On the second cohomology of nilpotent orbits in exceptional Lie algebras
aa r X i v : . [ m a t h . G R ] N ov ON THE SECOND COHOMOLOGY OF NILPOTENT ORBITS INEXCEPTIONAL LIE ALGEBRAS
PRALAY CHATTERJEE AND CHANDAN MAITY
Abstract.
In [BC], the second de Rham cohomology groups of nilpotent orbits in all the complexsimple Lie algebras are described. In this paper we consider non-compact non-complex exceptionalLie algebras, and compute the dimensions of the second cohomology groups for most of the nilpotentorbits. For the rest of cases of nilpotent orbits, which are not covered in the above computations,we obtain upper bounds for the dimensions of the second cohomology groups. Introduction
Let G be a connected real simple Lie group with Lie algebra g . An element X ∈ g is called nilpo-tent if ad( X ) : g → g is a nilpotent operator. Let O X := { Ad( g ) X | g ∈ G } be the corresponding nilpotent orbit under the adjoint action of G on g . Such nilpotent orbits form a rich class of ho-mogeneous spaces, and they are studied at the interface of several disciplines in mathematics suchas Lie theory, symplectic geometry, representation theory, algebraic geometry. Various topologicalaspects of such orbits have drawn attention over the years; see [CM], [M] and references thereinfor an account. In [BC, Proposition 1.2] for a large class of semisimple Lie groups a criterion isgiven for the exactness of the Kostant-Kirillov two form on arbitrary adjoint orbits which in turnled the authors asking the natural question of describing the full second cohomology groups ofsuch orbits. Towards this, in [BC], the second cohomology groups of nilpotent orbits in all thecomplex simple Lie algebras, under the adjoint actions of the corresponding complex groups, arecomputed. In this paper we continue the program of studying the second cohomology groups ofnilpotent orbits which was initiated in [BC]. We compute the second cohomology groups for mostof the nilpotent orbits in non-compact non-complex exceptional Lie algebras, and for the rest ofthe nilpotent orbits in non-compact non-complex exceptional Lie algebras we give upper boundsof the dimensions of second cohomology groups; see Theorems 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9,3.10, 3.11, 3.12, 3.13. In particular, our computations yield that the second cohomologies vanishfor all the nilpotent orbits in F − and E − .2. Notation and background
In this section we fix some general notation, and mention a basic result which will be used inthis paper. A few specialized notation are defined as and when they occur later.The center of a Lie algebra g is denoted by z ( g ). We denote Lie groups by capital letters, andunless mentioned otherwise we denote their Lie algebras by the corresponding lower case Germanletters. Sometimes, for convenience, the Lie algebra of a Lie group G is also denoted by Lie( G ).The connected component of a Lie group G containing the identity element is denoted by G ◦ . Fora subgroup H of G and a subset S of g , the subgroup of H that fixes S point wise is called the centralizer of S in H and is denoted by Z H ( S ). Similarly, for a Lie subalgebra h ⊂ g and a subset Mathematics Subject Classification.
Key words and phrases.
Nilpotent orbits, Exceptional Lie algebras, Second cohomology. S ⊂ g , by z h ( S ) we will denote the subalgebra consisting elements of h that commute with everyelement of S .If G is a Lie group with Lie algebra g , then it is immediate that the coadjoint action of G ◦ on z ( k ) ∗ is trivial; in particular, one obtains a natural action of G/G ◦ on z ( k ) ∗ . We denote by[ z ( g ) ∗ ] G/G ◦ the space of fixed points of z ( g ) ∗ under the action of G/G ◦ .For a real semisimple Lie group G , an element X ∈ g is called nilpotent if ad( X ) : g → g isa nilpotent operator. A nilpotent orbit is an orbit of a nilpotent element in g under the adjointrepresentation of G ; for a nilpotent element X ∈ g the corresponding nilpotent orbit Ad( G ) X isdenoted by O X .For a g be a Lie algebra over R a subset { X, H, Y } ⊂ g is said to be sl ( R ) -triple if X = 0,[ H, X ] = 2 X, [ H, Y ] = − Y and [ X, Y ] = H . It is immediate that, if { X, H, Y } ⊂ g is a sl ( R )-triple then Span R { X, H, Y } is a R -subalgebra of g which is isomorphic to the Lie algebra sl ( R ).We now recall the well-known Jacobson-Morozov theorem (see [CM, Theorem 9.2.1]) which ensuresthat if X ∈ g is a non-zero nilpotent element in a real semisimple Lie algebra g , then there exist H, Y in g such that { X, H, Y } ⊂ g is a sl ( R )-triple.To facilitate the computations in § Theorem 2.1.
Let G be an algebraic group defined over R which is R -simple. Let X ∈ Lie G ( R ) , X = 0 be a nilpotent element, and O X be the orbit of X under the adjoint action of the identitycomponent G ( R ) ◦ on Lie G ( R ) . Let { X, H, Y } be a sl ( R ) -triple in Lie G ( R ) . Let K be a maximalcompact subgroup in Z G ( R ) ◦ ( X, H, Y ) , and M be a maximal compact subgroup in G ( R ) ◦ containing K . Then H ( O X , R ) ≃ [( z ( k ) ∩ [ m , m ]) ∗ ] K/K ◦ . In particular, dim R H ( O X , R ) ≤ dim R z ( k ) . The above theorem follows from [CM, Lemma 3.7.3] and a description of the second cohomologygroups of homogeneous spaces which generalizes [BC, Theorem 3.3]. The details of the proof ofTheorem 2.1 and the generalization of [BC, Theorem 3.3], mentioned above, will appear elsewhere.3.
The second cohomology groups of nilpotent orbits
In this section we study the second cohomology of the nilpotent orbits in non-compact non-complex exceptional Lie algebras over R . The results in this section depend on the results of [D1,Tables VI-XV], [D2, Tables VII-VIII] and [K, Tables 1-12]. We refer to [CM, Chapter 9], [D1]and [D2] for the generalities required in this section. We begin by recalling the parametrization ofnilpotent orbits in this set-up.3.1. Parametrization of nilpotent orbits in exceptional Lie algebras
We follow the parametrization of nilpotent orbits in non-compact non-complex exceptional Liealgebras as given in [D1, Tables VI-XV] and [D2, Tables VII-VIII]. We consider the nilpotent orbitsin g under the action of Int g , where g is a non-compact non-complex real exceptional Lie algebra.We fix a semisimple algebraic group G defined over R such that g = Lie( G ( R )). Here G ( R ) denotesthe associated real semisimple Lie group of the R -points of G . Let G ( C ) be the associated complexsemisimple Lie group consisting of the C -points of G . It is easy to see that orbits in g under theaction of Int g are the same as the orbits in g under the action of G ( R ) ◦ . Thus in this set-up, fora nilpotent element X ∈ g , we set O X := { Ad( g ) X | g ∈ G ( R ) ◦ } . Let g = m + p be a Cartandecomposition and θ be the corresponding Cartan involution. Let g C be the Lie algebra of G ( C ).Then g C can be identified with the complexification of g . Let m C , p C be the C -spans of m and p in g C , respectively. Then g C = m C + p C . Let M C be the connected subgroup of G ( C ) with Lie N THE NILPOTENT ORBITS IN LIE ALGEBRAS 3 algebra m C . Recall that, if g is as above and g is different from both E − and E , then g isof inner type, or equivalently, rank m C = rank g C . When g is of inner type, the nilpotent orbitsare parametrized by a finite sequence of integers of length l where l := rank m C = rank g C . When g is not of inner type, that is, when g is either E − or E , then the nilpotent orbits areparametrized by a finite sequence of integers of length 4.Let X ′ ∈ g be a nonzero nilpotent element, and { X ′ , H ′ , Y ′ } ⊂ g be a sl ( R )-triple. Then { X ′ , H ′ , Y ′ } is G ( R )-conjugate to another sl ( R )-triple { e X, e H, e Y } in g such that θ ( e H ) = − e H , θ ( e X ) = − e Y . Set E := ( e H − i ( e X + e Y )) / F := ( e H + i ( e X + e Y )) / H := i ( e X − e Y ). Then { E, H, F } is a sl ( R )-triple and E, F ∈ p C and H ∈ m C . The sl ( R )-triple { E, H, F } is then calleda p C - Cayley triple associated to X ′ .3.1.1. Parametrization in exceptional Lie algebras of inner type.
We now recall from [D1, Column2, Tables VI-XV] the parametrization of non-zero nilpotent orbits in g when g is an exceptionalLie algebra of inner type. Let h C ⊂ m C be a Cartan subalgebra of m C such that h C ∩ m is a Cartansubalgebra of m . As g is of inner type, h C is a Cartan subalgebra of g C . Set h := h C ∩ i m . Let R, R be the root systems of ( g C , h C ) , ( m C , h C ), respectively. Let B := { α , . . . , α l } be a basis of R . Let B e := B ∪ { α } where α is the negative of the highest root of ( R, B ). Then there existsan unique basis of R , say B , such that B ⊂ B e . Let C be the closed Weyl chamber of R in h corresponding to the basis B . Let l be the rank of [ m C , m C ]. Then either l = l or l = l − l = l we set B ′ := B . If l = l − B ⊂ B ) we set B ′ := B . Clearly, B ′ = l . We enumerate B ′ := { β , . . . , β l } as in [D1, 7, p. 506 and Table IV]. Let X ∈ g bea nonzero nilpotent element, and { E, H, F } be a p C -Cayley triple (in g C ) associated to X . ThenAd( M C ) H ∩ C is a singleton set, say { H } . The element H is called the characteristic of the orbitAd( M C ) E as it determines the orbit M C · E uniquely. Consider the map from the set of nilpotentorbits in g to the set of integer sequences of length l , which assigns the sequence β ( H ) , . . . , β l ( H )to each nilpotent orbits O X . In view of the Kostant-Sekiguchi theorem (cf. [CM, Theorem 9.5.1]),this gives a bijection between the set of nilpotent orbits in g and the set of finite sequences of theform β ( H ) , . . . , β l ( H ) as above. We use this parametrization while dealing with nilpotent orbitsin exceptional Lie algebras of inner type.3.1.2. Parametrization in E − or E . We now recall from [D2, Column 1, Tables VII-VIII]the parametrization of non-zero nilpotent orbits in g when g is either E − or E . We need apiece of notation here : henceforth, for a Lie algebra a over C and an automorphism σ ∈ Aut C a ,the Lie subalgebra consisting of the fixed points of σ in a , is denoted by a σ . Let now h C be a Cartansubalgebra of g C (we point out the difference of our notation with that in [D2]; g and h of [D2, § g C and h C , respectively).Let g = E − . Let τ be the involution of g C as defined in [D2, p. 198 ] which keeps h C invariant. Then the subalgebra g τ C is of type F , and h τ C is a Cartan subalgebra of g τ C . Let G ( C ) τ be the connected Lie subgroup of G ( C ) with Lie algebra g τ C . Let { β , β , β , β } be the simpleroots of ( g τ C , h τ C ) as defined in [D2, (1), p. 198]. Let X ∈ E − be a nonzero nilpotent element.Let { E, H, F } be a p C -Cayley triple (in g C ) associated to X . Then H ∈ g τ C and E, F ∈ g − τ C . Wemay further assume that H ∈ h τ C . Then the finite sequence of integers β ( H ) , β ( H ) , β ( H ) , β ( H )determine the orbit Ad( G ( C ) τ ) E uniquely; see [D2, p. 204].Let g = E . Let τ ′ be the involution of g C as defined in [D2, p. 199 ] which keeps h C invariant.Then the subalgebra g τ ′ C is of type C , and h τ ′ C is a Cartan subalgebra of g τ ′ C . Let G ( C ) τ ′ be theconnected Lie subgroup of G ( C ) with Lie algebra g τ ′ C . Let { β , β , β , β } be the simple roots of( g τ ′ C , h τ ′ C ) as defined in [D2, p. 199]. Let X ∈ E be a nonzero nilpotent element. Let { E, H, F } be a p C -Cayley triple (in g C ) associated to X . Then H ∈ g τ ′ C and E, F ∈ g − τ ′ C . We may further P. CHATTERJEE AND C. MAITY assume that H ∈ h τ ′ C . It then follows that the finite sequence of integers β ( H ) , β ( H ) , β ( H ) , β ( H )determine the orbit Ad( G ( C ) τ ′ ) E uniquely; see [D2, p. 204].3.2. Nilpotent orbits of three types
For the sake of convenience of writing the proofs that appear in the later part §
3, it will beuseful to divide the nilpotent orbits in the following three types. Let X ∈ g be a nonzero nilpotentelement, and { X, H, Y } be a sl ( R )-triple in g . Let G be as in the beginning of § K be amaximal compact subgroup in Z G ( R ) ◦ ( X, H, Y ), and M be a maximal compact subgroup in G ( R ) ◦ containing K . A nonzero nilpotent orbit O X in g is said to be of(1) type I if z ( k ) = 0, K/K ◦ = Id and m = [ m , m ];(2) type II if either z ( k ) = 0, K/K ◦ = Id, m = [ m , m ]; or z ( k ) = 0, m = [ m , m ];(3) type III if z ( k ) = 0.In what follows we will use the next result repeatedly. Corollary 3.1.
Let g be a real simple non-compact exceptional Lie algebra. Let X ∈ g be a nonzeronilpotent element. (1) If the orbit O X is of type I, then dim R H ( O X , R ) = dim R z ( k ) . (2) If the orbit O X is of type II, then dim R H ( O X , R ) ≤ dim R z ( k ) . (3) If the orbit O X is of type III, then dim R H ( O X , R ) = 0 . Proof.
The proof of the corollary follows immediately from Theorem 2.1. (cid:3)
Let g be as above. In the proofs of our results in the following subsections we use the descriptionof a Levi factor of z g ( X ) for each nilpotent element X in g , as given in the last columns of [D1,Tables VI-XV] and [D2, Tables VII-VIII]. This enables us compute the dimensions dim R z ( k ) easily.We also use [K, Column 4, Tables 1-12] for the component groups for each nilpotent orbits in g .3.3. Nilpotent orbits in the non-compact real form of G Recall that up to conjugation there is only one non-compact real form of G . We denote it by G . There are only five nonzero nilpotent orbits in G ; see [D1, Table VI, p. 510]. Note thatin this case we have m = [ m , m ]. Theorem 3.2.
Let the parametrization of the nilpotent orbits be as in § X be a nonzeronilpotent element in G . (1) If the parametrization of the orbit O X is given by either or , then dim R H ( O X , R )= 1 . (2) If the parametrization of the orbit O X is given by any of , , , then dim R H ( O X , R )= 0 . Proof.
From [D1, Column 7, Table VI, p. 510] we have dim R z ( k ) = 1 and from [K, Column 4,Table 1, p. 247] we have K/K ◦ = Id for the nilpotent orbits as in (1). Thus these are of type I.We refer to [D1, Column 7, Table VI, p. 510] for the orbits as given in (2). These orbits are oftype III as dim R z ( k ) = 0. In view of the Corollary 3.1 the conclusions follow. (cid:3) Nilpotent orbits in non-compact real forms of F Recall that up to conjugation there are two non-compact real forms of F . They are denoted by F and F − . N THE NILPOTENT ORBITS IN LIE ALGEBRAS 5
Nilpotent orbits in F . There are 26 nonzero nilpotent orbits in F ; see [D1, Table VII,p. 510]. Note that in this case we have m = [ m , m ]. Theorem 3.3.
Let the parametrization of the nilpotent orbits be as in § X be a nonzeronilpotent element in F . (1) Assume the parametrization of the orbit O X is given by any of the sequences :
001 1 ,
001 3 ,
110 2 ,
111 1 ,
131 3 . Then dim R H ( O X , R ) = 1 . (2) Assume the parametrization of the orbit O X is given by any of the sequences :
100 2 ,
200 0 ,
103 1 ,
111 3 ,
204 4 . Then dim R H ( O X , R ) ≤ . (3) If the parametrization of the orbit O X is either
101 1 or
012 2 , then dim R H ( O X , R ) ≤ . (4) If O X is not given by the parametrizations as in (1), (2), (3) above ( dim R H ( O X , R ) = 0 . Proof.
For the Lie algebra F , we can easily compute dim R z ( k ) from the last column of [D1,Table VII, p. 510] and K/K ◦ from [K, Column 4, Table 2, pp. 247-248].For the orbits O X , as in (1), we have dim R z ( k ) = 1 and K/K ◦ = Id. Hence these are of type I.For the orbits O X , as in (2), we have dim R z ( k ) = 1 and K/K ◦ = Id; hence they are of type II. Forthe orbits O X , as in (3), we have dim R z ( k ) = 2 and K/K ◦ = Id. Hence these are also of type II.The rest of the 14 orbits, which are not given by the parametrizations in (1), (2), (3), are of typeIII as z ( k ) = 0. Now the theorem follows from Corollary 3.1. (cid:3) Nilpotent orbits in F − . There are two nonzero nilpotent orbits in F − ; see [D1, TableVIII, p. 511]. Theorem 3.4.
For all the nilpotent elements X in F − we have dim R H ( O X , R ) = 0 . Proof.
As the theorem follows trivially when X = 0 we assume that X = 0. We follow theparametrization of nilpotent orbits as in § z ( k ) = 0. Hence the nonzero nilpotent orbits are of type III. Using Corollary 3.1(3) we have dim R H ( O X , R ) = 0. (cid:3) Nilpotent orbits in non-compact real forms of E Recall that up to conjugation there are four non-compact real forms of E . They are denotedby E , E , E − and E − .3.5.1. Nilpotent orbits in E . There are 23 nonzero nilpotent orbits in E ; see [D2, Table VIII,p. 205]. Note that in this case we have m = [ m , m ]. Theorem 3.5.
Let the parametrization of the nilpotent orbits be as in § X be a nonzeronilpotent element in E . (1) If the parametrization of the orbit O X is given by either or or , then dim R H ( O X , R ) = 1 . (2) Assume the parametrization of the orbit O X is given by any of the sequences : , , , , . Then dim R H ( O X , R ) ≤ . (3) If O X is not given by the parametrizations as in (1), (2) above ( dim R H ( O X , R ) = 0 . Proof.
For the Lie algebra E , we can easily compute dim R z ( k ) from the last column of [D2,Table VIII, p. 205] and K/K ◦ from [K, Column 4, Table 4, p.253]. As pointed out in the 1 st paragraph of [K, p. 254], there is an error in row 5 of [D2, Table VIII, p. 205]. Thus when O X isgiven by the parametrization 2000 it follows from [K, p. 254] that z ( k ) = 0. P. CHATTERJEE AND C. MAITY
We have dim R z ( k ) = 1 and K/K ◦ = Id for the orbits given in (1). Thus these orbits are of typeI. For the orbits, as in (2), we have dim R z ( k ) = 1 and K/K ◦ = Z . Hence, the orbits in (2) are oftype II. For rest of the 15 nonzero nilpotent orbits, which are not given by the parametrizations of(1), (2), are of type III as dim R z ( k ) = 0. Now the results follow from Corollary 3.1. (cid:3) Nilpotent orbits in E . There are 37 nonzero nilpotent orbits in E ; see [D1, Table IX,p. 511]. Note that in this case we have m = [ m , m ]. Theorem 3.6.
Let the parametrization of the nilpotent orbits be as in § X be a nonzeronilpotent element in E . (1) Assume the parametrization of the orbit O X is given by any of the sequences : , , , , , , , .Then dim R H ( O X , R ) = 0 . (2) Assume the parametrization of the orbit O X is given by any of the sequences : , , , , , , , , .Then dim R H ( O X , R ) = 2 . (3) If the parametrization of the orbit O X is given by either or or ,then dim R H ( O X , R ) ≤ . (4) If the parametrization of the orbit O X is given by , then dim R H ( O X , R ) ≤ . (5) If O X is not given by the parametrizations as in (1), (2), (3), (4) above ( dim R H ( O X , R ) = 1 . Proof.
For the Lie algebra E , we can easily compute dim R z ( k ) from the last column of [D1,Table IX, p. 511] and K/K ◦ from [K, Column 4, Table 5, pp. 255-256].We have z ( k ) = 0 for the orbits, as given in (1), and these orbits are of type III. For the orbits,as given in (2), we have dim R z ( k ) = 2 and K/K ◦ = Id. Thus the orbits in (2) are of type I. For theorbits, as given in (3), we have dim R z ( k ) = 2 and K/K ◦ = Id, hence are of type II. For the orbits,as given in (4), we have dim R z ( k ) = 1 and K/K ◦ = Z . Thus this orbit is of type II. For the restof 16 orbits, which are not given in any of (1), (2), (3), (4), we have dim R z ( k ) = 1 and K/K ◦ = Id.Thus these orbits are of type I. Now the conclusions follow from Corollary 3.1. (cid:3) Nilpotent orbits in E − . There are 12 nonzero nilpotent orbits in E − ; see [D1, TableX, p. 512]. Note that in this case m ≃ so ⊕ R , and hence [ m , m ] = m . Theorem 3.7.
Let the parametrization of the nilpotent orbits be as in § X be a nonzeronilpotent element in E − . (1) If the parametrization of the orbit O X is given by − , then dim R H ( O X , R ) = 0 . (2) If O X is not given by the above parametrization ( dim R H ( O X , R ) ≤ . Proof.
For the Lie algebra E − , we can easily compute dim R z ( k ) from the last column of[D1, Table X, p. 512]. The orbit in (1) is of type III as z ( k ) = 0, and hence dim R H ( O X , R ) = 0.The other 11 orbits are of type II as dim R z ( k ) = 1 and m = [ m , m ]. Hence dim R H ( O X , R ) ≤ (cid:3) Nilpotent orbits in E − . There are two nonzero nilpotent orbits in E − ; see [D2, TableVII, p. 204]. Theorem 3.8.
For all the nilpotent element X in E − we have dim R H ( O X , R ) = 0 . N THE NILPOTENT ORBITS IN LIE ALGEBRAS 7
Proof.
As the theorem follows trivially when X = 0 we assume that X = 0. We follow theparametrization of the nilpotent orbits as given in § E − are of type III as z ( k ) = 0; see last column of [D2, Table VII, p. 204]. Hence, by Corollary3.1 (3) we conclude that dim R H ( O X , R ) = 0. (cid:3) Nilpotent orbits in non-compact real forms of E Recall that up to conjugation there are three non-compact real forms of E . They are denotedby E , E − and E − .3.6.1. Nilpotent orbits in E . There are 94 nonzero nilpotent orbits in E ; see [D1, Table XI,pp. 513-514]. Note that in this case we have m = [ m , m ]. Theorem 3.9.
Let the parametrization of the nilpotent orbits be as in § X be a nonzeronilpotent element in E . (1) If the parametrization of the orbit O X is given by , then dim R H ( O X , R ) = 3 . (2) Assume the parametrization of the orbit O X is given by any of the sequences : , , , , , , , , .Then dim R H ( O X , R ) = 2 . (3) Assume the parametrization of the orbit O X is given by any of the sequences : , , , , , , , , , , , , , , , , , , , , , , , .Then dim R H ( O X , R ) = 1 . (4) Assume the parametrization of the orbit O X is given by any of the sequences : , , , , , , , , , , . Then dim R H ( O X , R ) ≤ . (5) Assume the parametrization of the orbit O X is given by any of the sequences : , , , , , , , .Then dim R H ( O X , R ) ≤ . (6) If the parametrization of the orbit O X is given by either or , then dim R H ( O X , R ) ≤ . (7) If O X is not given by the parametrizations as in (1), (2), (3), (4), (5), (6) above ( dim R H ( O X , R ) = 0 . Proof.
For the Lie algebra E , we can easily compute dim R z ( k ) from the last column of [D1,Table XI, pp. 513-514] and K/K ◦ from [K, Column 4, Table 8, pp. 260-264].The orbit O X , as given in (1), is of type I as dim R z ( k ) = 3 and K/K ◦ = Id. For the orbits, asgiven in (2), we have dim R z ( k ) = 2 and K/K ◦ = Id. Hence these are also of type I. For the orbits,as given in (3), we have dim R z ( k ) = 1 and K/K ◦ = Id; hence they are of type I. For the orbits,as given in (4), we have dim R z ( k ) = 1 and K/K ◦ = Z . Thus these are of type II. For the orbits,as given in (5), we have dim R z ( k ) = 2 and K/K ◦ = Z . Hence these are also of type II. For theorbits, as given in (6), we have dim R z ( k ) = 3 and K/K ◦ = Id, hence they are of type II. Rest ofthe 39 orbits, which are not given by the parametrizations in (1), (2), (3), (4), (5), (6), are of typeIII as z ( k ) = 0. Now the results follow from Corollary 3.1. (cid:3) Nilpotent orbits in E − . There are 37 nonzero nilpotent orbits in E − ; see [D1, TableXII, p. 515]. Note that in this case m = [ m , m ]. P. CHATTERJEE AND C. MAITY
Theorem 3.10.
Let the parametrization of the nilpotent orbits be as in § X be a nonzeronilpotent element in E − . (1) If the parametrization of the orbit O X is given by either or , then dim R H ( O X , R ) = 2 . (2) Assume the parametrization of the orbit O X is given by any of the sequences : , , , , , , , , , , , . Then dim R H ( O X , R ) = 1 . (3) If the parametrization of the orbit O X is given by either or , then dim R H ( O X , R ) ≤ . (4) Assume the parametrization of the orbit O X is given by any of the sequences : , , , . Then dim R H ( O X , R ) ≤ . (5) If O X is not given by the parametrizations as in (1), (2), (3), (4) above ( dim R H ( O X , R ) = 0 . Proof.
For the Lie algebra E − , we can easily compute dim R z ( k ) from the last column of [D1,Table XII, pp. 515] and K/K ◦ from [K, Column 4, Table 9, pp. 266-268].For the orbit O X , as in (1), we have dim R z ( k ) = 2 and K/K ◦ = Id. Hence these orbits are oftype I. For the orbit O X , as in (2), we have dim R z ( k ) = 1 and K/K ◦ = Id. Hence these orbits arealso of type I. For the orbit O X , as in (3), we have dim R z ( k ) = 2 and K/K ◦ = Z , hence are oftype II. For the orbit O X , as in (4), we have dim R z ( k ) = 1 and K/K ◦ = Z . Hence these are alsoof type II. Rest of the 17 orbits, which are not given by the parametrizations in (1), (2), (3), (4),are of type III as z ( k ) = 0. Now the conclusions follow from Corollary 3.1. (cid:3) Nilpotent orbits in E − . There are 22 nonzero nilpotent orbits in E − ; see [D1, TableXIII, p. 516]. In this case we have m = [ m , m ]. Theorem 3.11.
Let the parametrization of the nilpotent orbits be as in § X be a nonzeronilpotent element in E − . (1) Assume the parametrization of the orbit O X is given by any of the sequences : , − , − , − , − , − , − , − , − , − . Then dim R H ( O X , R ) = 0 . (2) If O X is not given by any of the above parametrization ( ), then wehave dim R H ( O X , R ) ≤ . Proof.
Note that the parametrization of nilpotent orbits in E − as in [K, Table 10] is differentfrom [D1, Table X III, p. 516]. As the component group for all orbits in E − is Id; see [K,Column 4, Table 10, pp. 269-270], it does not depend on the parametrization. We refer to the lastcolumn of [D1, Table X III] for the orbits as given in (1). These are type III as z ( k ) = 0. For restof the 12 orbits we have dim R z ( k ) = 1; see last column of [D1, Table X III]. As m = [ m , m ], theseare of type II. Now the results follow from Corollary 3.1. (cid:3) Nilpotent orbits in non-compact real forms of E Recall that up to conjugation there are two non-compact real forms of E . They are denoted by E and E − .3.7.1. Nilpotent orbits in E . There are 115 nonzero nilpotent orbits in E ; see [D1, TableXIV, pp. 517-519]. Note that in this case we have m = [ m , m ]. N THE NILPOTENT ORBITS IN LIE ALGEBRAS 9
Theorem 3.12.
Let the parametrization of the nilpotent orbits be as in § X be a nonzeronilpotent element in E . (1) Assume the parametrization of the orbit O X is given by any of the sequences : , , , . Then dim R H ( O X , R ) = 2 . (2) Assume the parametrization of the orbit O X is given by any of the sequences : , , , , , , , , , , , , , , , , , , , , , , , , , . Then dim R H ( O X , R ) = 1 . (3) If the parametrization of the orbit O X is given , then dim R H ( O X , R ) ≤ . (4) Assume the parametrization of the orbit O X is given by any of the sequences : , , , , , , , .Then dim R H ( O X , R ) ≤ . (5) Assume the parametrization of the orbit O X is given by any of the sequences : , , , , , , , , , , , , , , , , , , , , , , , .Then dim R H ( O X , R ) ≤ . (6) If O X is not given by the parametrizations as in (1), (2), (3), (4), (5) above ( ), then we have dim R H ( O X , R ) = 0 . Proof.
For the Lie algebra E , we can easily compute dim R z ( k ) from the last column of [D1,Table XIV, pp. 517-519] and K/K ◦ from [K, Column 4, Table 11, pp. 271-275].For the orbits O X , as given in (1), we have dim R z ( k ) = 2 and K/K ◦ = Id. Hence these orbitsare of type I. For the orbits O X , as given in (2), we have dim R z ( k ) = 1 and K/K ◦ = Id. Hencethese orbits are also of type I. For the orbit O X , as given in (3), we have dim R z ( k ) = 3 and K/K ◦ = Id; hence they are of type II. For the orbits O X , as given in (4), we have dim R z ( k ) = 2and K/K ◦ = Id. Thus these orbits are of type II. For the orbits O X , as given in (5), we havedim R z ( k ) = 1 and K/K ◦ = Id. Hence these are of type II. Rest of the 52 orbits, which arenot given by the parametrizations of (1), (2), (3), (4), (5), are of type III as z ( k ) = 0. Now theconclusions follow from Corollary 3.1. (cid:3) Nilpotent orbits in E − . There are 36 nonzero nilpotent orbits in E − ; see [D1, TableXV, p. 520]. Note that in this case we have m = [ m , m ]. Theorem 3.13.
Let the parametrization of the nilpotent orbits be as in § X be a nonzeronilpotent element in E − . (1) Assume the parametrization of the orbit O X is given by any of the sequences : , , , , , , , , , , , , . Then dim R H ( O X , R ) = 1 . (2) If the parametrization of the orbit O X is given by either or , then dim R H ( O X , R ) ≤ . (3) If O X is not given by the parametrizations as in (1), (2) above ( dim R H ( O X , R ) = 0 . Proof.
For the Lie algebra E − , we can easily compute dim R z ( k ) from the last column of[D1, Table XV, p. 520] and K/K ◦ from [K, Column 4, Table 12, pp. 277-278]. For the orbits O X , as given in (1), we have dim R z ( k ) = 1 and K/K ◦ = Id, hence these are oftype I. For the orbits O X , as given in (2), we have dim R z ( k ) = 1 and K/K ◦ = Id. Hence theseorbits are of type II. Rest of the 21 orbits, which are not given by the parametrizations of (1), (2),are of type III as z ( k ) = 0. Now the conclusions follow from Corollary 3.1. (cid:3) Remark 3.14.
Here we make some observations about the first cohomology groups of the nilpotentorbits in non-compact non-complex real exceptional Lie algebras. To do this we begin by giving aconvenient description of the first cohomology groups of the nilpotent orbits. Following the set-upof Theorem 2.1 it can be shown thatdim R H ( O X , R ) = ( k + [ m , m ] $ m k + [ m , m ] = m . (3.1)The proof of the above result will appear elsewhere. As a consequences of (3.1), for all the nilpotentorbit O X in a simple Lie algebra g we have dim R H ( O X , R ) ≤
1. Recall that if g is a non-compactnon-complex real exceptional Lie algebra such that g E − and g E − then any maximalcompact subgroup of Int g is semisimple, and hence, using (3.1), it follows that dim R H ( O X , R ) = 0for all nilpotent orbit O X in g . We next assume g = E − or g = E − . Note that in boththe cases [ m , m ] $ m . We follow the parametrizations of the nilpotent orbits of g as given in [D1,Tables X, XIII]; see § g = E − we are able to conclude that dim R H ( O X , R ) = 1only for one orbit, namely, the orbit O X parametrized by 40000 −
2. In this case, from the lastcolumn and row 9 of [D1, Table X, p. 512] one has k = [ k , k ]. Thus k + [ m , m ] $ m , and (3.1)applies. For g = E − we obtain that dim R H ( O X , R ) = 1 when O X is parametrized by any ofthe following sequences : 000000 2, 000000 −
2, 000002 −
2, 200000 −
2, 200002 −
2, 400000 − −
6, 200002 −
6, 400004 −
6, 400004 −
10. For the above orbits, from the last column of [D1,Table XIII, p. 516] we have k = [ k , k ], and hence, using (3.1), analogous arguments apply. References [BC] I. Biswas and P. Chatterjee,
On the exactness of Kostant-Kirillov form and the second cohomology ofnilpotent orbits , Internat. J. Math. 23 (2012), no. 8, 1250086, 25 pp.[CM] D. H. Collingwood and W. M. McGovern,
Nilpotent orbits in semisimple Lie algebras , Van Nostrand Rein-hold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993.[D1] D.Z. Djokovic,
Classification of Nilpotent Elements in Simple Exceptional Real Lie Algebras of Inner Typeand Description of Their Centralizers , J. Alg. 112 (1988), 503-524.[D2] D.Z. Djokovic,
Classification of Nilpotent Elements in Simple Exceptional Real Lie Algebras E and E − and Description of Their Centralizers , J. Alg. 116 (1988), 196-207.[K] Donald King, The Component Groups of Nilpotents in Exceptional Simple Real Lie Algebras , Communica-tions in Algebra, 20(1), 219-284 (1992).[M] W. M. McGovern,
The adjoint representation and the adjoint action , in : Algebraic quotients. Torus actionsand cohomology. The adjoint representation and the adjoint action, 159–238, Encyclopaedia Math. Sci.,131, Springer, Berlin, 2002.
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