Reality of Unipotent elements in Simple Lie Groups
aa r X i v : . [ m a t h . G R ] J a n REALITY OF UNIPOTENT ELEMENTS IN SIMPLE LIE GROUPS
KRISHNENDU GONGOPADHYAY AND CHANDAN MAITY
Abstract.
The aim of this paper is to give a classification of real and strongly realunipotent elements in a classical simple Lie group. To do this we will introduce aninfinitesimal version of the notion of the classical reality in a Lie group. This notion hasbeen applied to classify real and strongly real unipotent elements in a classical simpleLie group. Also, the infinitesimally-real semisimple elements in complex simple classicalLie algebras are classified. Introduction
Reality and Strong Reality in Groups.
Recall that, an element g in a group G is called real or reversible if it is conjugate to g − in G , that is there is a h in G such that g − = hgh − . The element g is called strongly real or strongly reversible if g is a productof two involutions in G , equivalently if there is an involution h ∈ G so that g − = hgh − ,then g is strongly real. Thus a strongly real element in G is real, but a real element isnot necessarily strongly real.It has been a problem of wide interest to investigate the real and strongly real elementsin groups, see [FS] for an exposition of this theme. However, in the literature, most ofthe investigations address the converse question, that is when a real element is stronglyreal, e.g. in [ST] Singh and Thakur proved the equivalence of reality and strong realityin a connected adjoint semisimple algebraic group over a perfect field provided − Mathematics Subject Classification.
Primary 20E45; Secondary: 22E60, 17B08.
Key words and phrases.
Real element, Strongly real element, unipotent element.Gongopadhyay acknowledges partial support from SERB-DST MATRICS project: MTR/2017/000355.Maity was supported by an NBHM PDF during the course of this work.
Lie groups where concrete classifications are known are the general linear groups, compactgroups and real rank one classical groups, e.g. [FS, GP, BG].The aim of this paper is to give a complete classification of the real and strongly realunipotent elements in a classical simple Lie group. In the literature, the investigations ofthe reality problem have been done mostly from the viewpoint of linear algebra and thetheory of algebraic groups. In some geometric cases, e.g. real rank one, local geometry ofthe transformation have also seen a role. Even though there is a complete classificationfor simple Lie groups, we have not seen any work where general structure theory of Liegroups has been used to tackle the problem. This work is an attempt to fill this gapin the literature and initiate an investigation of reality using the adjoint representations.For this, we initiate an infinitesimal notion of reality, viz. Ad G -real given below. Thisinfinitesimal reality not only helps us to classify the unipotent classes, but it may be aproblem of independent interest to investigate these classes in their own rights.1.2. Adjoint Orbits and Reality.
Let G be a Lie group with Lie algebra g . Considerthe natural Ad( G )-representation of G on its Lie algebra g Ad : G −→ GL( g ) . For X ∈ g , the adjoint orbit of X in g is defined as O X := { Ad( g ) X | g ∈ G } . If X ∈ g isnilpotent (resp. semisimple), then O X is called nilpotent orbit (resp. semisimple orbit) .Many deep results about such orbits are available in the literature, e.g. [CM], [Mc]. Werelate this theory with the above problem of reality in Lie groups. To connect the twothemes, we define the following. Definition 1.1.
An element X ∈ g is called Ad G -real if − X ∈ O X . An Ad G -real elementis called strongly Ad G -real if − X = Ad( τ ) X for some τ ∈ G so that τ = Id.We will show that the classical reality of the unipotent elements in a Lie group andAd G -reality of the nilpotent elements are equivalent but this is not true in general. Tosee this let I , = diag(1 , −
1) = exp Y ∈ GL ( C ), where Y = diag( √− π, √− π ). ThenI , is strongly real but Y is not Ad GL ( C ) -real in gl ( C ).In this paper by a real element in the Lie algebra g we mean the Ad G -real elementwhenever the underline Lie group G is understood from the context. Similarly, by thereal element in the Lie group G we mean the classical reality. We shall classify the Ad G -real nilpotent classes in a simple Lie algebra and that leads us to the classification of realunipotent classes in the corresponding Lie group using the exponential map, see Section2.1.3. Summary of our results.
First, we shall consider the complex simple Lie groups,i.e. the corresponding Lie algebra g is a simple Lie algebra over C . We prove that everyunipotent element in a complex semi-simple Lie group G is real, see Proposition 5.2.However, not every unipotent is strongly real in general, and we classify precisely whichunipotent elements are strongly real. Next we will consider the real simple Lie groups, andclassify real and strongly real unipotent elements in those groups. As mentioned above, wehave not seen any place in the literature where the problem of reality has been addressedusing Lie theoretic tools. Rather it has been using tools from geometric algebra. So, tofollow that tradition, we divide the real and complex Lie groups as per their geometricalgebra classification and summarize our results in the context of each class. EALITY OF UNIPOTENT ELEMENTS 3
In the following let D = R , C or H unless stated otherwise. Let GL n ( D ) be the generallinear group over D . Other than GL n ( D ), we have worked with the classical Lie groupsand Lie algebras as listed in Section 3.2. In the following we give summary of our resultsconcerning these groups.1.3.1. The general linear group.
The real and strongly real elements in GL n ( D ) are well-understood in the literature, e.g. [Wo], [FS]. However, we do not know any literaturefor GL n ( H ). We have given a uniform approach to prove that every unipotent element inGL n ( D ) is strongly real, see Proposition 5.4.1.3.2. The special linear group.
The classification of real and strongly real elements inSL n ( D ) is more tricky than the general linear case. It is easy to see that the unipotentelements in SL ( R ) are not even real. General investigation of involution length in SL n ( F ),for a field F , has been done by many authors, e.g. [KN2], [KN3]. However, in view of theclassification problem of real elements the best result that we know states that for n n ( F ) is real if and only if it is strongly real, where F is a field ofcharacteristic not 2, see [ST, Theorem 3.1.1]. In this work, we ask for a precise count ofthe strongly real unipotent elements. We have given a complete characterization of realand strongly real unipotent elements in SL n ( D ). For D = R or C , not every unipotentelement is strongly real, and we classify precisely when a unipotent is strongly real, seeTheorem 5.6 and Theorem 6.1. For SL n ( H ), we show that every unipotent element isstrongly real, see Theorem 6.3.1.3.3. The symplectic group.
Let F = R or C . Let Sp( n, F ) denote the symplectic groupover F . It is known that not every element is strongly real in Sp( n, F ). It follows from theliterature that every element in Sp( n, F ) is a product of two skew-involutions, i.e. everyelement in PSp( n, F ) is strongly real, see [ST], [Wo]. In a recent work Ellers and Villa [EV]have proved decomposition of a symplectic map into a product of six involutions and haveclassified certain strongly real elements under some strong hypothesis, see [EV, Corollary9]. In [Cr], de la Cruz has given a characterization of strongly real elements in Sp( n, C ). Itis proved in [AD] that every element in Sp(2 , R ) is a product of four involutions. However,strongly real classes are not very well-understood in Sp( n, R ). In this work, we have givena very first classification of unipotent strongly real elements in Sp( n, R ), see Theorem6.16. Classification of semisimple strongly real elements in Sp( n, R ) remains open.1.3.4. The orthogonal and unitary groups.
Product of involutions in orthogonal groups(over general fields) have been characterized in the literature, e.g. [FZ], [KN1], [KN2].However, we do not know any concrete classification of real elements in O( p, q ) or SO( p, q )other than when p = 1 , q >
0, see [FS], [Go], [BM]. The same can be said about unitarygroups U( p, q ). In [DM], it has been proved that every element in U( p, q ) is a product offinitely many reflections and an estimate of the reflection length was given. However, aconcrete classification is known only for U( n,
1) and SU( n, n,
1) has been obtained very recently in [BG].Other than the rank one groups, not much is known about a precise classification of thereal elements in SU( p, q ) or Sp( p, q ).In this paper, we completely classify real and strongly real unipotent elements inSU( p, q ) and Sp( p, q ). We prove that every unipotent element in Sp( p, q ) is real, see
K. GONGOPADHYAY AND C. MAITY
Corollary 6.18. As we shall see that is not the case in the groups SU( p, q ). Though realityand strong reality are equivalent in this group, not every unipotent element is real. Weask for a precise list of real or strongly real unipotent elements in SU( p, q ), Sp( p, q ) andclassify such elements, see Theorem 6.4 and Theorem 6.19.A particular subtle case is the identity component of the group O( p, q ). It is easy tosee that every unipotent element in SO( p, q ) is strongly real, but as it is easy to see inthe case of SO(2 , , the case is more tricky in the identity component SO( p, q ) . Wecompletely classify the real or strongly real unipotent elements in the identity componentSO( p, q ) , see Theorem 6.14.1.3.5. The group SO ∗ (2 n ) . The group SO ∗ (2 n ) is a real form of the complex orthogonalgroup SO(2 n, C ), and is a subgroup of SL n ( H ). We have not seen any place in theliterature where reality properties in SO ∗ (2 n ) has been addressed. In this paper we give avery first account of the classification of real and strongly real unipotent elements in thisgroup, see Theorem 6.15. We hope to address the semisimple case in a later investigation.A natural problem following Definition 1.1 is classification of the Ad G -real elements in g . For completeness, we have completely classified Ad G -reality of the semi-simple andnilpotent elements in complex simple Lie algebras. The situation for real Lie algebrasseems more tricky. We hope to address this in subsequent work.1.4. Key tools.
The main strategy of our proof depends on the theory of the nilpotentorbits. More specifically, we will use certain bases which were constructed to describecentralizers of the nilpotent elements in classical simple Lie algebras as in [SS], [BCM].For base field R or C , a specific basis was obtained in [SS] to describe the reductive parts ofcentralizers of the unipotent elements. In [BCM], construction of such basis was extendedto H , and was obtained in a unified way to D using the Jacobson-Morozov theoremand the structure of the related sl ( R )-representations. These special bases are used toconstruct conjugating elements. Another key tool to classify the Ad G -real elements isthe parametrization of nilpotent orbits in simple Lie algebras involving the signed Youngdiagrams. Explicit descriptions of centralizers of the nilpotent elements are also used inthe classification of the strongly Ad G -real elements.1.5. Structure of the paper.
The paper is organized as follows. In Section 1.2 thenotion of Ad G -real in the Lie algebra is introduced. Relationship between Ad G -real andclassical reality is established in Section 2. In Section 3 we fix the notation and terminologyand recall some background. Section 4 deals with the Ad G -real and strongly Ad G -realsemi-simple elements in the complex simple Lie algebras of classical type. In Section 5Ad G -real and strongly Ad G -real nilpotent elements in complex simple classical Lie algebrasare classified. Finally, in Section 6 we have classified Ad G -real and strongly Ad G -realnilpotent elements in simple classical Lie algebras over R .2. Ad G -reality and Classical Reality In this section we will introduce an infinitesimal version of the notion of classical realityin a Lie group. An element g ∈ G is called exponential if g = exp X for some X ∈ g . ALie group is called exponential if every element is exponential. First, we will show that EALITY OF UNIPOTENT ELEMENTS 5 Ad G -real in the Lie algebra implies the classical reality of the corresponding element inthe Lie group. Lemma 2.1.
Let G be a Lie group with Lie algebra g . Let X ∈ g be an Ad G -real (resp.strongly Ad G -real) element, then exp X is real (resp. strongly real) in G . There is a G -equivariant bijection between the set of nilpotent elements in g and theset of the unipotent elements in G via the exponential map. Corollary 2.2.
Let G be a semi-simple Lie group, and u ∈ G be a unipotent element sothat u = exp X . Then u is real (resp. strongly real) if and only if X is Ad G -real (resp.strongly Ad G -real ) in g . Corollary 2.3.
The classical reality of a connected unipotent Lie group G is determinedby the Ad G -reality in g . For the converse of Lemma 2.1, the following terminology will be needed. A normalneighbourhood U is a subset of G such that there is a symmetric neighbourhood V of theorigin in g , and exp | V is a diffeomorphism between V and U . Proposition 2.4.
Let G be a Lie group with Lie algebra g . Suppose g = exp X , theelement g /m belongs to a normal neighbourhood U for some m ∈ N , and g /m is real(resp. strongly real) in G . Then X is Ad G -real (resp. Ad G -strongly real) in g . Proof. As g /m ∈ U , and real in G , it follows that g /m = exp( m X ) and exp( − m X ) = σ (exp m X ) σ − = exp( σ m Xσ − ) ∈ U for some σ ∈ G . Hence, − X = σXσ − . (cid:3) It is not true in general for a Lie group G that g = exp X ∈ G is real (resp. stronglyreal) if and only if X is Ad G -real (resp. strongly Ad G -real). We give examples to illustratethe situation that the exponentiality condition on the Lie group G is necessary but notsufficient in the Proposition 2.4. Example 2.5. (i)
The exponentiality condition in Proposition 2.4 is necessary forstrongly reality. The Lie group SL ( C ) is not exponential. Let g = − Id ∈ SL ( C )and X = diag( √− π, −√− π ). Then g = e X . Note that g is strongly real in SL ( C ).But X is not a strongly Ad G -real element in sl ( C ). (ii) Next we will consider the compact Lie group SO which is exponential. Theelement σ = − I = exp (cid:18) π − π (cid:19) ∈ SO is real. But (cid:18) π − π (cid:19) is not Ad G -real in the Liealgebra so . Thus, exponentiality is not sufficient in Proposition 2.4.3. Notation and background
In this section, we fix some notation and recall some known results which will be usedin this paper. We will follow the notation of [BCM].Once and for all fix a square root of − √−
1. The Lie groups will bedenoted by the capital letters, while the Lie algebra of a Lie group will be denoted bythe corresponding lower case German letter. Sometimes, for notational convenience, theLie algebra of a Lie group G is also denoted by Lie( G ). The connected component of aLie group G containing the identity element is denoted by G . For a subset S of g , thesubgroup of G that fixes S point wise under the adjoint action is called the centralizer K. GONGOPADHYAY AND C. MAITY of S in G ; the centralizer of S in G is denoted by Z G ( S ). Similarly, for a Lie subalgebra g and a subset S ⊂ g , by z g ( S ) we will denote the subalgebra of g consisting of all theelements that commute with every element of S .3.1. Associated Lie groups and Lie algebras.
Let D = R , C or H . Let V be aright vector space of finite dimension over D . Let End D ( V ) be the real algebra of right D -linear maps from V onto V . After choosing a basis for V , the elements of End D ( V ) canbe represented by matrices over D . For a D -linear map T ∈ End D ( V ) and an ordered D -basis B of V , the matrix of T with respect to B is denoted by [ T ] B . Let GL(V) denotethe group of invertible right D -linear maps from End D ( V ).When D = R or C , let tr : End D ( V ) −→ D and det : End D V −→ D , respectivelybe the usual trace and determinant maps. DefineSL( V ) := { g ∈ GL( V ) | det( g ) = 1 } and sl ( V ) := { T ∈ End D ( V ) | tr( T ) = 0 } . If D = H then recall that A = End D ( V ) is a central simple R -algebra. Let Nrd A : A −→ R be the reduced norm on A , and let Trd A : A −→ R be the reduced trace on A .DefineSL( V ) := { g ∈ GL( V ) | Nrd
End D V ( g ) = 1 } and sl ( V ) := { T ∈ End D ( V ) | Trd
End D ( V ) ( T ) = 0 } . Let D = R , C or H , as above. Let σ be either the identity map Id or an involution of D ,that is, σ is a real linear anti-automorphism of D with σ = Id. Let ǫ = ±
1. Following[Bo, § h· , ·i : V × V −→ D a ǫ - σ Hermitian form if(i) h u + v, w i = h u, w i + h v, w i ,(ii) h v, u i = ǫσ ( h u, v i ), and(iii) h vα, u i = σ ( α ) h v, u i for all v, u, w ∈ V and for all α ∈ D .Recall that A ǫ - σ Hermitian form h· , ·i is non-degenerate if for every non-zero u in V ,there is a non-zero v such that h v, u i 6 = 0. We defineU( V, h· , ·i ) := { g ∈ GL( V ) | h gv, gu i = h v, u i ∀ v, u ∈ V } (3.1)and u ( V, h· , ·i ) := { T ∈ End D ( V ) | h T v, u i + h v, T u i = 0 ∀ v, u ∈ V } . We next defineSU( V, h· , ·i ) := U( V, h· , ·i ) ∩ SL( V ) and su ( V, h· , ·i ) := u ( V, h· , ·i ) ∩ sl ( V ) . It is well-known that su ( V, h· , ·i ) is a simple Lie algebra (cf. [Kn, Chapter I, Section 8]).3.2. Classical simple Lie groups and Lie algebras.
For a suitable choice of V and h· , ·i , we have the following classical Lie groups and Lie algebras which will be used here.We define the usual conjugations σ c on C by σ c ( x + √− x ) = x − √− x , and on H by σ c ( x + i x + j x + k x ) = x − i x − j x − k x , x i ∈ R for i = 1 , · · · ,
4. For P = ( p ij ) ∈ M r × s ( D ), P t denotes the transpose of P . If D = C or H , then define P := ( σ c ( p ij )). Let I p,q := (cid:18) I p − I q (cid:19) , J n := (cid:18) − I n I n (cid:19) . (3.2) EALITY OF UNIPOTENT ELEMENTS 7
We will work with the following classical simple Lie groups and Lie algebras:SL n ( C ) := { g ∈ GL n ( C ) | det( g ) = 1 } , sl n ( C ) := { z ∈ M n ( C ) | tr( z ) = 0 } ;SO( n, C ) := { g ∈ SL n ( C ) | g t g = I n } , so ( n, C ) := { z ∈ sl n ( C ) | z t I n + I n z = 0 } ;Sp( n, C ) := { g ∈ SL n ( C ) | g t J n g = J n } , sp ( n, C ) := { z ∈ sl n ( C ) | z t J n + J n z = 0 } ;SL n ( R ) := { g ∈ GL n ( R ) | det( g ) = 1 } , sl n ( R ) := { z ∈ M n ( R ) | tr( z ) = 0 } ;SL n ( H ) := { g ∈ GL n ( H ) | Nrd M n ( H ) ( g ) = 1 } , sl n ( H ) := { z ∈ M n ( H ) | Trd M n ( H ) ( z ) = 0 } ;SU( p, q ) := { g ∈ SL p + q ( C ) | g t I p,q g = I p,q } , su ( p, q ) := { z ∈ sl p + q ( C ) | z t I p,q + I p,q z = 0 } ;SO( p, q ) := { g ∈ SL p + q ( R ) | g t I p,q g = I p,q } , so ( p, q ) := { z ∈ sl p + q ( R ) | z t I p,q + I p,q z = 0 } ;Sp( p, q ) := { g ∈ SL p + q ( H ) | g t I p,q g = I p,q } , sp ( p, q ) := { z ∈ sl p + q ( H ) | z t I p,q + I p,q z = 0 } ;Sp( n, R ) := { g ∈ SL n ( R ) | g t J n g = J n } , sp ( n, R ) := { z ∈ sl n ( R ) | z t J n + J n z = 0 } ;SO ∗ (2 n ) := { g ∈ SL n ( H ) | g t j I n g = j I n } , so ∗ (2 n ) := { z ∈ sl n ( H ) | z t j I n + j I n z = 0 } . Partitions and (signed) Young diagrams.
For two ordered sets ( v , . . . , v n ) and( w , . . . , w m ), the ordered set ( v , . . . , v n , w , . . . , w m ) will be denoted by( v , . . . , v n ) ∨ ( w , . . . , w m ) . A partition of a positive integer n is an object of the form d := [ d t d , . . . , d t ds s ], where t i , d i ∈ N , 1 ≤ i ≤ s , such that P si =1 t d i d i = n , t d i ≥ d > · · · > d s >
0; see [CM, § P ( n ) be the set of all partitions of n . For a partition d = [ d t d , . . . , d t ds s ]of n , define N d := { d i | ≤ i ≤ s } , E d := N d ∩ (2 N ) , O d := N d \ E d . (3.3)Further define O d := { d | d ∈ O d , d ≡ } and O d := { d | d ∈ O d , d ≡ } . (3.4)Following [CM], a partition d of n will be called even if N d = E d . Let P even ( n ) be thesubset of P ( n ) consisting of all even partitions of n . We call a partition d of n to be very even if d is even, and t η is even for all η ∈ N d . Let P v . even ( n ) be the subset of P ( n )consisting of all very even partitions of n . Now define P − ( n ) := { d ∈ P ( n ) | t θ is even for all θ ∈ O d } , (3.5) P ( n ) := { d ∈ P ( n ) | t η is even for all η ∈ E d } . (3.6)Next, we will define the Young diagram and signed Young diagram, see [CM, p. 140]. Forour convenience, we will follow the modified definition of the signed Young diagram usedin [BCM, § Definition 3.1. • A Young diagram is a left-justified array of rows of empty boxes arranged so thatno row is shorter than the one below it; the size of a Young diagram is the numberof empty boxes appearing in it. There is an obvious correspondence between theset of Young diagrams of size n and the set P ( n ) of partitions of n . Hence the setof Young diagrams of size n is also denoted by P ( n ). K. GONGOPADHYAY AND C. MAITY • A signed Young diagram is a Young diagram in which every box is labelled with+1 or − • Two signed Young diagrams are equivalent if and only if each can be obtainedfrom the other by permuting rows of equal length. • The signature of a signed Young diagram is the ordered pair of integers ( p, q )where p -many +1 and q -many − Remark 3.2.
Let p d (resp. q d ) be the number of +1 (resp. −
1) in the 1 st column ofthe rectangular block of size t d × d . Then a signed Young diagram of size n is uniquelydetermined by a partition d = [ d t d , . . . , d t ds s ] of n , and a pair of integer ( p d , q d ) for all d ∈ N d so that p d + q d = t d .3.4. Jacobson-Morozov Theorem and associated results.
For a Lie algebra g over R , a subset { X, H, Y } ⊂ g is said to be a sl -triple if X = 0, [ H, X ] = 2 X , [ H, Y ] = − Y and [ X, Y ] = H . Note that for a sl -triple { X, H, Y } in g , Span R { X, H, Y } is isomorphicto sl ( R ). We now recall a famous result due to Jacobson and Morozov. Theorem 3.3 (Jacobson-Morozov, cf. [CM, Theorem 9.2.1]) . Let X ∈ g be a non-zeronilpotent element in a semisimple Lie algebra g over R . Then there exist H, Y ∈ g suchthat { X, H, Y } is a sl -triple. Let V be a right D -vector space of dimension n , where D is, as before, R or C or H .Let { X, H, Y } ⊂ sl ( V ) be a sl ( R )-triple. Note that V is also a R -vector space usingthe inclusion R ֒ → D . Hence V is a module over Span R { X, H, Y } ≃ sl ( R ). For anypositive integer d , let M ( d −
1) denote the sum of all the R -subspaces W of V such that • dim R W = d , and • W is an irreducible Span R { X, H, Y } -submodules of V .Then M ( d −
1) is the isotypical component of V containing all the irreducible submodulesof V with highest weight d −
1. As
X, H, Y of V are D -linear, the R -subspaces M ( d − V are also D -subspace. Let L ( d −
1) := ker Y ∩ M ( d − , and t d := dim D L ( d − . (3.7)Consider the non-zero irreducible Span R { X, H, Y } -submodules of V . Let { d , . . . , d s } ,with d > · · · > d s , be the integers that occur as R -dimension of such Span R { X, H, Y } -modules. Then it follows that P si =1 t d i d i = dim D V = n . Thus, d := (cid:2) d t d , . . . , d t ds s (cid:3) ∈ P ( n ) . (3.8)Recall that for a partition d ∈ P ( n ), N d , E d and O d are defined in (3.3). Proposition 3.4. [BCM, Proposition A.2]
Let { X, H, Y } ⊂ sl ( V ) be a sl ( R ) -triple,where V is a right D -vector space of dimension n . For all d ∈ N d and for any D -basis of L ( d − , say, { v dj | ≤ j ≤ t d := dim D L ( d − } the following two hold:(1) X d v dj = 0 and H ( X l v dj ) = X l v dj (2 l + 1 − d ) for ≤ j ≤ t d , ≤ l ≤ d − , d ∈ N d . EALITY OF UNIPOTENT ELEMENTS 9 (2) For all d ∈ N d , the set { X l v dj | ≤ j ≤ t d , ≤ l ≤ d − } is a D -basis of M ( d − . In particular, { X l v dj | ≤ j ≤ t d , ≤ l ≤ d − , d ∈ N d } is a D -basis of V . Proposition 3.5. [BCM, Proposition A.6]
Let V be a right D -vector space of dimension n , ǫ = ± , σ : D −→ D is either the identity map or it is σ c when D is C or H . Let h· , ·i : V × V −→ D be a ǫ - σ Hermitian form. Let { X, H, Y } ⊂ su ( V, h· , ·i ) be a sl ( R ) -triple. Let d ∈ N d and t d := dim D L ( d − . Then for all d ∈ N d , there exists a D -basis { v dj | ≤ j ≤ t d } of L ( d − such that the following three hold:(1) X d v dj = 0 and H ( X l v dj ) = X l v dj (2 l + 1 − d ) for all ≤ j ≤ t d , ≤ l ≤ d − and d ∈ N d .(2) For all d ∈ N d , the set { X l v dj | ≤ j ≤ t d , ≤ l ≤ d − } is a D -basis of M ( d − . In particular, { X l v dj | ≤ j ≤ t d , ≤ l ≤ d − , d ∈ N d } is a D -basis of V .(3) The value of h· , ·i on any pair of the above basis vectors is , except in the followingcases: • If σ = σ c , then h X l v dj , X d − − l v dj i ∈ D ∗ . • If σ = Id and ǫ = 1 , then h X l v dj , X d − − l v dj i ∈ D ∗ when d ∈ O d , and h X l v dj , X d − − l v dj +1 i ∈ D ∗ when d ∈ E d and j is odd. • If σ = Id and ǫ = − , then h X l v dj , X d − − l v dj i ∈ D ∗ when d ∈ E d , and h X l v dj , X d − − l v dj +1 i ∈ D ∗ when d ∈ O d and j is odd. In the following remark, we will simplify the basis of V as in Proposition 3.5 case bycase which will be used later. We need to consider only the following cases: Remark 3.6 (cf. [BCM, Remark A.8]) . The D -basis elements { v dj | ≤ j ≤ t d } of L ( d −
1) in Proposition 3.5 are modified as follows:(1) If D = R and ǫ = 1, by suitable rescaling each element of { v dj | ≤ j ≤ t d } wemay assume that • h v dj , X d − v dj i = ± d ∈ O d , and • h v dj , X d − v dt d / j i = 1 when d ∈ E d and 1 ≤ j ≤ t d / D = R and ǫ = −
1, analogously we may assume that the elements of the R -basis { v dj | ≤ j ≤ t d } of L ( d −
1) in Proposition 3.5 satisfy the conditionthat • h v dj , X d − v dj i = ± d ∈ E d , and • h v dj , X d − v dt d / j i = 1 when d ∈ O d and 1 ≤ j ≤ t d / D = C , ǫ = 1 and σ = σ c , rescaling the elements of the C -basis { v dj | ≤ j ≤ t d } we may assume that • h v dj , X d − v dj i = ± d ∈ O d , and • h v dj , X d − v dj i = ±√− d ∈ E d .(3) If D = H , ǫ = 1 and σ = σ c , after rescaling and conjugating the elements of the H -basis { v dj | ≤ j ≤ t d } of L ( d −
1) by suitable scalars the elements of the H -basis, we may assume that • h v dj , X d − v dj i = ± d ∈ O d , and • h v dj , X d − v dj i = j when d ∈ E d . If D = H , ǫ = − σ = σ c , analogously we may assume that the elements ofthe H -basis { v dj | ≤ j ≤ t d } of L ( d −
1) satisfy • h v dj , X d − v dj i = ± d ∈ E d , and • h v dj , X d − v dj i = j when d ∈ O d .The following basic result will be used to prove Theorem 5.9. Lemma 3.7 (cf. [BCM, Lemma 2.6]) . Let { X, H, Y } be a sl -triple in the Lie algebra g of a Lie group G . Then Z G ( X, H ) = Z G ( X, H, Y ) . Next, we will state a well-known result of Springer-Steinberg regarding the reductivepart of the centralizer of a nilpotent element in the Lie group Sp( n, C ). Theorem 3.8 (Springer-Steinberg, cf. [CM, Theorem 6.1.3]) . Let X be a nilpotent ele-ment in sp ( n, C ) which corresponds to the partition d = [ d t d , . . . , d t ds s ] ∈ P − (2 n ) . Let { X, H, Y } be a sl -triple in sp ( n, C ) containing X . Then Z Sp( n, C ) ( X, H, Y ) ∼ = Y η ∈ E d O( t η ) × Y θ ∈ O d Sp( t θ , C ) . The next result follows from [BCM, Lemma 4.4(4)].
Proposition 3.9.
Let X ∈ sp ( p, q ) be a non-zero nilpotent element, and { X, H, Y } be a sl -triple in sp ( p, q ) . Let the nilpotent orbit O X corresponds to the signed Young diagramof signature ( p, q ) where p θ (resp. q θ ) denotes the number of +1 (resp. − ) in the st column of the block of size t θ × θ for θ ∈ O d . Then Z Sp( p,q ) ( X, H, Y ) ≃ Y η ∈ E d SO ∗ (2 t η ) × Y θ ∈ O d Sp( p θ , q θ ) . Next we include another basic result which will be needed to prove Theorem 6.1.
Lemma 3.10 ([Ma, Lemma 4.1.1 ]) . Let X ∈ sl n ( R ) be a non-zero nilpotent elementso that the corresponding partition d ∈ P even ( n ) . Suppose that gX = Xg for some g ∈ GL n ( R ) . Then det g > . An ordered basis of V . Here we will fix a suitable ordering of the basis as describedin Propositions 3.4, 3.5 of the underlying vector space V . This ordering is motivated bya similar ordering used in [BCM, (4.4)]. After starting with their initial ordering, we willslightly modify it to serve our purpose.Let ( v d , . . . , v dt d ) be the ordered D -basis of L ( d −
1) as in Proposition 3.5 for d ∈ N d . Then, as done in [BCM, (4.3)], it follows from Proposition 3.5 that B l ( d ) :=( X l v d , . . . , X l v dt d ) is an ordered D -basis of X l L ( d −
1) for 0 ≤ l ≤ d − d ∈ N d .Define B ( j ) := B d − j ( d ) ∨ · · · ∨ B d s − j ( d s ) , and B := B (1) ∨ · · · ∨ B ( d ) . (3.9)It follows from the definition of B (1) that X | B (1) = 0. Suppose τ ∈ GL( V ) so that τ Xτ − = ǫX , where ǫ = ±
1. Then τ keeps the vector spaces Span C {B d − ( d ) ∨ · · · ∨B d r − ( d r ) } for r = 1 , . . . , s and Span C {B (1) ∨ · · · ∨ B ( t ) } invariant for t = 1 , . . . , d .Also, τ will be determined uniquely by it’s action on L ( d −
1) for d ∈ N d . The matrix[ τ ] B is a block upper triangular matrix with ( d + · · · + d s )-many diagonal blocks. Write EALITY OF UNIPOTENT ELEMENTS 11 [ τ ] B = ( A ij ), where A ij are block matrices. Then A ij = 0 for i > j . The order of the first s -many diagonal blocks A jj of [ τ ] B , is t d j × t d j , where t d j = dim L ( d j −
1) for j = 1 , . . . , s .Any diagonal block of [ τ ] B is of the form ǫA jj for j = 1 , . . . , s . Moreover, if τ = Id, thennecessarily A jj = Id for j = 1 , . . . , s . We will illustrate the above in the Appendix 6.6with a particular example.3.6. Semi-simple orbits in complex simple Lie algebra.
Let g be a complex semisim-ple Lie algebra, h ⊂ g be a Cartan subalgebra, and W g be the Weyl group of g . Then W g = N g ( h ) / z g ( h ). Thus, the Weyl group W g has a natural action on h and h /W g denotesthe set of orbits. We next state an important result regarding the parametrization of thesemi-simple orbits in semi-simple Lie algebras over C . Theorem 3.11 (cf. [CM, Theorem 2.2.4]) . Let g be a complex semi-simple Lie algebra, h be a Cartan subalgebra, and W g be the Weyl group of g . Then the semi-simple orbits areparametrized by h /W g . Reality for semisimple elements in simple Lie algebra over C We will recall some well-known basic facts regarding semisimple elements and Cartansubalgebra in a Lie algebra. Let H ∈ g be a semisimple element in a semisimple Liealgebra over C and h be a Cartan subalgebra in g . Then gHg − ∈ h for some g ∈ G .4.1. Semi-simple elements in sl n ( C ) . Let g = sl n ( C ) and h , all diagonal matrices in g , be a Cartan subalgebra in g . The Weyl group W sl n ( C ) of sl n ( C ) is isomorphic to thepermutation group S n . Let φ ∈ W sl n ( C ) , and H = diag( h , . . . , h n ) ∈ h . Considering φ asa permutation of n , we have φ · H = diag( h φ (1) , . . . , h φ ( n ) ). Lemma 4.1.
Let H = diag( h , . . . , h n ) ∈ h . Then gHg − = − H for some g ∈ SL n ( C ) ifand only if { h , . . . , h n } = {− h , . . . , − h n } . Theorem 4.2.
Let H be a semisimple element in gl n ( C ) . Then H is a real element in gl n ( C ) if and only if λ is an eigen value of H , then − λ is also an eigen value of H . Proof.
Use Theorem 3.11 and Lemma 4.1. (cid:3)
Corollary 4.3.
A semisimple element H ∈ gl n ( C ) is real if and only if H is real in sl n ( C ) . Remark 4.4.
The analogous statements of the Corollary 4.3 is also true in the classicalsense (in the group level), see [FS, p. 77].
Theorem 4.5.
Every real semisimple element in
Lie( PSL n ( C )) is strongly Ad PSL n ( C ) -real. Proof.
Let H ∈ Lie( PSL n ( C )) be a real semisimple element. Without loss of general-ity, we can assume H = diag( h , . . . , h m , − h , . . . , − h m , , . . . , σ = diag(J m , I s )will conjugate H and − H , where 2 m + s = n , and J n as in (3.2). (cid:3) Example 4.6.
Consider the semi-simple element H = diag( x , − x ). Let g ∈ GL ( C ) sothat gH = − Hg . Then g is of the form (cid:18) bc (cid:19) . Hence H is a real element in sl ( C ) butnot strongly real. Note that g ∈ Sp(1 , C ) if bc = −
1. Similarly, H is a real element in sp (1 , C ) but not strongly real. Next we will consider Ad SL n ( C ) -real in the Lie algebra sl n ( C ). Proposition 4.7.
A real semisimple element in sl n ( C ) is strongly real if and only if either is an eigen value or n . Proof.
Let H ∈ sl n ( C )) be a real semisimple element. If 0 is an eigen value of H ,then H is strongly real. Suppose 0 is not an eigen value of H , and n n = 4 m for m ∈ N and we can assume H = diag( h , . . . , h m , − h , . . . , − h m ). As Weylgroup permutes the diagonal entries, H and − H will be conjugated by g = (cid:18) I m I m (cid:19) .Next assume that H is strongly real and 0 is not an eigen value of H . We will show n ∈ N . Without loss of generality, we can assume H = diag( h , . . . , h m , − h , . . . , − h m )and gH = − Hg for some involution. Let e j be the standard column vector in C n with 1in j th place and 0 elsewhere. For 1 ≤ j ≤ m , Hge j = − gHe j = − gh j e j = − h j ge j . Hence ge j ∈ C e m + j and similarly ge m + j ∈ C e j . Let V j := C e j ⊕ C e m + j and C j := { e j , e m + j } . Set C := C ∨ · · · ∨ C m . Then the matrix [ g ] C is a 2 × g ] C = ( − m . As det g = 1, it follows that m ∈ N . This completes the proof. (cid:3) Semi-simple elements in o ( n, C ) and so ( n, C ) . Upto conjugacy, any semisimpleelement in o ( n, C ) or so ( n, C ) belongs to the following Cartan subalgebra h , see [Kn, p.127]: h := ( diag( H , . . . , H m ,
0) if n = 2 m + 1diag( H , . . . , H m ) if n = 2 m , where H j = (cid:18) x j − x j (cid:19) , x j ∈ C . (4.1) Example 4.8.
Consider H = (cid:18) x − x (cid:19) ∈ so (2 , C ) = o (2 , C ), where x ∈ C . Let g ∈ GL(2 , C ) with gHg − = − H . Then g is of the form (cid:18) a bb − a (cid:19) . Hence, det g = − a − b ,and gg t = g = ( − det g )I . Thus, one can choose g ∈ O(2 , C ) with det g = −
1. Thisshows that H is a real element (in fact strongly real) in o (2 , C ) but not a real element in so (2 , C ). Proposition 4.9.
Every semisimple element in o ( n, C ) is strongly real. Proof.
Enough to consider the elements of h . For the elements in h one can easilyconstruct required involution using the Example 4.8. (cid:3) Theorem 4.10.
Let H ∈ so ( n, C ) be a semisimple element. Then TFAE :(1) H is real.(2) H is strongly real.(3) Either is an eigen value of H or n . Proof.
We omit the proof as it is similar to that of Proposition 4.7. (cid:3)
EALITY OF UNIPOTENT ELEMENTS 13
Semi-simple elements in sp ( n, C ) . We fix a Cartan subalgebra h in sp ( n, C ) as h := { diag( h , . . . , h m , − h , . . . , − h m ) | h j ∈ C } . Recall that in Example 4.6, the elementdiag( x, − x ) is not strongly real but we can choose a conjugating element g from PSp( n, C ).Thus, we will first consider the semisimple element in the Lie algebra of PSp( n, C ). Theanalogous result for the compact group PSp( n ) has been studied in [BG, Proposition 3.1]. Theorem 4.11.
Every semisimple element in
Lie (PSp( n, C )) is strongly Ad PSp( n, C ) -real. Proof. As h is a Cartan subalgebra in sp ( n, C ), one can define required involutionusing Example 4.6. (cid:3) The following corollary is immediate.
Corollary 4.12.
Every semisimple element in sp ( n, C ) is Ad Sp( n, C ) -real. Theorem 4.13.
A semisimple element in sp ( n, C ) is strongly Ad Sp( n, C ) -real if and onlyif the multiplicity of each non-zero eigen value is even. Proof.
Let H be a strongly real semisimple element in sp ( n, C ). We can assume H = diag( h , . . . , h n , − h , . . . , − h n ), and gH = − Hg for some involution g ∈ Sp( n, C ).Suppose h j = 0 and multiplicity of h j is m . Let C j := { e j , . . . , e j m } be an ordered basis ofthe eigen space of H corresponding to the eigen value h j . Then C n + j := { e n + j , . . . , e n + j m } is an ordered basis of the eigen space corresponding to the eigen value − h j . Let V j be the C -span of C j ∨ C n + j . Then the involution g keeps V j invariant and ( g | V j ) t J m ( g | V j ) = J m ,where J m is as in (3.2). As H ( ge j l ) = − h j ge j l for 1 ≤ l ≤ m , we can write[ g ] C j ∨C n + j := (cid:18) BC (cid:19) , for some B, C ∈ GL m ( C ) . (4.2)Since g | V j is an involution and ( g | V j ) t J m ( g | V j ) = J m , it follows that C − = B = − B t .Hence, m (the multiplicity of h j ) has to be even.Next assume that multiplicity of each non-zero eigen value is even. Can assumeany semisimple element H = diag( h , . . . , h n , − h , . . . , − h n ), where h j − = h j for j = 1 , . . . , n/
2. In this case to define a required involution g , set B := J ...J in(4.2). This completes the proof. (cid:3) Reality for nilpotent elements in simple Lie algebra over C Using the Jacobson-Morozov Theorem 3.3 and Mal’cev Theorem [CM, Theorem 3.4.12],it follows that nilpotent elements are real in simple Lie algebra over C . We will provide anelementary proof without using those theorems and also construct an explicit conjugatingelement for the nilpotent element. The following basic lemma is useful for this. Lemma 5.1.
Let X ∈ g be a non-zero nilpotent element in a semi-simple Lie algebra g over C . Then X ∈ [ g , X ] . Proof.
See [CM, (3 . . (cid:3) Proposition 5.2.
Every unipotent element in a complex semi-simple Lie group G is real. Proof.
It is enough to prove that every nilpotent element is real in g , see Corollary2.2. Let X ∈ g be a non-zero nilpotent element. Using Lemma 5.1, we have X = [ H, X ]for some H ∈ g . Let α ∈ C so that e α = −
1. Then [ αH, X ] = αX andAd (exp αH )( X ) = e ad( αH ) ( X ) = X n (cid:0) ad ( αH ) (cid:1) n n ! X = X n α n n ! X = − X .
Hence X and ( − X ) are conjugate via exp αH . This completes the proof. (cid:3) An analogous result may not true over R , see Example 5.3. Example 5.3.
Let us consider the nilpotent element X = (cid:18) (cid:19) . A simple computationshows that X is not real in sl (2 , R ), real in sl ( C ) but not strongly real in sl ( C ), andfinally, strongly real in gl ( C ).In view of the Proposition 5.2, we want to classify the strongly real nilpotent elementsin a simple Lie algebra over C which will be done in the rest of the section. Moreover,we will construct an explicit element in G corresponding to the nilpotent element X ∈ g which conjugates X and − X .5.1. Nilpotent elements in sl n ( C ) . First we will consider the Lie algebra gl n ( D ) where D = R , C , H . Proposition 5.4.
Every nilpotent element in gl n ( D ) is strongly real when D = R , C , H . Proof.
Let X ∈ gl n ( D ) be a non-zero nilpotent element. Then X ∈ sl n ( D ). UsingProposition 3.4, D n has a basis of the form { X l v dj | ≤ j ≤ t d , ≤ l ≤ d − , d ∈ N d } .Now define g ∈ GL n ( D ) as follows: g ( X l v dj ) = ( ( − l X l v dj d ∈ E d ∪ O d , ( − l +1 X l v dj d ∈ O d . (5.1)Then g = Id and Xg = − gX . This completes the proof. (cid:3) Now we will consider the nilpotent elements in sl n ( C ). Every nilpotent element isreal in sl n ( C ), see Proposition 5.2. In Example 5.3, we have observed that any nilpotentelement in sl ( C ) is not strongly real. In general every nilpotent element in Lie( PSL n ( C ))is strongly real, but not in sl n ( C ). The following terminology is needed for the next result.Define E d := { η ∈ E d | η ≡ } , where E d is as in (3.3), and f P e ( n ) := { d ∈ P even ( n ) \ P v.even ( n ) | X η ∈ E d t η is odd } . (5.2) Theorem 5.5.
Every nilpotent element is strongly real in
Lie( PSL n ( C )) . Proof.
Let X ∈ sl n ( C ) be a non-zero nilpotent element. Using Proposition 3.4, C n has a basis of the form { X l v dj | ≤ j ≤ t d , ≤ l ≤ d − , d ∈ N d } . We now dividethe proof in two parts:- Case 1: d f P e ( n ). Set g ∈ GL n ( C ) as in (5.1). Then det g = 1, g = Id, Xg = − gX . Case 2: d ∈ f P e ( n ). In this case define g ∈ GL n ( C ) as follows: g ( X l v η ) = ( − l √− X l v η η ∈ E d . EALITY OF UNIPOTENT ELEMENTS 15
Then det g = 1, g = − Id and Xg = − gX . This completes the proof. (cid:3) Theorem 5.6.
Let X ∈ sl n ( C ) be a nilpotent element, and d ∈ P ( n ) be the correspondingpartition. Then X is strongly real if and only if the partition d f P e ( n ) . Proof.
Let X ∈ sl n ( C ) be a non-zero nilpotent element, and { X, H, Y } be a sl -triplein sl n ( C ) containing X . Then C n has a basis of the form { X l v i | i, l } , see Proposition 3.4.First assume that d f P e ( n ). In this case, define an involution g as in (5.1). This showsthat X is strongly real.Next assume that X is strongly real, and d ∈ f P e ( n ). The converse part of this theoremwill follow if we arrive a contradiction. Let τ ∈ SL n ( C ) be an involution so that τ Xτ − = − X . We will write τ with respect to the ordered basis B constructed in (3.9) so that [ τ ] B is block upper triangular matrix and follow the notation of § τ = Id, A jj = Idfor 1 ≤ j ≤ s . It follows form the definition of the ordered basis B , that A jj and − A jj both occur d j / ≤ j ≤ s . This contradicts thefact that det τ = 1, as1 = det τ = ( − P η ∈ E d t η (det A jj ) d j = ( − P η ∈ E d t η = − . This completes the proof. (cid:3)
Remark 5.7.
The same proof will work for sl n ( R ). But we will provide more straight-forward proof in Theorem 6.1 for sl n ( R ).5.2. Nilpotent elements in so ( n, C ) . Throughout this subsection h· , ·i denotes the sym-metric form on C n defined by h x, y i = x t y for x, y ∈ C n . Recall the definition of P ( n ) asin (3.6). Theorem 5.8.
Every nilpotent element is strongly real in so ( n, C ) for n ≥ . Proof.
Let X ∈ so ( n, C ) be a non-zero nilpotent element. Let { X, H, Y } be a sl -triplein so ( n, C ). Let { X l v dj | ≤ j ≤ t d , ≤ l ≤ d − , d ∈ N d } is a C -basis of C n as inProposition 3.5. The partition d ∈ P ( n ) for X , see [CM, Theorems 5.1.2, 5.1.4]. Define g ∈ GL( V ) as follows: g ( X l v dj ) := ( − l X l v dj if d ∈ O d , ( − l +1 X l v dj if d ∈ O d , ( − l X l v dj + t d / if d ∈ E d , ≤ j ≤ t d / , ( − l X l v dj − t d / if d ∈ E d , t d / < j ≤ t d . It follows from the definition that gX = − Xg , g = 1, and det g = 1. Moreover, h g ( X l v dj ) , g ( X d − l − v dj ) i = h X l v dj , X d − l − v dj i for d ∈ O d , h g ( X l v dj ) , g ( X d − l − v dj ) i = h X l v dj + t d / , X d − l − v dj + t d / i for d ∈ E d , ≤ j ≤ t d / . In view of Proposition 3.5, h gx, gy i = h x, y i for all x, y ∈ C n . Hence, g ∈ SO( n, C ). (cid:3) Nilpotent elements in sp ( n, C ) . Throughout this subsection h· , ·i denotes the skewsymmetric form on C n defined by h x, y i = x t J n y for x, y ∈ C n , where J n is as in (3.2).Recall the definition of P − (2 n ) as in (3.5). In the case of sp ( n, C ), every nilpotent element is not strongly real. For example,any nilpotent element in sp (1 , C ) is not strongly real. Nilpotent orbits in sp ( n, C ) areparametrized by the partition of 2 n in which odd parts occurs with even multiplicity, see[CM, Theorem 5.1.3]. Let X ∈ sp ( n, C ) be a nilpotent element which corresponds to thepartition d := [ d t d , . . . , d t ds s ] ∈ P − (2 n ). Recall that in view of Proposition 5.2, everynilpotent element in sp ( n, C ) is real. Here we will construct an element g ∈ Sp( n, C )so that gX = − Xg for nilpotent X ∈ sp ( n, C ). Let { X l v dj | ≤ j ≤ t d , ≤ l ≤ d − , d ∈ N d } is a C -basis of C n as in Proposition 3.5. Define g ∈ GL ( C n ) as follows: g ( X l v dj ) := ( ( − l X l v dj if d ∈ O d , ( − l √− X l v dj if d ∈ E d . (5.3)Note that gX = − Xg . Using Proposition 3.5, we have h gx, gy i = h x, y i for all x, y ∈ C n .This shows that g ∈ Sp( n, C ). Theorem 5.9.
A nilpotent element X in sp ( n, C ) is strongly real if and only if t η is evenfor all η ∈ E d . Proof.
Let { X l v dj | ≤ j ≤ t d , ≤ l ≤ d − , d ∈ N d } is a C -basis of C n as inProposition 3.5. Suppose that t i is even for even d i . Define g ∈ GL ( C n ) as follows: g ( X l v dj ) := ( − l X l v dj if d ∈ O d , ( − l √− X l v dj + t d / if d ∈ E d , ≤ j ≤ t d / , ( − l +1 √− X l v dj − t d / if d ∈ E d , t d / < j ≤ t d . It follows that gX = − Xg , g = 1. In view of Proposition 3.5, h gx, gy i = h x, y i for all x, y ∈ V . This shows that g ∈ Sp( n, C ).Next assume that X is a strongly real nilpotent element in sp ( n, C ). i.e. νXν − = − X for some involution ν ∈ Sp( n, C ). For any g ∈ Sp( n, C ) so that gXg − = − X , ν ∈ gZ G ( X ) . (5.4)Let τ ∈ Z G ( X ). Recall that B is the ordered basis of V as in (3.9). Following § τ ] B = ( A ij ), and τ D be the block diagonal part of [ τ ] B , i.e., consists of only the matrices( A jj ) in the diagonal. Then τ D X = Xτ D and τ D H = Hτ D . Using Lemma 3.7, it followsthat τ D ∈ Z G ( X, H, Y ). Because of Theorem 3.8, we conclude that for 1 ≤ j ≤ sA jj ∈ ( O( t d j , C ) if d j even , Sp( t d j / , C ) if d j odd . (5.5)Next define g ∈ Sp( n, C ) as in (5.3). Then the matrix [ g ] B becomes a diagonal matrix.The first t d + · · · + t d s diagonal entries of [ g ] B is of the form:diag (cid:0) D , . . . , D s (cid:1) , where D j = ( ( − d j − √− t dj if d j even , ( − d j − I t dj if d j odd . (5.6)As ν is an involution, using (5.4) it follows that A jj = − I t dj when d j is even and 1 ≤ j ≤ s .Since det A jj = ±
1, we conclude that t d j is an even integer. (cid:3) We have a stronger result for nilpotent elements in Lie(PSp( n, C )). Theorem 5.10.
Every nilpotent element in
Lie(PSp( n, C )) is strongly real. EALITY OF UNIPOTENT ELEMENTS 17
Proof.
Let X ∈ Lie(PSp( n, C )) be a non-zero nilpotent element. Let { X l v dj | ≤ j ≤ t d , ≤ l ≤ d − , d ∈ N d } be a C -basis of C n as in Proposition 3.5. Recall that t d is even for d ∈ O d . Define g ∈ GL ( C n ) as follows: g ( X l v dj ) := ( − l √− X l v dj if 1 ≤ j ≤ t d / , d ∈ O d , ( − l +1 √− X l v dj if t d / < j ≤ t d , d ∈ O d , ( − l √− X l v dj if d ∈ E d . It follows that gX = − Xg , g = − Id. Using Proposition 3.5, we conclude that h gx, gy i = h x, y i for all x, y ∈ C n . This shows that g ∈ Sp( n, C ). (cid:3) Reality for nilpotent elements in simple Lie algebra over R In this section, we will consider the nilpotent elements in simple Lie algebra over R ofclassical type.6.1. The Lie algebras sl n ( R ) & sl n ( H ) . In this subsection first we will consider nilpotentelements in sl n ( R ). Recall the definition of f P e ( n ) as in (5.2). Theorem 6.1.
Let X ∈ sl n ( R ) be a nilpotent element. Suppose that X corresponds tothe partition d ∈ P ( n ) . Then TFAE :(1) X is a real element.(2) X is a strongly real element.(3) d f P e ( n ) . Proof.
Using Proposition 3.4, { X l v dj | ≤ j ≤ t d , ≤ l ≤ d − , d ∈ N d } is a R -basis of R n . Define g ∈ GL n ( R ) as (5.1). Then g = Id, Xg = − gX , and det g = ± Case 1: If d ∈ P v . even ( n ), then det g = 1. Hence, X is strongly real. Case 2:
Next assume that d ∈ P ( n ) \ P even ( n ). Moreover, assume that det g = − d i ∈ O d . i.e., d i is an odd integer. Then define e g ∈ GL n ( R ) as follows: e g ( X l v dj ) = ( ( − l X l v dj if either d = d i or when d = d i , j > − l +1 X l v dj if d = d i , j = 1 . Then det e g = − det g = 1, e g = Id, X e g = − e gX . Hence, in this case also X is stronglyreal. Case 3 :
Finally assume that d ∈ P even ( n ) \ P v . even ( n ). In view of Lemma 3.10, X isreal if and only if det g = 1. Therefore, in this case if X is real, then it is strongly real. Itfollows from the definition of g as in (5.1) that det g = ( − P η ∈ E d t η .Now (1) ⇒ (3) follows from Case 3 . (2) always implies (1). Finally, (3) ⇒ (2) followsfrom Case 1, Case 2 and
Case 3 . This completes the proof. (cid:3)
As a corollary we have the following known result:
Corollary 6.2 ([ST, Theorem 3.1.1]) . Let X ∈ sl n ( R ) be a nilpotent element. Supposethat n . Then TFAE:(1) X is a real element. (2) X is a strongly real element. Proof.
Condition (3) of Theorem 6.1 always true for n (cid:3) Next we will consider the nilpotent elements in the Lie algebra sl n ( H ). As the Liealgebra sl ( H ) is isomorphic to su (2) which is a compact Lie algebra, we will furtherassume that n > Theorem 6.3.
Every nilpotent element in sl n ( H ) is strongly real. Proof.
The proof follows from Proposition 5.4. (cid:3)
The Lie algebra su ( p, q ) . As we would like to address nilpotent elements; so itwill be assumed that p > q >
0. Here h· , ·i denotes the Hermitian form on C p + q defined by h x, y i := x t I p,q y , where I p,q is as in (3.2). We will follow notation as definedin § X ∈ su ( p, q ) be a non zero nilpotent element, and let { X, H, Y } ⊂ su ( p, q ) bea sl ( R )-triple. We now apply Proposition 3.5, Remark 3.6(2). Let { d , . . . , d s } , with d > · · · > d s , be the finite ordered set of integers that arise as R -dimension of non-zero irreducible Span R { X, H, Y } -submodules of C p + q . Let t d r := dim C L ( d r −
1) for1 ≤ r ≤ s . Then we have d := [ d t d , . . . , d t ds s ] ∈ P ( p + q ). Let p d := ( { j | h v dj , X d − v dj i = 1 } if d ∈ O d , { j | √− h v dj , X d − v dj i = 1 } if d ∈ E d ; and q d := t d − p d . (6.1)Recall that the signed Young diagram is uniquely determined by the sign of 1 st column,see Definition 3.1 and Remark 3.2. Now we associate a signed Young diagram for theelement X by setting +1 in the p d -many boxes and − q d -many boxes in the1 st column of the rectangular block of size t d × d . It follows that d ∈ P ( n ) and ( p d , q d )for d ∈ N d are conjugation invariant and hence, they determined a well-defined signedYoung diagram for the nilpotent orbit O X . In fact, the nilpotent orbits in su ( p, q ) arein bijection with the signed Young diagram of signature ( p, q ), see [CM, Theorem 9.3.3],[BCM, Theorem 4.10]. Theorem 6.4.
Let X ∈ su ( p, q ) be a non-zero nilpotent element. Let p d (resp. q d ) be thenumber of +1 occurred in the st column of the block of size t d d × t d d in the signed Youngdiagram corresponding to the orbit O X of X . Then TFAE :(1) X is real in su ( p, q ) .(2) X is strongly real in su ( p, q ) .(3) p η = q η for all η ∈ E d . Proof.
We will show (3) ⇒ (2) ⇒ (1) ⇒ (3). Let { X, H, Y } ⊂ su ( p, q ) be a sl -triple. Then C p + q has a C -basis of the form { X j v dj | d ∈ N d } which satisfy Proposition3.5 (3). First assume that (3) holds. i.e., p d = q d for all d ∈ E d . After suitable reorderingwe may assume that h v dj , X d − v dj i = √− ≤ j ≤ t d / , d ∈ E d −√− t d / < j ≤ t d , d ∈ E d +1 1 ≤ j ≤ p d , d ∈ O d − p d < j ≤ t d , d ∈ O d EALITY OF UNIPOTENT ELEMENTS 19
Define g ∈ GL( C p+q ) as follows: g ( X l v dj ) := ( − l X l v dj if d ∈ O d , ( − l +1 X l v dj if d ∈ O d , ( − l X l v d i j + t d / if d ∈ E d , ≤ j ≤ t d / , ( − l X l v d i j − t d / if d ∈ E d , t d i / < j ≤ t d i . Then g ∈ SU( p, q ), g = Id, and gX = − Xg . This prove (3) ⇒ (2).Note that (2) ⇒ (1) is always true. Thus, it remains to show (1) ⇒ (3). Suppose(1) holds. X is real in su ( p, q ), i.e., − X ∈ O X , the nilpotent orbit of X. In view ofRemark 3.2, the signed Young diagram of O X is determined by d ∈ P ( n ) and ( p d , q d ) forall d ∈ N d where p d and q d is given by (6.1). Note that {− X, H, − Y } is a sl -triple in su ( p, q ) containing − X and the partition d ∈ P ( p + q ) for the nilpotent element − X issame as X , see Section 3.4. Following the construction as done for X , let p ′ d := ( { j | h v dj , ( − X ) d − v dj i = 1 } if d ∈ O d , { j | √− h v dj , ( − X ) d − v dj i = 1 } if d ∈ E d ; and q ′ d := t d − p ′ d . (6.2)Comparing (6.1) and (6.2), we conclude that( p d , q d ) = ( p ′ d , q ′ d ) for d ∈ O d , ( p d , q d ) = ( q ′ d , p ′ d ) for d ∈ E d . As − X ∈ O X , the signed Young diagrams of O X and O − X coincide, see [CM, Theorem9.3.3]. Hence, p d = q d for all d ∈ E d . This completes the proof. (cid:3) The Lie algebra so ( p, q ) . Next we will consider the real simple Lie algebra so ( p, q ),see § p, q > h· , ·i denotes the symmetric form on R p + q defined by h x, y i := x t I p,q y , where I p,q is as in (3.2). Proposition 6.5.
Every nilpotent element in so ( p, q ) is strongly Ad SO( p,q ) -real. Proof.
Let X ∈ so ( p, q ) be a non zero nilpotent element. Then R p + q has a basis of theform { X l v jd | ≤ l ≤ d − , ≤ j ≤ t d , d ∈ N d } which satisfies Proposition 3.5(3) andRemark 3.6(1). Now define g ∈ GL( R p + q ) as follows : g ( X l v dj ) := ( − l X l v dj if d ∈ O d , ( − l +1 X l v dj if d ∈ O d , ( − l X l v dj + t d / if d ∈ E d , ≤ j ≤ t d / , ( − l X l v dj − t d / if d ∈ E d , t d / < j ≤ t d . (6.3)Note that det g = 1 , gX = − Xg and g ∈ SO( p, q ). (cid:3) Now we will consider the Ad -action of SO( p, q ) on the nilpotent elements in so ( p, q ).The following example shows that every nilpotent element in so (2 ,
1) is not strongly realunder the action of SO(2 , , see also [Go, Theorem 1.1, Corollary 1.3]. Example 6.6.
Let X ∈ so (2 ,
1) be a nilpotent element whose signed Young diagram is − g ∈ SO(2 , so that gXg − = − X . Thisis easy to see using the upper half space model of the hyperbolic space, where such a nilpotent element corresponds to the translation z z + 1 in the isometry group and cannot be conjugated to z z − , R ).Let X ∈ so ( p, q ) be a non zero nilpotent element. Let { X, H, Y } ⊂ so ( p, q ) be a sl ( R )-triple. Let V := R n be the right R -vector space of column vectors. We consider V as a Span R { X, H, Y } -module via its natural so ( p, q )-module structure. Recall that eachrow of a signed Young diagram will represent an invariant Span R { X, H, Y } -module V dj which admits a cyclic basis of the form B d ( j ) := { X l v dj | ≤ l < d } . When d is odd,the symmetric form h· , ·i is non-degenerate on V dj × V dj , and the signs of that row in thediagram will correspond with the signature of the form restricted to V dj × V dj . When d iseven, t d is even and the form h· , ·i is non-degenerate on V dj × V dj + t d / for 1 ≤ j ≤ t d /
2, seeRemark 3.6, also [BCM, Remark A.13] for more details in this regard. Let V E := M η ∈ E d M ( η − , V O := M θ ∈ O d M ( θ − , V O := M ζ ∈ O d M ( ζ − . (6.4)Using Proposition 3.5, we have the following orthogonal decomposition of V : V = V E ⊕ V O ⊕ V O . (6.5)Let g ∈ SO( p, q ) be as in (6.3). Then g ( V θj ) ⊂ V θj and g ( V ηj ⊕ V ηj + t η / ) ⊂ V ηj ⊕ V ηj + t η / for θ ∈ O d , η ∈ E d . In particular, g ( V E ) ⊂ V E , g ( V O ) ⊂ V O and g ( V O ) ⊂ V O . In the next lemma, we wantto study g | V E , g | V O d , g | V O d . Lemma 6.7.
Let X ∈ so ( p, q ) be a non-zero nilpotent element, and d = [ d t d , . . . , d t ds s ] ∈ P ( p + q ) be the corresponding partition for X . Let p θ (resp. q θ ) be the number of +1 (resp. − ) in the st column of the rectangular block of size t θ × θ in the signed Youngdiagram of X . Let g ∈ SO( p, q ) be as in (6.3) . Then the following hold :(1) For η ∈ E d , g | V ηj ⊕ V ηj + tη/ ∈ ( SO ( η, η ) if η ≡ η, η ) \ SO ( η, η ) if η ≡ .In particular , g | V E ∈ ( SO ( p E , p E ) if P η ≡ t η / p E , p E ) \ SO ( p E , p E ) if P η ≡ t η / odd , where p E = P η ∈ E d ηt η / .(2) For ζ ∈ O d , g | V ζj ∈ ( SO ( ζ +12 , ζ − ) if h v ζj , X ζ − v ζj i = 1SO ( ζ − , ζ +12 ) if h v ζj , X ζ − v ζj i = − .In particular, g | V O d ∈ SO ( p O , q O ) , where ( p O , q O ) is the signature of the form h· , ·i| V O d × V O d .(3) For θ ∈ O d , g | V θj ∈ ( SO( θ − , θ +12 ) \ SO ( θ − , θ +12 ) if h v θj , X θ − v θj i = 1 , SO( θ +12 , θ − ) \ SO ( θ +12 , θ − ) if h v θj , X θ − v θj i = − . Proof.
Proof of (1) :- Recall that B d ( j ) is an ordered basis of the vector space V dj .Now we will construct an orthogonal ordered basis of V dj ⊕ V dj + t η / . For 1 ≤ j ≤ t η / EALITY OF UNIPOTENT ELEMENTS 21 ≤ l ≤ η , define w η + jl := (cid:0) X η − l v ηj + t η / + ( − X ) l − v ηj (cid:1) √ w η − jl := (cid:0) X l − v ηj + ( − X ) η − l v ηj + t η / (cid:1) √ C η + j := { w η + jl | ≤ l ≤ η } and C η − j := { w η − jl | ≤ l ≤ η } . Then C η + j ∨ C η − j is an suitableorthogonal basis so that h w η + jl , w η + jl i = 1 and h w η − jl , w η − jl i = − . The matrix [ g ] C η + j = [ g ] C η − j = ... ∈ ( SO( η if η ≡ , O( η ) \ SO( η ) if η ≡ . Hence, [ g ] C η + j ∨C η − j = [ g ] C + j [ g ] C − j ! ∈ ( SO ( η, η ) if η ≡ , SO( η, η ) \ SO ( η, η ) if η ≡ . This completes the proof of (1) as V E = P ≤ j ≤ t η / ( V dj ⊕ V dj + t η / ). Proof of (2) :- In these cases we will also construct an orthogonal basis of the vectorspace V θj using the basis B θ ( j ), for θ ∈ O d , see [BCM, Lemma A.9(2)]. For θ ∈ O d ,define w θjl := (cid:0) X l v θj + X θ − − l v θj (cid:1) √ if 0 ≤ l < ( θ − / X l v θj if l = ( θ − / (cid:0) X θ − − l v θj − X l v θj (cid:1) √ if ( θ − / < l ≤ θ − . (6.6)Then h w θjl , w θjl ′ ) i = 0 for θ ∈ O d , l = l ′ and 0 ≤ l, l ′ ≤ θ − h w θjl , w θjl i = ( − l h v θj , X θ − v θj i if 0 ≤ l < ( θ − / − l h v θj , X θ − v θj i if l = ( θ − / − l +1 h v θj , X θ − v θj i if ( θ − / < l ≤ θ − . For ζ ∈ O d , we need to consider two cases. First assume that h v ζj , X ζ − v ζj i = +1. Set C ζ + j := { w ζjl | l even , ≤ l ≤ ζ − } ∨ { w ζj ζ − } ∨ { w ζjl | l odd , ζ − < l ≤ ζ − }C ζ − j := { w ζjl | l odd , ≤ l ≤ ζ − } ∨ { w ζjl | l even , ζ − < l ≤ ζ − } . (6.7)Since, ζ ∈ O d , ζ − is even. Hence,[ g | V ζj ] C ζ + j ∨C ζ − j = Id ζ +12 − Id ζ − ! ∈ SO (cid:0) ζ + 12 , ζ − (cid:1) . Next assume that h v ζj , X ζ − v ζj i = −
1. In this case set C ζ + j := { w ζjl | l odd , ≤ l ≤ ζ − } ∨ { w ζjl | l even , ζ − < l ≤ ζ − }C ζ − j := { w ζjl | l even , ≤ l ≤ ζ − } ∨ { w ζj ζ − } ∨ { w ζjl | l odd , ζ − < l ≤ ζ − } . Hence, [ g | V ζj ] C ζ + j ∨C ζ − j = − Id ζ − Id ζ +12 ! ∈ SO (cid:0) ζ − , ζ + 12 (cid:1) . This completes the proof of (2).
Proof of (3) :- For θ ∈ O d , first assume that h v θj , X θ − v θj i = +1. Set C θ + j := { w θjl | l even , ≤ l ≤ θ − } ∨ { w θjl | l odd , θ − < l ≤ θ − }C θ − j := { w θjl | l odd , ≤ l ≤ θ − } ∨ { w θj θ − } ∨ { w θjl | l even , θ − < l ≤ θ − } . Since θ ∈ O d , θ − is odd. Note that [ g | Span C θ − j ] C θ − j ∈ SO (cid:0) θ +12 (cid:1) Hence,[ g | V θj ] C θ + j ∨C θ − j = − Id θ − [ g | Span C θ − j ] C θ − j ! ∈ SO (cid:0) θ − , θ + 12 (cid:1) \ SO (cid:0) θ − , θ + 12 (cid:1) . We omit the case when h v θj , X θ − v θj i = −
1, as this can be done in a similar way. (cid:3)
The following corollaries are immediate from the above Lemma 6.7.
Corollary 6.8.
Let X ∈ so ( p, q ) and g ∈ SO( p, q ) be as in Lemma 6.7. If P θ ∈ O d t θ iseven, then g | V O d ∈ SO ( p O , q O ) . Corollary 6.9.
Let X ∈ so ( p, q ) and g ∈ SO( p, q ) be as in Lemma 6.7. Suppose that P θ ∈ O d t θ is odd. If either p θ = 0 or q θ = 0 , ∀ θ ∈ O d , then g | V O d ∈ SO( p O , q O ) \ SO ( p O , q O ) . Corollary 6.10.
Let X ∈ so ( p, q ) and g ∈ SO( p, q ) be as in Lemma 6.7. Suppose thateither p θ = 0 or q θ = 0 , ∀ θ ∈ O d .(1) If either both P θ ∈ O d t θ and P η ≡ t η / is even or odd, then g ∈ SO ( p, q ) .(2) If one of P θ ∈ O d t θ and P η ≡ t η / is even and another one is odd, then g ∈ SO( p, q ) \ SO ( p, q ) . In the next result we will show that a certain class of nilpotent elements in so ( p, q ) isstrongly Ad SO( p,q ) -real. Proposition 6.11.
Let X ∈ so ( p, q ) be a non-zero nilpotent element, and d = [ d t d , . . . , d t ds s ] ∈ P ( p + q ) be the corresponding partition for X . Let p θ (resp. q θ ) be the number of +1 (resp. − ) in the st column of the rectangular block of size t θ × θ in the signed Youngdiagram of X . Suppose that there exist θ, θ ′ ∈ O d so that p θ = 0 and q θ ′ = 0 . Then X isstrongly Ad SO( p,q ) -real. Proof.
Let g ∈ SO( p, q ) be as in (6.3). If g ∈ SO ( p, q ), then we are done. Supposethat g SO ( p, q ). Recall that B d ( j ) is an ordered basis of the vector space V dj . Wecan assume that h v θj θ , X θ − v θj θ i = +1 and h v θj θ ′ , X θ ′ − v θ ′ j θ ′ i = − θ, θ ′ ∈ O d . Define EALITY OF UNIPOTENT ELEMENTS 23 e g ∈ SO( p, q ) as follows: e g ( X l v dj ) := ( − g ( X l v θj ) if d = θ, θ ′ ; j = j θ , , j θ ′ ,g ( X l v θj ) otherwise . Since, g ∈ SO( p, q ) \ SO ( p, q ), it follows that e g ∈ SO ( p, q ) is an involution and − X = e gX e g − . This completes the proof. (cid:3) The proof of the next proposition is implicit in the proof of [BCM, Theorem 4.25].
Proposition 6.12.
Let X ∈ so ( p, q ) be a non-zero nilpotent element, and d ∈ P ( p + q ) be the corresponding partition for X . Let p θ (resp. q θ ) be the number of +1 (resp. − ) inthe st column of the rectangular block of size t θ × θ in the signed Young diagram of X .If either p θ = 0 or q θ = 0 for all θ ∈ O d , then Z SO( p,q ) ( X ) = Z SO( p,q ) ( X ) . Proof.
As Lie( Z SO( p,q ) ( X, H, Y )) = Lie( Z SO( p,q ) ( X, H, Y )) ≃ z so ( p,q ) ( X ), to provethis proposition, it is enough to show Z SO( p,q ) ( X, H, Y ) = Z SO( p,q ) ( X, H, Y ) . Let h· , ·i E := h· , ·i| V E × V E and h· , ·i O := h· , ·i| V O × V O . Let X E := X | V E , X O := X | V O , H E := H | V E , H O := H | V O , Y E := Y | V E and Y O := Y | V O . We have the following naturalisomorphism Z SO( p,q ) ( X, H, Y ) ≃ Z SO( V E , h· , ·i E ) ( X E , H E , Y E ) × Z SO( V O , h· , ·i O ) ( X O , H O , Y O ) , (6.8)where Z SO( V E , h· , ·i E ) ( X E , H E , Y E ) ≃ Q η ∈ E d Sp( t η / , R ) is connected. Also, it follows from[BCM, (4.31)] that Z SO( p,q ) ◦ ( X, H, Y ) ≃ Z SO( V E , h· , ·i E ) ( X E , H E , Y E ) × Z SO( V O , h· , ·i O ) ◦ ( X O , H O , Y O ) . (6.9)We now appeal to [BCM, Proposition 4.21 and 4.21] to conclude under the given assump-tion on the nilpotent element X that Z SO( V O , h· , ·i O ) ( X O , H O , Y O ) ≃ Z SO( V O , h· , ·i O ) ( X O , H O , Y O ) . This completes the proof. (cid:3)
Proposition 6.13.
Let X ∈ so ( p, q ) be a non-zero nilpotent element as in Proposition6.12, i.e., either p θ = 0 or q θ = 0 for all θ ∈ O d . Then X is not Ad SO( p,q ) -real if one of P η ≡ t η / and P θ ∈ O d t θ is odd and another one is even. Proof.
Suppose that X is a real element, i.e., − X = τ Xτ − for some τ ∈ SO ( p, q ).Let g ∈ SO ( p, q ) be as in (6.3). Then using Proposition 6.12 we have g ∈ τ Z SO( p,q ) ( X ) = τ Z SO ( p,q ) ( X ) ⊂ SO ( p, q ) , which contradicts Corollary 6.10(2). This completes the proof. (cid:3) Theorem 6.14.
Let X ∈ so ( p, q ) be a non-zero nilpotent element, and d = [ d t d , . . . , d t ds s ] ∈ P ( p + q ) be the corresponding partition for X . Let p θ (resp. q θ ) be the number of +1 (resp. − ) in the st column of the rectangular block of size t θ × θ in the signed Youngdiagram of X . Then TFAE :(1) X is Ad SO( p,q ) -real.(2) X is strongly Ad SO( p,q ) -real.(3) The signed Young diagram of X does not satisfy both the following conditions : • Either p θ = 0 or q θ = 0 for all θ ∈ O d . • One of P η ≡ t η / and P θ ∈ O d t θ is odd and another one is even. Proof.
First note that (2) ⇒ (1) is obvious. (1) ⇒ (3) follows from Proposition 6.13.Using Proposition 6.11 and Corollary 6.10(1), we conclude that (3) ⇒ (2). (cid:3) The Lie algebra so ∗ (2 n ) . Here we will consider the real simple Lie algebra so ∗ (2 n ).Recall that H = R + i R + j R + k R . Throughout this subsection h· , ·i denotes the skew-Hermitian form on H n defined by h x, y i := x t j I n y , for x, y ∈ H n . We will follow notationas defined in § X ∈ so ∗ (2 n ) be a non zero nilpotent element, and let { X, H, Y } ⊂ so ∗ (2 n ) be a sl ( R )-triple. We now apply Proposition 3.5, Remark 3.6(2). Following Section 3.5, let t d r := dim H L ( d r −
1) for 1 ≤ r ≤ s , and d := [ d t d , . . . , d t ds s ] ∈ P ( n ) be the partitionof n corresponding to the element X . Let p d := ( { j | h v dj , X d − v dj i = 1 } when d ∈ O d , { j | √− h v dj , X d − v dj i = 1 } when d ∈ E d ; and q d := t d − p d . (6.10)Now we associate a signed Young diagram for the element X by setting +1 in the p η -manyboxes and − q η -many boxes in the 1 st column of the rectangular block of size t η × η for η ∈ E d , and place +1 in the left most box of odd length rows. It follows thatthis association does not depend on the conjugacy class. The nilpotent orbits in so ∗ (2 n ) is parametrized by the signed Young diagram of size n and any signature in which rowsof odd length have their left most boxes labelled +1, see [CM, Theorem 9.3.4], also [BCM,Theorem 4.27]. Theorem 6.15.
Let X be a nilpotent element in so ∗ (2 n ) . Then TFAE :(1) X is real.(2) X is strongly real.(3) p η = q η for all η ∈ E d . Proof.
We will show (3) ⇒ (2) ⇒ (1) ⇒ (3). Let { X, H, Y } ⊂ g be a sl -triplecorresponding to X . Then H n has a H -basis of the form { X j v dj | d ∈ N d } which satisfyProposition 3.5. First assume that p d = q d for all d ∈ E d . In view of Remark 3.6, we canassume that h v dj , X d − v dj i = ≤ j ≤ p d d ∈ E d , − p d < j ≤ t d d ∈ E d , j d ∈ O d . Define g ∈ GL( H n ) as follows: g ( X l v dj ) := ( − l X l v dj if d ∈ O d , ( − l +1 X l v dj if d ∈ O d , ( − l X l v dj + t d / if d ∈ E d , ≤ j ≤ t d / , ( − l X l v dj − t d / if d ∈ E d , t d / < j ≤ t d . Then g ∈ SO ∗ (2 n ), g = Id, and gX = − Xg . This proves (2).Note that (2) ⇒ (1) is always true. Thus, it remains to show (1) ⇒ (3). Suppose(1) holds, i.e., − X ∈ O X . In view of Remark 3.2, the signed Young diagram of O X is EALITY OF UNIPOTENT ELEMENTS 25 determined by d ∈ P ( n ) and ( p d , q d ) for all d ∈ N d where p d and q d is given by (6.10). Notethat {− X, H, − Y } is a sl -triple in so ∗ (2 n ) containing − X and the partition d ∈ P ( n )for the nilpotent element − X is same as X , see Section 3.4. Following the constructionas done for X , let p ′ d := ( { j | h v dj , ( − X ) d − v dj i = 1 } if d ∈ O d , { j | √− h v dj , ( − X ) d − v dj i = 1 } if d ∈ E d ; and q ′ d := t d − p ′ d . (6.11)Comparing (6.10) and (6.11), we conclude that( p d , q d ) = ( p ′ d , q ′ d ) for d ∈ O d , ( p d , q d ) = ( q ′ d , p ′ d ) for d ∈ E d . Since, − X ∈ O X , the signed Young diagrams of O X and O − X coincide, see [CM, Theorem9.3.4]. Hence, p d = q d for all d ∈ E d . This completes the proof. (cid:3) The Lie algebra sp ( n, R ) . Here we will consider the real simple Lie algebra sp ( n, R ).Throughout this subsection h· , ·i denotes the symplectic form on R n defined by h x, y i := x t J n y , x, y ∈ R n , where J n is as in (3.2).Let X ∈ sp ( n, R ) be a non zero nilpotent element. We now apply Proposition 3.5,Remark 3.6(2). Following Section 3.5, let t d r := dim R L ( d r −
1) for 1 ≤ r ≤ s , and d := [ d t d , . . . , d t ds s ] ∈ P (2 n ) be the partition corresponding to the element X . Let p η := { j | h v ηj , X η − v ηj i = 1 } , and q η := t η − p η for η ∈ E d . Now we associate a signed Young diagram for the element X by setting +1 in the p η -manyboxes and − q η -many boxes in the 1 st column of the rectangular block of size t η × η for η ∈ E d , and place +1 in the left most box of odd length rows. It follows thatthis association does not depend on the conjugacy class. The nilpotent orbits in sp ( n, R ) is parametrized by the signed Young diagram of size n and any signature in which rowsof odd length have their left most boxes labelled +1 and occur with even multiplicity , see[CM, Theorem 9.3.5], also [BCM, Theorem 4.27]. Theorem 6.16.
Let X be a nilpotent element in sp ( n, R ) . Then TFAE :(1) X is real.(2) X is strongly real.(3) p η = q η for all η ∈ E d . Proof.
We will show (3) ⇒ (2) ⇒ (1) ⇒ (3). Let { X, H, Y } ⊂ sp ( n, R ) be a sl -triple corresponding to X . Then R n has a basis of the form { X j v dj | d ∈ N d } whichsatisfy Proposition 3.5 (3). In view of Remark 3.6(1), we can assume that h v dj , X d − v dj i = ( ≤ j ≤ p d − p d < j ≤ t d , d ∈ E d ; h v dj , X d − v dt d / j i = 1 for d ∈ O d , ≤ j ≤ t d / . First assume that p d = q d for all d ∈ E d . Define g ∈ GL( R ) as follows: g ( X l v dj ) := ( − l X l v dj if d ∈ O d , ( − l +1 X l v dj if d ∈ O d , ( − l X l v dj + t d / if d ∈ E d , ≤ j ≤ t d / , ( − l X l v dj − t d / if d ∈ E d , t d / < j ≤ t d . Then g ∈ Sp( n, R ), g = Id, and gX = − Xg . This proves (2).Note that (2) ⇒ (1) is always true. Thus, it remains to show (1) ⇒ (3). Let the signedYoung diagram for the orbit O − X be determined by d ′ ∈ P ( n ) and ( p ′ d , q ′ d ) for d ∈ N d .Using Proposition 3.5 and Remark 3.6(3), it follows that d = d ′ ; ( p d , q d ) = ( p ′ d , q ′ d ) for d ∈ O d , ( p d , q d ) = ( q ′ d , p ′ d ) for d ∈ E d . Now (1) implies − X ∈ O X . Therefore, the signed Young diagrams of O X and O − X coincide, see [CM, Theorem 9.3.5]. Hence, p d = q d for all d ∈ E d . This completes theproof. (cid:3) The Lie algebra sp ( p, q ) . Here we will deal with the nilpotent elements in the Liealgebra sp ( p, q ). We will further assume that p, q >
0. Recall that H = R + i R + j R + k R .Throughout this subsection h· , ·i denotes the hermitian form on H n defined by h x, y i :=¯ x t I p,q y , where I p,q is as in (3.2). Recall the definition of N d and E d as in (3.3). Theorem 6.17.
Every non-zero nilpotent element in
Lie(PSp( p, q )) is real. Proof.
Let X ∈ sp ( p, q ) be a non zero nilpotent element. Then H p + q has a H -basis ofthe form { X l v jd | d ∈ N d } which satisfies Proposition 3.5(3). Now define g ∈ GL( V ) asfollows : g ( X l v dj ) := ( − l X l v dj i , d ∈ N d . Then g = − Id , gX = − Xg and g ∈ Sp( p, q ). This completes the proof. (cid:3)
Now we have an immediate corollary which extends [BG, Theorem 1.1] for any unipotentelement in the higher rank situation.
Corollary 6.18.
Every non-zero nilpotent element in sp ( p, q ) is real. Every nilpotent element in sp ( p, q ) corresponds to a partition d = [ d t d , . . . , d t ds s ] ∈P ( n ), see [CM, Theorem 9.3.5], also [BCM, Theorem 4.38]. Next we classify strongly realnilpotent elements in sp ( p, q ). Theorem 6.19.
Let X ∈ sp ( p, q ) be a nilpotent. Then X is a strongly real element ifand only if t η is even for all η ∈ E d . Proof.
Let X ∈ sp ( p, q ) be a non zero nilpotent element. Then H p + q has a H -basis ofthe form { X l v jd | d ∈ N d } which satisfies Proposition 3.5(3). Suppose that t η is even forall η ∈ E d . Define g ∈ GL( V ) as follows : g ( X l v dj ) := ( − l X l v dj if d ∈ O d , ( − l +1 X l v dj if d ∈ O d , ( − l ( X l v dj + t d / ) i if d ∈ E d , ≤ j ≤ t d / , ( − l +1 ( X l v dj − t d / ) i if d ∈ E d , t d / < j ≤ t d . Then g = Id , gX = − Xg and g ∈ Sp( p, q ). Hence, X is strongly real.Next assume that X is a strongly real nilpotent element in sp ( p, q ). Let τ ∈ Z Sp( p,q ) ( X ).Recall that B is the ordered basis of H p + q as in (3.9). Following § τ ] B = ( A ij ),and τ D be the block diagonal part of [ τ ] B , i.e., consists of only the matrices ( A jj ) in EALITY OF UNIPOTENT ELEMENTS 27 the diagonal. Then τ D X = Xτ D and τ D H = Hτ D . Using Lemma 3.7, it follows that τ D ∈ Z Sp( p,q ) ( X, H, Y ). Using Proposition 3.9, we conclude that for 1 ≤ j ≤ sA jj ∈ SO ∗ (2 t d j ) if d j even . (6.12)Next define g ∈ Sp( p, q ) as in the proof of Theorem 6.17. Then the matrix [ g ] B becomesa diagonal matrix. The first t d + · · · + t d s diagonal entries of [ g ] B is of the form:diag (cid:0) D , . . . , D s (cid:1) , where D j = ( − d j − i I t dj for 1 ≤ j ≤ s . Since X is strongly real, νXν − = − X for some involution ν ∈ Sp( p, q ). Then ν ∈ gZ G ( X ). It follows that when d j is even and 1 ≤ j ≤ s , the matrices A jj also satisfy( A jj i ) = I t dj . (6.13)Using (6.12) and (6.13) we conclude that A jj = − ¯ A tjj Let λ be a right eigen value of A jj (ref for existence of eigen value). Then it follows that − ¯ λ − is also a right eigen value which is distinct from λ . This proves that t d j , the orderof the matrix A jj , must be even. This completes the proof. (cid:3) Appendix
In this section we will deal with a particular example in support of the Section 3.5. Let D be either R or C or H . Let X be a nilpotent element in sl ( D ) where D is R or C or H which correspondence to the partition d := [3 , , ]. Let { X l v d r j | ≤ j ≤ t d r , ≤ l ≤ d r − , ≤ r ≤ } be a D -basis of D as in Proposition 3.4. Here, d = 3 , d = 2 , d = 1and t d = 2 , t d = 2 , t d = 4. Recall that B l ( d r ) := ( X l v d r , . . . , X l v d r t dr ). Set B (1) := B ( d ) ∨ B ( d ) ∨ B (1), B (2) := B ( d ) ∨ B ( d ), and B (3) := B ( d ). Finally define, B := B (1) ∨ B (2) ∨ B (3) . (6.14)We would like to write explicitly the matrix [ τ ] B . Suppose τ ( v ) = c X v + c X v + d X v + d X v + e v + e v + e v + e v + c X v + c X v + d v + d v + c v + c v , (6.15) τ ( v ) = c X v + c X v + d X v + d X v + e v + e v + e v + e v + c X v + c X v + d v + d v + c v + c v , (6.16) τ ( v ) = c X v + c X v + d X v + d X v + e v + e v + e v + e v + c X v + c X v + d v + d v , (6.17) τ ( v ) = c X v + c X v + d X v + d X v + e v + e v + e v + e v + c X v + c X v + d v + d v , (6.18)Since τ Xτ − = ǫX , using (6.15) and (6.16) we have τ ( Xv ) = ǫ ( c X v + c X v + d Xv + d Xv + c Xv + c Xv ) , τ ( Xv ) = ǫ ( c X v + c X v + d Xv + d Xv + c Xv + c Xv ,τ ( v ) = c X v + c X v + d X v + d X v + e v + e v + e v + e v ,τ ( v ) = c X v + c X v + d X v + d X v + e v + e v + e v + e v ,τ ( v ) = c X v + c X v + d X v + d X v + e v + e v + e v + e v ,τ ( v ) = c X v + c X v + d X v + d X v + e v + e v + e v + e v ,τ ( Xv ) = ǫ ( c X v + c X v + d v + d v ) , (6.19) τ ( Xv ) = ǫ ( c X v + c X v + d Xv + d Xv ) , (6.20)Finally, we have τ ( X v ) = c X v + c X v , (6.21) τ ( X v ) = c X v + c X v . (6.22)Hence, the matrix [ τ ] B is of the following form: c c ǫc ǫc c c c c ǫc ǫc c c c c c c ǫc ǫc c c c c ǫc ǫc c c c c ǫd ǫd d d d d ǫd ǫd d d d d ǫd ǫd d d d d ǫd ǫd d d d d e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e ǫc ǫc c c c c ǫc ǫc c c c c d d d d d d d d c c c c . Acknowledgements.
We thank Pralay Chatterjee, Dipendra Prasad, Anupam Singhand Maneesh Thakur for their comments on a first draft of this article.
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