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Journal of Algebra www.elsevier.com/locate/jalgebra

Hermitian real forms of contragredient Lie superalgebras Meng-Kiat Chuah a , Rita Fioresi b a

Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan b Dipartimento di Matematica, University of Bologna, Piazza Porta San Donato 5, 40126 Bologna, Italy

a r t i c l e

i n f o

Article history: Received 30 April 2014 Available online 25 May 2015 Communicated by Alberto Elduque MSC: 17B20 17B22

a b s t r a c t For a complex semisimple Lie algebra g with Hermitian real form gR = kR + pR , there exists a positive system of roots such that the adjoint k-representation on p stabilizes the positive and negative root spaces. In this article, we extend this result to contragredient Lie superalgebras g, and study the number of irreducible components of the k-representation. We also discuss the complex structure on gR /kR . © 2015 Elsevier Inc. All rights reserved.

Keywords: Hermitian real forms Contragredient Lie superalgebras

1. Introduction Symmetric spaces of semisimple Lie groups were originally introduced and then classiﬁed by É. Cartan. Harish-Chandra uncovered their importance and subsequently elucidated the representation theory of semisimple Lie groups. Each Harish-Chandra module is assumed to be of highest weight and is realized as the space of sections of a certain holomorphic vector bundle on the associated symmetric space. These symmetric spaces are then called Hermitian and they are singled out precisely by the property of E-mail addresses: [email protected] (M.-K. Chuah), rita.ﬁ[email protected] (R. Fioresi). http://dx.doi.org/10.1016/j.jalgebra.2015.04.018 0021-8693/© 2015 Elsevier Inc. All rights reserved.

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admitting an invariant complex structure. This complex structure is surprisingly linked to some inﬁnitesimal properties, namely the existence of an admissible system for the complex Lie algebra together with its Cartan decomposition corresponding to the given symmetric space [7]. Our paper represents the ﬁrst step towards a complete generalization to supergeometry of the Harish-Chandra picture. It must be clear that for a thorough treatment of this theory, much more than just the existence of an admissible system is required and the full ﬂedge of supergeometry together with a certain class of inﬁnite dimensional representations of Lie superalgebras must come into the play. This will be achieved in a series of papers to come [1]. A real semisimple Lie group GR is said to be Hermitian if its Riemannian symmetric space GR /KR has a GR -invariant complex structure such that the Riemannian structure is Hermitian. In this case we say that its Lie algebra gR is Hermitian. We drop the subscript R to denote complexiﬁcation, so g is the complexiﬁcation of gR . Hermitian Lie groups and Lie algebras play important roles in geometry, and their properties are well-studied. One property is that given a Cartan decomposition gR = kR + pR , we can choose a positive system of roots such that the adjoint k-representation on p stabilizes p± . In this article, we extend this result to contragredient Lie superalgebras g and study the number of irreducible components of the k-representation on p+ . We also discuss the complex structure of gR /kR , which has potential applications in complex supergeometry and representation theory of supergroups. We now explain our projects in more details. Let g = g¯0 + g¯1 be a complex contragredient Lie superalgebra which is not a Lie algebra. So g is one of sl(m, n), B(m, n), C(n), D(m, n), D(2, 1; α), F (4) and G(3) [8, 2.5]. Its real forms gR and their symmetric spaces have been classiﬁed and studied by M. Parker [10] and V. Serganova [11]. In Deﬁnition 3.3, we introduce Hermitian real forms gR = g¯0,R + g¯1,R of g. By (4.1), there is a corresponding Cartan decomposition gR = kR + pR . Here k ⊂ g¯0 and p = p¯0 + p¯1 , where p¯0 = p ∩ g¯0 and p¯1 = g¯1 . Also, k, g¯0 and g have equal rank, and they share a common Cartan subalgebra h. Given a positive system of roots on h, we write p = p+ + p− for the positive and negative root spaces of p. Theorem 1.1. There exists a positive system of roots such that [k, p± ] ⊂ p± and [p± , p± ] ⊂ p± . The positive system which satisﬁes Theorem 1.1 is called admissible. This is an extension of the classical case, whose admissible positive system satisﬁes [k, p± ] ⊂ p± and ¯ ¯ 0 0 ± ± [p¯0 , p¯0 ] = 0. We shall always use an admissible positive system. It implies that p+ ¯ 0 + and p+ are k-modules. In the classical case if g is simple, then p is k-irreducible. But ¯ 0 ¯ ¯ 1 0 here p+ is not k-irreducible in general. We next compute the number of its irreducible ¯ 1 subrepresentations. The contragredient Lie superalgebras are represented by distinguished extended Dynkin diagrams D, whose vertices α ∈ D are equipped with speciﬁc positive integers mα [2, Fig. 1]. Their Hermitian real forms are represented by diagrams obtained by circling certain vertices of D; these are known as Vogan superdiagrams [3, Def. 1.2].

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Fig. 1. Hermitian real forms of sl(m, n).

Fig. 2. Hermitian real forms of B(n, m) and D(n, m).

Fig. 3. Hermitian real forms of C(m).

Fig. 4. Hermitian real forms of D(2, 1; α).

Fig. 5. Hermitian real forms of F (4).

We list these Vogan superdiagrams in Figs. 1–6. For each diagram, we indicate g¯0,R and specify a vertex ϕ as the lowest root. Let |p+ | denote the number of irreducible ¯ 1 k − k-subrepresentations in p+ . We need not consider p because |p− | = |p+ |. ¯ ¯ ¯ ¯ 1 1 1 k 1 k Theorem 1.2. In the Vogan superdiagram of gR , let C denote the circled vertices in D\ϕ. Then + β∈C (mβ + 1) if C = ∅, |p¯1 |k = 1 if C = ∅.

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Fig. 6. Hermitian real form of G(3).

Theorem 1.2 allows us to compute |p+ | via Vogan superdiagrams. We apply it to ¯ 1 k Figs. 1–6 and obtain the following corollary. Corollary 1.3. The values of |p+ | are as follows. ¯ 1 k g sl(m, n) B(n, m), D(n, m)

C(m) D(2, 1; α) F (4) G(3)

g¯0,R su(p, m − p) ⊕ su(n) ⊕ iR su(p, m − p) ⊕ su(r, n − r) ⊕ iR sp(m, R) so(p) ⊕ sp(m, R) so∗ (2n) ⊕ sp(m) so(q, 2) ⊕ sp(m, R) sp(m, R) ⊕ so(2) sl(2, R) ⊕ sl(2, R) ⊕ sl(2, R) su(2) ⊕ su(2) ⊕ sl(2, R) sl(2, R) ⊕ so(7) su(2) ⊕ so(5, 2) sl(2, R) ⊕ gC

|p+ | ¯ 1 k 2 4 1 1 1 3 2 4 1 1 1 1

In the above table, each gR is determined by indicating g and g¯0,R [8, Prop. 5.3.2]. For B(n, m), we have p = 2n + 1 and q = 2n − 1. For D(n, m), we have p = 2n and q = 2n − 2. In the last entry, gC refers to the compact real form of G2 . Just as in the classical setting, we also have an invariant complex structure on gR/kR . Let g = h + Δ gα be the root space decomposition. Then b = h + Δ− gα is a Borel subalgebra of g. Theorem 1.4. There is a parabolic subalgebra b ⊂ q ⊂ g such that the natural map gR → g g/q leads to an isomorphism gR /kR ∼ = g/q. In this way gR /kR acquires a complex structure. The complex structure on gR /kR is invariant under the adjoint action and is very important from a geometrical point of view. In fact if one considers the supergroups GR and KR corresponding respectively to gR and kR (see [5,6] for their constructions and [4] for the compact form), then the complex structure on gR /kR endows GR /KR with an almost complex structure. It allows us to consider holomorphic vector bundles and to construct Harish-Chandra representations on the superspace of their sections. Ultimately this can lead to the unitary representations of GR , which occupy a preeminent role in the applications to physics. A compact subalgebra of gR is deﬁned by the ﬁxed point

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set kR = gθR of a Cartan automorphism θ (see (3.4)) as opposed to the topology of some supergroup. Every nonzero element of g¯1,R has order 4 under θ because the invariant form B of gR is skew-symmetric on g¯1,R , and we need −B(·, θ·) to be an inner product (see (3.3)). Consequently, in our setting the compact subalgebra kR always lies in the even part. A related setting are the supersymmetric spaces introduced by Serganova [11] as the quotients G/Gϕ of the real supergroup G by its closed subgroup Gϕ , where ϕ has order 2 or 4. Hermitian supersymmetric spaces are then deﬁned as those admitting invariant complex structures. Our notion of Hermitian real forms concerns (g, gθ ) which carries complex structure, whereas [11] works out all ϕ of order 2 or 4 and does not single out those G/Gϕ with complex structure. Consequently, the list in [11] is longer than ours. We are concerned with the existence of admissible systems, and we take a more conceptual viewpoint in solving this problem, essentially considering only the root system properties and not the concrete realizations of the Lie superalgebras. The sections in this article are organized as follows. In Section 2, we recall some standard properties of complex semisimple Lie algebras and their real forms. In Section 3, we introduce the Hermitian real forms of contragredient Lie superalgebras, and study the corresponding Cartan automorphisms and Vogan superdiagrams. In Section 4, we prove Theorem 1.1. In Section 5, we prove Theorem 1.2 and Corollary 1.3. In Section 6, we prove Theorem 1.4. Finally in Section 7, we provide an example to illustrate these theorems. 2. Semisimple Lie algebras In this section, we review some basic facts on complex semisimple Lie algebras, with focus on their Hermitian real forms. Let s be a complex semisimple Lie algebra, and inv(s) denotes involutions on s. We agree that involutions have to be C-linear and not conjugate linear. Also, we include the identity map as an involution. Let B be the Killing form of sR . An involution θ on a real form sR of s is called a Cartan involution if −B(·, θ·) is an inner product on sR . It leads to the Cartan decomposition sR = kR + pR , where kR and pR are the ±1-eigenspaces of θ. We drop the subscript R to denote complexiﬁcation, so s = k + p. We extend θ uniquely to s by complex linearity, and let θ denote the involution on sR as well as s. We shall identify involutions with real forms. For this purpose, we do not distinguish s-involutions which are conjugate by an s-automorphism, and we do not distinguish isomorphic real forms of s. This gives a bijective correspondence {real forms sR ⊂ s} ←→ {θ ∈ inv(s)}.

(2.1)

In this correspondence θ stabilizes sR , and the restriction of θ to sR is a Cartan involution.

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Deﬁnition 2.1. If s is simple, we say that sR or θ is Hermitian if the adjoint k-representation on p has two irreducible components, or equivalently if k has a 1-dimensional center. If s has multiple simple ideals, we say that sR or θ is Hermitian if each ideal of sR is compact or Hermitian. The above terminology is motivated by the fact that if SR is a Lie group whose Lie algebra is sR , then sR is Hermitian if and only if the Riemannian symmetric space SR /KR has an SR -invariant complex structure such that the Riemannian structure is Hermitian. Let θ ∈ inv(s) be Hermitian. There exists a Cartan subalgebra h of s such that θ acts as identity on h. So h is also a Cartan subalgebra of k = sθ . Let Δ ⊂ h∗ be the root system, and let s = h + Δ sα be the root space decomposition. Since θ acts as identity on h, it stabilizes each root space sα . A choice of simple system of Δ leads to the Dynkin diagram Ds of s. We circle some vertices α ∈ Ds so that α

not circled if θ = 1 on sα , circled if θ = −1 on sα .

(2.2)

This circling gives a diagram called the Vogan diagram of θ or sR [9, VI-8]. We use the circling instead of painting (i.e. white and black vertices) because the black vertices will denote some odd roots later. The general deﬁnition of Vogan diagram involves diagram involutions on Ds , but here our diagram involution is trivial because θ stabilizes each root space. Given a positive system on Δ, we write Δ± for the positive and negative roots. Since h is a Cartan subalgebra of both k and s, its root system consists of the disjoint union ± ± Δ = Δk ∪ Δp . Let Δ± sα . p = Δp ∩ Δ , and let p = Δ± p Each connected component C of Ds is the Dynkin diagram of a simple ideal of s. Let {nα }C be positive integers such that

nα α is the highest root with respect to C.

(2.3)

C

An involution may be represented by distinct Vogan diagrams due to distinct choices of simple systems. Since θ is Hermitian, it can be represented by a Vogan diagram such that each connected component C of Ds has at most one circled vertex β, and nβ = 1. For such Vogan diagram, (a)

[k, p± ] ⊂ p± ,

(b)

[p± , p± ] = 0.

(2.4)

The positive system which satisﬁes (2.4) is called admissible, and it always exists. For + a given Δ+ k , the admissible positive system containing Δk is unique up to inverting

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+ positivity of noncompact roots of ideals of g. Namely if Δ+ k ∪ Δp is admissible and kj + pj is an ideal of g, then another admissible positive system is obtained by replacing − + Δ+ pj with Δpj ; and this method exhausts all admissible positive systems containing Δk . From now on, we ﬁx an admissible positive system.

Example 2.2. Consider the real form sR = su(2, 1) of s = sl(3, C). Write Δ = {±α, ±β, ±(α + β)}, where Δk = {±α}. Consider the two positive systems (a)

− Δ+ = {α, β, α + β} , Δ+ p = {β, α + β} , Δp = {−β, −α − β},

(b)

− Δ+ = {−β, α + β, α} , Δ+ p = {−β, α + β} , Δp = {β, −α − β}.

(2.5)

The positive system in (2.5)(a) is admissible because Δ± p satisfy (2.4). Its Vogan diagram has one circled vertex β. The positive system in (2.5)(b) is not admissible because Δ± p do not satisfy (2.4). Its Vogan diagram has two circled vertices −β, α + β. Let sR be Hermitian. Write sR = cR +

r

sj,R ,

1

where cR is a compact ideal and sj,R = kj,R +pj,R are simple Hermitian ideals. A maximal r compact subalgebra of sR is kR = cR + 1 kj,R . Proposition 2.3. For any a1 , . . . , ar ∈ R, there exists iξ ∈ hR such that: (a) α(ξ) = 0 for all α ∈ Δk , (b) β(ξ) = aj for all β ∈ Δ+ pj . Proof. By Deﬁnition 2.1, each kj has a 1-dimensional center z(kj ). Hence Δkj (ξj ) = 0 for + all ξj ∈ z(kj ). Let βj be the lowest root of Δ+ pj . Since pj is an irreducible kj -module, all + roots in Δpj are of the form βj + α, where α is a sum of kj -roots. So for all ξj ∈ z(kj ), + Δ+ pj (ξj ) is a constant. The center of sj is trivial, so Δpj does not annihilate z(kj ). Hence for any c ∈ C, there exists ξj ∈ z(kj ) such that Δ+ pj (ξj ) = c.

(2.6)

Since hR is compact, α(ξ) is real-valued for all iξ ∈ hR and α ∈ Δ. So for c = aj ∈ R, r we can pick iξj ∈ hR ∩ z(kj ) such that (2.6) holds. Let ξ = 1 ξj , and the proposition follows. 2 In the following sections, we study complex contragredient Lie superalgebras g = g¯0 + g¯1 . The commutator subalgebra gss = [g¯0 , g¯0 ] is semisimple, so our results in this ¯ 0 section can be applied to gss . ¯ 0

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3. Hermitian real forms Let g = g¯0 + g¯1 be a contragredient Lie superalgebra that is not a Lie algebra. Let aut2,4 (·) denote automorphisms of order 2 on the even part and order 4 on the odd part. The main result of [3] is that, up to equivalence, there exist bijective correspondences real forms of g ←→ aut2,4 (g) ←→ Vogan superdiagrams.

(3.1)

In this section, we introduce Hermitian real forms of g, and study the corresponding automorphisms and Vogan superdiagrams under (3.1). The resulting Vogan superdiagrams are listed in Figs. 1–6. = [g¯0 , g¯0 ] be the commutator subalgebra, and let z(g¯0 ) be the center of g¯0 . Let gss ¯ 0 Here g¯0 is reductive, so gss is semisimple and ¯ 0 g¯0 = gss ¯ 0 ). 0 + z(g¯ Let h be a Cartan subalgebra of g with root system Δ ⊂ h∗ and root space decomposition g = h + Δ gα . Given a simple system Π ⊂ Δ, let ϕ denote the corresponding lowest root. This leads to the extended Dynkin diagram D. Its vertices represent Π ∪ ϕ with colors white, grey or black, together with edges drawn according to [8, p. 54–55]. Let D¯0 denote the subdiagram formed by the white vertices, and let D¯1 be the dark (i.e. grey or black) vertices. There are distinct D due to the choice of Π, but the following theorem picks out a preferred one. Theorem 3.1. [2, Thm. 1.1] There exists an extended Dynkin diagram D such that (a) D¯0 is the Dynkin diagram of gss , ¯ 0 (b) |D¯1 | − 1 = dim z(g¯0 ), (c) D¯1 are the lowest roots of the adjoint g¯0 -representation on g¯1 . We always choose D that satisﬁes Theorem 3.1. There are unique positive integers {mα }D without nontrivial common factor such that

mα α = 0.

(3.2)

D

The list of D together with mα are given in [2, Fig. 1]. They satisfy D¯1 mγ = 2. This property and Theorem 3.1 divide the contragredient Lie superalgebras into the following two types.

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Deﬁnition 3.2. (a) We say that g is of Type 1 if D¯1 = {γ1 , γ2 } and mγ1 = mγ2 = 1. Here g¯0 has a 1-dimensional center and its adjoint representation on g¯1 has two irreducible components. (b) We say that g is of Type 2 if D¯1 = {γ} and mγ = 2. Here g¯0 is semisimple and its adjoint representation on g¯1 is irreducible. Thus sl(m, n) and C(n) are of Type 1, while the remaining contragredient Lie superalgebras are of Type 2. There is a nondegenerate invariant bilinear form B on g which is symmetric on g¯0 and skew-symmetric on g¯1 [8, §2.5]. Let gR be a real form of g. Given θ ∈ aut2,4 (gR ), we deﬁne Bθ (X, Y ) = −B(X, θY ).

(3.3)

If Bθ is an inner product on gR , we say that θ is a Cartan automorphism of gR . This generalizes the Cartan involution of a real semisimple Lie algebra. Just like (2.1), we do not distinguish g-involutions which are conjugate by a g-automorphism, and we do not distinguish isomorphic real forms of g. Then (2.1) extends to a bijective correspondence {real forms gR ⊂ g} ←→ {θ ∈ aut2,4 (g)}

(3.4)

[3, Thm. 1.1]. In this correspondence θ stabilizes gR , and the restriction of θ to gR is a Cartan automorphism. Since θ has order 2 on g¯0,R , we have the decomposition g¯0,R = kR + p¯0,R ,

(3.5)

where kR and p¯0,R are the ±1-eigenspaces of θ on g¯0,R . Deﬁnition 3.3. We say that gR is Hermitian if (a) gss is Hermitian, and ¯ 0,R (b) rank k = rank g¯0 . Actually the equal rank condition in Deﬁnition 3.3(b) is automatic in most cases. We shall discuss its signiﬁcance in Remark 3.4. We claim that the Hermitian real forms of g correspond to θ¯0 ∈ inv(g¯0 ) such that (a)

the restriction of θ¯0 to gss ¯ 0 is Hermitian,

(b)

θ¯0 = 1 on z(g¯0 ),

(c)

θ¯0 extends to aut2,4 (g).

(3.6)

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In (3.4), we can identify the real forms with θ¯0 ∈ inv(g¯0 ) which satisfy (3.6)(c). This is because if θ¯0 has several extensions to aut2,4 (g), then the corresponding real forms are equivalent [8, Prop. 5.3.2]. Clearly Deﬁnition 3.3(a) and (3.6)(a) are equivalent. Suppose that they hold. There exists a Cartan subalgebra h of gss such that θ¯0 = 1 on h , namely h ⊂ k. Then ¯ 0 h + z(g¯0 ) is a Cartan subalgebra of g¯0 . So Deﬁnition 3.3(b) holds if and only if z(g¯0 ) ⊂ k, which is equivalent to (3.6)(b). We conclude that the Hermitian real forms correspond to θ¯0 ∈ inv(g¯0 ) which satisfy (3.6), as claimed. Let us interpret (3.6) diagrammatically by the notion of Vogan superdiagrams. A Vogan superdiagram on D is a circling on the white vertices D¯0 together with a colorpreserving diagram symmetry on D, such that additional conditions are satisﬁed (see [3, Def. 1.2]). The Vogan superdiagrams represent real forms of g, or equivalently aut2,4 (g) [3, Thm. 1.3]. So they correspond to θ¯0 ∈ inv(g¯0 ) which satisfy (3.6)(c). Let nα be the integers in (2.3). By the discussion in Section 2, if θ¯0 is Hermitian on gss , the circling on D¯0 can be chosen such that ¯ 0 at most one vertex β on each connected component of D¯0 is circled, and nβ = 1.

(3.7)

It amounts to an admissible positive system on the even roots, namely (2.4) holds for g¯0 = k + p¯0 . So (3.6)(a) is equivalent to the existence of the circling (3.7). Condition (3.6)(b) is equivalent to the diagram symmetry of D acts trivially on the dark vertices.

(3.8)

If g is of Type 2, then z(g¯0 ) = 0, and D has only one dark vertex (Theorem 3.1(b)), so the color-preserving diagram symmetry has to ﬁx it. So (3.6)(b) and (3.8) are vacuous for g of Type 2. We conclude that Hermitian real forms of g correspond to the Vogan superdiagrams which satisfy (3.7) and (3.8). All Vogan superdiagrams are listed in [3, Figs. 2–7]. We select the ones that satisfy (3.7) and (3.8), and present them in Figs. 1–6. We also indicate g¯0,R with the ﬁgures. In Fig. 6, gC refers to the compact real form of G2 . There can be distinct simple systems Π, Π such that Π ∪ ϕ = Π ∪ ϕ = D. In fact any α ∈ D with mα = 1 can be the lowest root ϕ of a simple system. In Figs. 1–6, we specify our preferred simple system by indicating ϕ. Our choice of ϕ satisﬁes: (a)

For g of Type 1, ϕ is grey.

(b)

For g of Type 2, ϕ is white and circled.

(3.9)

We end this section by a discussion on the signiﬁcance of Deﬁnition 3.3(b). Remark 3.4. The rank of k. Suppose that a real form gR satisﬁes Deﬁnition 3.3(a) but not Deﬁnition 3.3(b). It corresponds to θ¯0 which satisﬁes (3.6)(a, c) but not (3.6)(b). So

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Fig. 7. Real forms that satisfy Deﬁnition 3.3(a) but not Deﬁnition 3.3(b). g = sl(2, 2).

it also corresponds to a Vogan superdiagram which satisﬁes (3.7) but not (3.8). The only such Vogan superdiagrams are given by Fig. 7(a, b). Thus Deﬁnition 3.3(a) automatically implies Deﬁnition 3.3(b) except for the two real forms in Fig. 7(a, b). 4. Admissible positive systems In this section, we prove Theorem 1.1. Let g = g¯0 + g¯1 be a contragredient Lie superalgebra. Let gR = g¯0,R + g¯1,R be a Hermitian real form of g in the sense of Deﬁnition 3.3. Let p¯1,R = g¯1,R . By (3.5), gR = kR + p¯0,R + p¯1,R = kR + pR ,

(4.1)

where pR = p¯0,R + p¯1,R . By Deﬁnition 3.3(b), kR , g¯0,R and gR have equal rank, namely they share a common Cartan subalgebra hR . Its root system consists of the disjoint unions Δ = Δk ∪ Δp = Δk ∪ Δp¯0 ∪ Δp¯1 . The roots α satisfy α(hR ) ⊂ iR. ± ± Given a positive system on Δ, we deﬁne Δ± p , Δp¯0 and Δp¯1 accordingly, and their ± ± ± root spaces form p , p¯0 and p¯1 . We choose the positive system so that the lowest root ϕ is indicated in Figs. 1–6, and hence satisﬁes (3.9). Each connected component of D¯0 has at most one circled vertex, and it gives an admissible positive system on gss in the ¯ 0 sense of (2.4). By Deﬁnition 3.2, g is of Type 1 or 2. The next two propositions provide separate discussions for these two types. Proposition 4.1. For g of Type 1, there exists iξ ∈ hR such that (a) α(ξ) = 0 for all α ∈ Δk , (b) α(ξ) > 0 for all α ∈ Δ+ p. Proof. The cases for g of Type 1 occur in Figs. 1 and 3. Here the lowest root ϕ is grey. Let γ be the other grey vertex, and let βj be the circled white vertices in D. Hence: βj are the lowest roots of p+ , ¯ 0 γ is the lowest root of p+ , ¯ 1 ϕ is the lowest root of p− and g. ¯ 1

(4.2)

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Here gss is a semisimple Lie algebra with Cartan subalgebra h ∩gss . By Proposition 2.3, ¯ ¯ 0 0 ss there exists iξ ∈ hR ∩ g¯0,R such that Δk (ξ ) = 0 , βj (ξ ) = 1.

(4.3)

Since g is of Type 1, dimC z(g¯0 ) = 1. Let iξ satisfy iξ ∈ z(g¯0,R ) , γ(ξ ) > |γ(ξ )|.

(4.4)

Let ξ = ξ + ξ . By (4.3) and (4.4), Δk (ξ) = 0 , βj (ξ) = βj (ξ ) = 1 , γ(ξ) > 0.

(4.5)

The proposition follows from (4.2) and (4.5). 2 Proposition 4.2. For g of Type 2, there exists iξ ∈ hR such that (a) α(ξ) = 0 for all α ∈ Δk , (b) α(ξ) > 0 for all α ∈ Δ+ p. Proof. The cases for g of Type 2 occur in Figs. 2, 4, 5 and 6. Here the lowest root ϕ is a circled white vertex. Let βj be the remaining circled white vertices (possibly empty), and let γ be the unique dark vertex. Hence: βj are the lowest roots of p+ , ¯ 0 , γ is the lowest root of p+ ¯ 1 and g. ϕ is the lowest root of p− ¯ 0

(4.6)

Here g¯0 is a semisimple Lie algebra with Cartan subalgebra h. By Proposition 2.3, there exists iξ ∈ hR such that Δk (ξ) = 0 , βj (ξ) = 1 , ϕ(ξ) 0.

(4.7)

Let mβ be the integers in (3.2). In (4.7), it suﬃces to take ϕ(ξ) <

− 0

j

mβj

if {βj } = ∅, if {βj } = ∅.

We have 0 = ( D mα α)(ξ) = 2γ(ξ) + ϕ(ξ) + j mβj , so γ(ξ) > 0. The proposition follows from (4.6), (4.7) and (4.8). 2

(4.8)

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Proof of Theorem 1.1. Let ξ be given by Propositions 4.1 and 4.2, so that − Δk (ξ) = 0 , Δ+ p (ξ) > 0 , Δp (ξ) < 0.

(4.9)

Let α ∈ Δk and β ∈ Δ+ p . By (4.9), (α + β)(ξ) = β(ξ) > 0, and therefore + α ∈ Δk , β ∈ Δ+ / Δ. p =⇒ α + β ∈ Δp or α + β ∈

(4.10)

By (4.10), [k, p+ ] ⊂ p+ . Similarly, [k, p− ] ⊂ p− . This proves the ﬁrst assertion of Theorem 1.1. Next we want to show that + β, δ ∈ Δ+ / Δ. p =⇒ β + δ ∈ Δp or β + δ ∈

(4.11)

We check (4.11) for the following three cases, (a)

β, δ ∈ Δ+ p¯0 ,

(b)

+ β ∈ Δ+ p¯0 , δ ∈ Δp¯1 ,

(c)

β, δ ∈ Δ+ p¯1 .

(4.12)

In (4.12)(a), we have (β + δ)(ξ) > 0 by (4.9). Since [p¯0 , p¯0 ] ⊂ k and Δk (ξ) = 0, it follows that β + δ ∈ / Δ and (4.11) is satisﬁed. In (4.12)(b), we have [p+ , p+ ] ⊂ p+ so (4.11) is satisﬁed. ¯ ¯ ¯ 0 1 1 In (4.12)(c), β + δ cannot be a k-root because (β + δ)(ξ) > 0 and Δk (ξ) = 0 by (4.9). So either β + δ ∈ Δ+ / Δ, and (4.11) is satisﬁed. p¯0 or β + δ ∈ This proves (4.11) for all cases. It implies that [p+ , p+ ] ⊂ p+ . Similarly, [p− , p− ] ⊂ p− . This completes the proof of Theorem 1.1. 2 5. Irreducible k-subrepresentations in p+ ¯ 1 In this section, we study the adjoint k-representation on p+ , and show that the number ¯ 1 of its irreducible subrepresentations is given by Theorem 1.2. Proof of Theorem 1.2 and Corollary 1.3. Let γ, ϕ carry the same meaning as (4.2) and (4.6), namely γ is the lowest root of p+ , and ϕ is the lowest root of g. Let C denote ¯ 1 the circled vertices in D\ϕ; it may be empty. A p+ -root can be uniquely written as ¯ 1 γ+

β∈C

bβ β +

aα α ∈ Δ+ p¯1 ,

(5.1)

α∈D\{γ,ϕ,C}

where 0 ≤ bβ ≤ mβ and 0 ≤ aα . Here β are the lowest roots of p+ , and α ∈ Δ+ ¯ k . Let 0 p+ = ¯ 1

j

Vj

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be a decomposition such that each Vj is a sum of root spaces, and gμ , gν lie in the same Vj if and only if the expressions of μ, ν in (5.1) have the same bβ for all β ∈ C. This is equivalent to the condition that ν − μ is a sum of k-roots. It follows that each Vj is an irreducible k-module. Therefore, the number of irreducible k-subrepresentations in p+ is ¯ 1 the number of possibilities of {bβ }C . If C = ∅, then each β ∈ C has mβ + 1 possible values 0, 1, . . . , mβ , so there are β∈C (mβ + 1) possibilities of {bβ }C . If C = ∅, then (5.1) becomes γ + D\{γ,ϕ,C} aα α, + and p¯1 is an irreducible k-module. This proves Theorem 1.2. Corollary 1.3 is obtained by checking the number of possibilities of {bβ }β∈C for all Vogan superdiagrams in Figs. 1–6. 2 6. Complex structure In this section, we prove Theorem 1.4, which equips gR /kR with a complex structure. We keep the same positive system Δ+ used in the previous section. Let n = Δ− gα . We have the Iwasawa decomposition g = gR + ihR + n.

(6.1)

Then b = h + n is a Borel subalgebra of g. Let ξ ∈ h, and let Δξ = {α ∈ Δ ; α(ξ) = 0}. Then a parabolic subalgebra of g is q=b+

gα .

(6.2)

Δ+ ξ

Proof of Theorem 1.4. Given ξ ∈ hR , we deﬁne gξR = {X ∈ gR , [X, ξ] = 0}. Then gξR = gR ∩ (h +

gα ).

(6.3)

Δξ

For the ξ given by Propositions 4.1 and 4.2, we have gξR = kR . Hence gR /kR = gR /(gR ∩ (h + Δξ gα )) by (6.3) = g/(ihR + n + hR + Δ+ gα ) by (6.1) ξ

= g/q.

by (6.2)

This proves Theorem 1.4. 2 7. Example In this section, we study an example to illustrate the theorems of this article. We consider the real form gR of g = D(4, 2), where g¯0,R = so(6, 2) ⊕ sp(2, R). Fig. 8 shows

M.-K. Chuah, R. Fioresi / Journal of Algebra 437 (2015) 161–176

175

Fig. 8. Example of a Hermitian real form of g = D(4, 2).

its Vogan superdiagram together with the integers mα of (3.2). The vertices β, γ, ϕ carry the same meaning as (4.6): β is the lowest root of p+ , ¯ 0 , γ is the lowest root of p+ ¯ 1 − ϕ is the lowest root of p¯0 and g. We ﬁrst illustrate Theorem 1.2 and Corollary 1.3. Let |p+ | be the number of irre¯ 1 k + . By Theorem 1.2, |p | = mβ + 1 = 3, and this is ducible k-subrepresentations in p+ k ¯ ¯ 1 1 indeed given in Corollary 1.3. Let αi be the four uncircled white vertices in Fig. 8. They belong to Δk . A root of p+ ¯ 1 is of the form γ + bβ β +

aj αj ∈ Δ+ p¯1 ,

(7.1)

where bβ = 0, 1, 2 and aj ≥ 0. We have p+ = V0 + V1 + V2 , where Vj are the root spaces ¯ 1 of the roots in (7.1) with bβ = j. Then two root spaces gμ , gν lie in the same Vi if and only if ν − μ is a sum of k-roots. Therefore, each Vj is an irreducible k-subrepresentation. This explains |p+ | = 3. ¯ 1 k ] ⊂ p+ and [p+ , p+ ] ⊂ p+ . Suppose Next we illustrate Theorem 1.1, in particular [k, p+ ¯ ¯ ¯ ¯ ¯ 1 1 1 1 0 + that δ is the sum of a k-root bj αj and a p¯1 -root of (7.1), namely δ = γ + bβ β +

(aj + bj )αj .

Here {γ, β, αj } is a simple system of g. So if δ is a root, then δ ∈ Δ+ p¯1 because γ has coeﬃcient 1 in δ. We have shown that the sum of a k-root and a p+ -root is either a ¯ 1 + + + p¯1 -root or not a root. This implies that [k, p¯1 ] ⊂ p¯1 . , p+ ] ⊂ p+ by this example. Let μ be a sum of two p+ -roots. Next we explain [p+ ¯ ¯ ¯ ¯ 1 1 0 1 By (7.1), μ is of the form μ = 2γ + bβ β + where bβ , aj ≥ 0. Since

D

aj αj ,

(7.2)

mα α = 0, (7.2) becomes

mj αj ) + bβ β + aj αj = −ϕ + (bβ − 2)β + (aj − mj )αj .

μ = (−ϕ − 2β −

(7.3)

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Here {ϕ, β, αj } is a simple system of g¯0 . So if μ is a root, then the term −ϕ ∈ Δ+ p¯0 in + the last expression of (7.3) implies that μ ∈ Δp¯0 . We have shown that the sum of two p+ -roots is either a p+ -root or not a root. This implies that [p+ , p+ ] ⊂ p+ . ¯ ¯ ¯ ¯ ¯ 1 0 1 1 0 Acknowledgments We thank Prof. V.S. Varadarajan for suggesting the problem and providing comments, and Prof. V. Serganova for helpful discussions. The Referee pointed out important motivations and background references, and consequently improved its presentation. M.K. Chuah is supported by a grant from the Ministry of Science and Technology, Taiwan, and thanks the Department of Mathematics at the University of Bologna for the wonderful hospitality. R. Fioresi similarly thanks the NTHU Department of Mathematics for the great hospitality. References [1] C. Carmeli, R. Fioresi, V.S. Varadarajan, Highest weight Harish-Chandra supermodules and their geometric realizations I. The inﬁnitesimal theory, arXiv:1503.03828, 2015. [2] M.K. Chuah, Finite order automorphisms on contragredient Lie superalgebras, J. Algebra 351 (2012) 138–159. [3] M.K. Chuah, Cartan automorphisms and Vogan superdiagrams, Math. Z. 273 (2013) 793–800. [4] R. Fioresi, Compact forms of complex Lie supergroups, J. Pure Appl. Algebra 218 (2014) 228–236. [5] R. Fioresi, F. Gavarini, On the construction of Chevalley supergroups, in: S. Ferrara, R. Fioresi, V.S. Varadarajan (Eds.), Supersymmetry in Mathematics and Physics, UCLA, Los Angeles, USA 2010, in: Lecture Notes in Math., vol. 2027, Springer-Verlag, Berlin–Heidelberg, 2011. [6] R. Fioresi, F. Gavarini, Chevalley Supergroups, Memoirs Amer. Math. Soc., vol. 215, 2012. [7] Harish-Chandra, Representations of semisimple Lie groups IV, Amer. J. Math. 77 (1955) 743–777. [8] V.G. Kac, Lie superalgebras, Adv. Math. 26 (1977) 8–96. [9] A.W. Knapp, Lie Groups Beyond an Introduction, 2nd. ed., Progr. Math., vol. 140, Birkhäuser, Boston, 2002. [10] M. Parker, Classiﬁcation of real simple Lie superalgebras of classical type, J. Math. Phys. 21 (1980) 689–697. [11] V. Serganova, Classiﬁcation of real simple Lie superalgebras and symmetric superspaces, Funct. Anal. Appl. 17 (1983) 200–207.

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