subgroups with respective orbit complexification map, and the above discussion then establishes k k Lie algebras and . The Kostant-Sekiguchi Correspondence is one that has no simple summand of the form. Conversely, assume that is

Page 6 of 6 quasi-split and has no simple summand of the form. By Proposition Birkhuser Boston, Inc., Boston, MA 495: 0-8176-3792-3. 2 (ii), each simple summand of is quasi-split. Furthermore, the above 8. Jiro S (1987) Remarks on real nilpotent orbits of a symmetric pair. J Math Soc discussion implies that the only quasi-split simple real Lie algebras Japan 39 (1): 127-138. with non-surjective orbit complexification maps are those of the form. 9. Robert KE (1982) Rationalconjugacy classes in reductive groups. Duke Math Hence, each simple summand of has a surjective orbit complexification J 49(4): 785-806. map, and Proposition 2 (i) implies that is surjective. 10. Gerstenhaber M (1961) Dominance over the classical groups. Ann of Math Acknowledgements 74(2): 532-569. The author is grateful to John Scherk for discussions that prompted much of 11. Richard SP (1999) Enumerative Combinatorics. Vol. 2. With a foreword by this work. The author also acknowledges Lisa Jeffrey and Steven Rayan for their Gian-Carlo Rota and appendix 1 by Sergey Fomin. Cambridge Studies in considerable support. This work was partially funded by NSERC CGS and OGS Advanced Mathematics, 62. Cambridge University Press, Cambridge (1999). awards. 581: 0-521-56069-1; 0-521-78987-7. References 12. David CH, William GM (1993) Nilpotent Orbits in Semisimple Lie Algebras. Van Nos-trand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New 1. Wilfried S, Kari V (1999) On the geometry of nilpotent orbits. Sir Michael Atiyah: York 186: 0-534-18834-6. a great mathematician of the twentieth century. Asian J Math 3 (1): 233-274. 13. Dokovic, Dragomir Z (1998) Explicit Cayley triples in real forms of G2, F4, and 2. Dragomir DZ (2003) The closure diagram for nilpotent orbits of the split real E6. Pacific J Math 184(2): 231-255. form of E8. Cent. Eur J Math 1(4): 573-643.

3. Dragomir DZ (2001) The closure diagrams for nilpotent orbits of real forms of 14. Rothschild LP (1972) Orbits in a real reductive Lie algebra. Trans. Amer. Math. E6. J. Lie Theory 11(2): 381-413. Soc 168: 403-421.

4. Brieskorn E (1971) Singular elements of semi-simple algebraic groups. Actes 15. Anthony KW (2002) Lie groups Beyond an Introduction. Second edition. du CongrÃ¨s International des MathÃ©maticiens (Nice, 1970), Gauthier-Villars, Progress in Mathematics, 140. Birkhuser Boston, Inc., Boston, MA 812: Paris 2: 279-284. 0-8176-4259-5.

5. Kraft H, Claudio P (1982) On the geometry of conjugacy classes in classical 16. Barbasch Dan, Mark SR (1998) Closure ordering and the Kostant-Sekiguchi groups. Comment Math Helv 57(4): 539-602. correspondence. Proc. Amer Math Soc 126(1): 311-317.

6. Peter S (1980) Simple singularities and simple algebraic groups. Lecture Notes 17. Michele V (1995) Instantonset correspondence de Kostant-Sekiguchi. C. R. in Mathematics, 815. Springer, Berlin 175: 3-540-10026-1. Acad. Sci. Paris Sr. I Math 320 (8): 901-906.

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This article was originally published in a special issue, Recent Advances of Lie Theory in differential Geometry, in memory of John Nash handled by Editor. Dr. Princy Randriambololondrantomalala, Unversity of Antananarivo, Madagascar

Recent Advances of Lie Theory in J Generalized Lie Theory Appl differential Geometry, in memory of ISSN: 1736-4337 GLTA, an open access journal John Nash