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(Bi)Parabolic Subalgebras in the Reductive Lie Algebras. Karin Baur, Anne Moreau

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(Bi)Parabolic Subalgebras in the Reductive Lie Algebras. Karin Baur, Anne Moreau Quasi-reductive (bi)parabolic subalgebras in the reductive Lie algebras. Karin Baur, Anne Moreau To cite this version: Karin Baur, Anne Moreau. Quasi-reductive (bi)parabolic subalgebras in the reductive Lie algebras.. 2008. hal-00348974v1 HAL Id: hal-00348974 https://hal.archives-ouvertes.fr/hal-00348974v1 Preprint submitted on 22 Dec 2008 (v1), last revised 30 Jun 2010 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. QUASI-REDUCTIVE (BI)PARABOLIC SUBALGEBRAS IN THE REDUCTIVE LIE ALGEBRAS. KARIN BAUR AND ANNE MOREAU Abstract. Let g be a finite dimensional Lie algebra, and z its center. We say that g is quasi- reductive if there is f ∈ g∗ such that g(f)/z is a reductive Lie algebra whose center consists of semisimple elements, where g(f) denotes the stabilizer of f in g for the coadjoint action. If g is reductive, then g is quasi-reductive itself, and also every Borel or Levi subalgebra of g. However the parabolic subalgebras of g are not always quasi-reductive (except in types A or C, see [P03]). Biparabolic (or seaweed) subalgebras are the intersection of two parabolic subalgebras whose sum is g. The classification of quasi-reductive biparabolic subalgebras in the classical case has been achieved recently in unpublished work of Duflo et al. ([DKT]). In this paper, we investigate the quasi-reductivity of biparabolic subalgebras of reductive Lie algebras. As a main result, we complete the classification of quasi-reductive parabolic subalgebras in the reductive Lie algebras. Introduction Let q be a finite dimensional Lie algebra over a field K of characteristic zero. Denote by Q the adjoint group of q. The group Q acts on q and on its dual q∗ by the adjoint and the coadjoint action. A classical problem is to describe the unitary dual Q of Q, i.e. the equivalence classes of unitary irreducible representations of Q. In this context,b the coadjoint orbits in q∗ play a fundamental role, as has been illustrated by A. Kirillov more than 40 years ago, [Ki68]. His so-called “orbit method” aims to relates the set Q to the set q∗/Q of coadjoint orbits. This correspondence is especially simple when Q is ab connected, simply connected, nilpotent Lie group. In this case Q and q∗/Q are naturally in bijection. But the orbit method does not work forb reductive groups in general; for such Q, the classification of Q is still an open problem. When Q is a real algebraic Lie group, there is a description of theb set Q2 of the equivalence classes of irreducible unitary representations of Q modulo the center Zbof Q which are square integrable: The elements of Q2 are parameterized by linear forms of compact type, [HC65], [HC66]. Thus, the set Q2 isb a nonempty set if and only there exist linear forms of compact type. A linear form f isb of compact type if the quotient Q(f)/Z of the stabilizer of f in Q by the center is compact. Classifying the set of linear forms of compact type is a difficult problem in general. The corresponding problem for complex algebraic Lie groups is more approachable, as we will see here. The complex analogues of the linear forms of compact type are the linear forms of reductive type. They are the linear forms on q whose quotient Q(f)/Z is a reductive subgroup of GL(q). 1991 Mathematics Subject Classification. 17B20, 17B45, 22E60. Key words and phrases. reductive Lie algebra, quasi-reductive Lie algebra, index, biparabolic Lie algebra, seaweed algebras, regular linear form. 1 The analogues of real Lie groups with a linear form of compact type are thus complex Lie groups with a linear form of reductive type. Such Lie groups and their Lie algebras are called quasi-reductive. This notion goes back to M. Duflo. He was motivated by applications in harmonic analysis, see [Du82], when he initiated the study of such Lie algebras. For a precise definition of linear forms of reductive type and of quasi-reductive Lie algebras we refer the reader to Section 1. Reductive Lie algebras are obviously quasi-reductive Lie algebras since in that case, 0 is a linear form of reductive type. Parabolic subalgebras can be viewed as the first class of non-reductive Lie algebras. More generally, the so-called biparabolic subalgebras form a very interesting class of non- reductive Lie algebras. They naturally extend the classes of parabolic subalgebras and of Levi subalgebras. The latter are clearly quasi-reductive since they are reductive subalgebras. Biparabolic subalgebras were introduced by V. Dergachev and A. Kirillov in the case q = sln, see [DK00]. A biparabolic subalgebra or seaweed subalgebra (of a semisimple Lie algebra) is the intersection of two parabolic subalgebras whose sum is the total Lie algebra. In this article, we are interested in the classification of quasi-reductive (bi)parabolic subalgebras. Note that it is enough to consider the case of (bi)parabolic subalgebras of the simple Lie algebras, cf. Remark 2.5. In the classical cases, various results are already known: All biparabolic subalgebras of sln and sp2n are quasi-reductive as has been proven by D. Panyushev in [P03]. The case of orthogonal Lie algebras is more complicated: On one hand, there are parabolic subalgebras of orthogonal Lie algebras which are not quasi-reductive, as P. Tauvel and R. Yu have shown (Section 3.2 of [TY04a]). On the other hand, D. Panyushev and A. Dvorsky exhibit many quasi-reductive parabolic subalgebras in [Dv03] and [P03] by constructing linear forms with the desired properties. Recently, M. Duflo, M. Khalgui and C. Torasso have obtained the classification of quasi-reductive biparabolic subalgebras of the orthogonal Lie algebras in unpublished work, [DKT]. They were able to characterize quasi-reductive parabolic subal- gebras in terms of the flags stabilized by the subalgebras. The main result of this paper is the completion of the classification of quasi-reductive par- abolic subalgebras of simple Lie algebras and thus of reductive Lie algebras (cf. Remark 2.5). This is done in Section 8. Our goal is ultimately to describe all quasi-reductive biparabolic subalgebras. Thus, in the first three sections we present results concerning biparabolic sub- algebras to remain in a general setting as far as possible. In fact, it is often better to deal with biparabolic subalgebras since they form a class of subalgebras stable under intersection. For the remainder of the introduction, let g be a complex semisimple Lie algebra. The paper is organized as follows: in Section 1 we give the definition and first properties of quasi-reductive Lie algebras. After that we include a short review of known results about biparabolic subalgebras (Section 2), including the description of quasi-reductive parabolic subalgebras in the orthogonal Lie algebras. Theorem 3.8 allows us to exhibit a large collection of quasi-reductive biparabolic subalgebras (Section 3). 2 Starting with Section 4, we concentrate on parabolic subalgebras. We first describe two methods of reduction, namely the transitivity result (Theorem 4.1) and the additivity property (Theorem 4.13). In Section 5 we characterize the case of certain “small” parabolic subalgebras (i.e. where the Levi factor has small dimension): Theorem 5.2 classifies the quasi-reductive parabolic subalgebras associated to a single root and Theorem 5.8 allows us to deal with the cases where the parabolic subalgebra is given by a simple a subsystem of rank 2. In Section 6 we discuss parabolic subalgebras obtained from a certain k-form of g where k is a subfield of C, cf. Theorem 6.2. They give rise to quasi-reductive parabolic subalgebras of g. Using a reduction process and the results from Sections 3, 5 and 6 we are able to deal with a large number of parabolic subalgebras of the exceptional Lie algebras. The remaining cases are dealt with in Section 7. As a conclusion, we are then able to list all parabolic subalgebras of the simple Lie algebras which are not quasi-reductive (see Table 6 and Subsection 8.2). This completes the classification of quasi-reductive parabolic subalgebras of g. Contents Introduction 1 1. Quasi-reductive Lie algebras 4 2. Biparabolic Lie algebras 4 3. Construction of quasi-reductive biparabolic subalgebras 12 4. Methods of reduction 19 5. Small parabolic subalgebras 27 6. Minimal parabolic subalgebras 33 7. The remaining cases 34 8. Classification 37 Appendix A. 39 References 40 Acknowledgment: We thank M. Duflo for introducing us to the subject of quasi-reductive subalgebras. We also thank P. Tauvel and R. Yu for useful discussions. We thank W. De Graaf and J. Draisma for helpful hints in the use of GAP. 3 1. Quasi-reductive Lie algebras Recall that q is a Lie algebra over a field K of characteristic 0 and that Q is the adjoint group of q. For a linear form f on q, we denote by Q(f) its stabilizer in Q and by Q.f its coadjoint orbit; q(f) is the Lie algebra of Q(f). Thus, q(f)= {x ∈ q | (ad∗x)(f)=0} , where ad∗ is the coadjoint representation of q defined by ad∗x : q∗ → q∗, f 7→ f([·, x]), for any x ∈ q.
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