L’INSTITUTFOURIER DE ANNALES

(1)

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Karin BAUR & Anne MOREAU

Quasi-reductive (bi)parabolic subalgebras in reductive Lie algebras.

Tome 61, n^o2 (2011), p. 417-451.

<http://aif.cedram.org/item?id=AIF_2011__61_2_417_0>

L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

cedram

Article mis en ligne dans le cadre du

(2)

QUASI-REDUCTIVE (BI)PARABOLIC SUBALGEBRAS IN REDUCTIVE LIE ALGEBRAS.

by Karin BAUR & Anne MOREAU (*)

Abstract. — We say that a finite dimensional Lie algebra is quasi-reductive if it has a linear form whose stabilizer for the coadjoint representation, modulo the center, is a reductive Lie algebra with a center consisting of semisimple elements.

Parabolic subalgebras of a semisimple Lie algebra are not always quasi-reductive (except in types A or C by work of Panyushev). The classification of quasi-reductive parabolic subalgebras in the classical case has been recently achieved in unpublished work of Duflo, Khalgui and Torasso. In this paper, we investigate the quasi- reductivity of biparabolic subalgebras of reductive Lie algebras. Biparabolic (or seaweed) subalgebras are the intersection of two parabolic subalgebras whose sum is the total Lie algebra. As a main result, we complete the classification of quasi- reductive parabolic subalgebras of reductive Lie algebras by considering the exceptional cases.

Résumé. — Une algèbre de Lie de dimension finie est dite quasi-réductive si elle possède une forme linéaire dont le stablisateur pour la représentation coadjointe, modulo le centre, est une algèbre de Lie réductive avec un centre formé d’éléments semi-simples. Les sous-algèbres paraboliques d’une algèbre de Lie semi-simple ne sont pas toujours quasi-réductives (sauf en types A ou C d’après un résultat de Pa- nyushev). Récemment, Duflo, Khalgui and Torasso ont terminé la classification des sous-algèbres paraboliques quasi-réductives dans le cas classique. Dans cet article nous étudions la quasi-réductivité des sous-algèbres biparaboliques des algèbres de Lie réductives. Les sous-algèbres biparaboliques sont les intersections de deux sous-algèbres paraboliques dont la somme est l’algèbre de Lie ambiante. Notre prin- cipal résultat est la complétion de la classification des sous-algèbres paraboliques quasi-réductives des algèbres de Lie réductives.

Keywords:Reductive Lie algebras, quasi-reductive Lie algebras, index, biparabolic Lie algebras, seaweed algebras, regular linear forms.

Math. classification:17B20, 17B45, 22E60.

(*) We thank M. Duflo for introducing us to the subject of quasi-reductive subalgebras.

We also thank P. Tauvel and R.W.T. Yu for useful discussions. Furthermore, we thank W. de Graaf and J. Draisma for helpful hints in the use ofGAP. At this point, we also want to thank the referee for the very useful comments and suggestions. This work was supported by the Swiss National Science Foundation (grant PP0022-114794).

(3)

Introduction

LetGbe a complex connected linear algebraic Lie group. Denote bygits Lie algebra. The groupGacts on the dualg^∗ ofgby the coadjoint action.

Forf ∈g^∗, we denote by G(f) its stabilizer inG; it always contains the centerZofG. One says that a linear formf ∈g^∗hasreductive type if the quotient G(f)/Z is a reductive subgroup of GL(g). The Lie algebra g is calledquasi-reductive if it has linear forms of reductive type. This notion goes back to M. Duflo. He initiated the study of such Lie algebras because of applications in harmonic analysis, see [3]. For more details about linear forms of reductive type and 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. 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 g = sl_n, see [2]. 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 1.4.

In the classical cases, various results are already known: All biparabolic subalgebras ofsl_nandsp_2nare quasi-reductive as has been proven by D. Pa- nyushev in [19]. 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.W.T. Yu have shown (Section 3.2 of [20]). On the other hand, D. Panyushev and A. Dvorsky exhibit many quasi-reductive parabolic subalgebras in [6] and [19] by con- structing linear forms with the desired properties. Recently, M. Duflo, M.S. Khalgui and P. Torasso have obtained the classification of quasi- reductive parabolic subalgebras of the orthogonal Lie algebras in unpublished work, [4]. They were able to characterize quasi-reductive parabolic subalgebras in terms of the flags stabilized by the subalgebras.

The main result of this paper is the completion of the classification of quasi-reductive parabolic subalgebras of simple Lie algebras. This is done in Section 5 (Theorem 5.1 and Theorem 5.2). Our goal is ultimately to describe all quasi-reductive biparabolic subalgebras. Thus, in the first sections

(4)

we present results concerning biparabolic subalgebras to remain in a gen- eral setting as far as possible. For the remainder of the introduction,gis a finite dimensional complex semisimple Lie algebra.

The paper is organized as follows:

In Section 1 we introduce the main notations and definitions. We also include in this section a short review of known results about biparabolic subalgebras, including the description of quasi-reductive parabolic subalgebras in the classical Lie algebras (Subsection 1.4). In Section 2, we describe two methods of reduction, namely the transitivity property (Theorem 2.1) and the additivity property (Theorem 2.11). As a first step of our classification, we exhibit in Section 3 a large collection of quasi-reductive biparabolic subalgebras ofg (Theorem 3.6). Next, in Section 4, we consider the non quasi-reductive parabolic subalgebras ofg, for simplegof exceptional type (Theorems 4.1, 4.3 and 4.6). This is a crucial part of the paper. Indeed, to study the quasi-reductivity, we can make explicit computations (cf. Sec- tion 5) while it is much trickier to prove that a Lie algebra is not quasi- reductive. Using the results of Sections 2, 3 and 4, we are able to cover a large number of parabolic subalgebras. The remaining cases are dealt with in Section 5 (Theorem 5.6, Propositions 5.8 and 5.9). This completes the classification of quasi-reductive parabolic subalgebras of g(Theorems 5.1 and 5.2, see also Tables 5.1 and 5.2).

At this place, we also want to point out that in [16], O. Yakimova and the second author study themaximal reductive stabilizersof quasi-reductive parabolic subalgebras of g. This piece of work yields an alternative proof of Proposition 5.9 which is not based on the computer programmeGAP, see Remark 5.10.

1. Notations, definitions and basic facts

In this section, we recall a number of known results that will be used in the sequel.

1.1.Letgbe a complex Lie algebra of a connected linear algebraic Lie groupG. Denoting by g(f) the Lie algebra ofG(f), we have g(f) ={x∈ g| (ad^∗x)(f) = 0} where ad^∗ is the coadjoint representation of g. Recall that a linear formf ∈g^∗ is of reductive typeifG(f)/Z is a reductive Lie subgroup of GL(g). We can reformulate this definition as follows:

(5)

Definition 1.1. — An elementf of g^∗ is said to be of reductive type if g(f)/z is a reductive Lie algebra whose center consists of semisimple elements ofgwherezis the center ofg.

Recall that a linear form f ∈ g^∗ is regular if the dimension of g(f) is as small as possible. By definition, theindex ofg, denoted by indg, is the dimension of the stabilizer of a regular linear form. The index of various special classes of subalgebras of reductive Lie algebras has been studied by several authors, cf. [18], [25], [14], [15]. For the index of seaweed algebras, we refer to [17], [6], [20], [22], [10] and [11].

Recall that gis calledquasi-reductive if it has linear forms of reductive type. From Duflo’s work [3, §§I.26-27] one deduces the following result about regular linear forms of reductive type:

Proposition 1.2. — Suppose thatgis quasi-reductive. The set of regular linear forms of reductive type forms a Zariski open dense subset ofg^∗. 1.2.From now on, gis a complex finite dimensional semisimple Lie algebra. The dual ofgis identified withgthrough the Killing form ofg. For u∈ g, we denote by ϕ_u the corresponding element of g^∗. For u ∈g, the restriction ofϕu, to any subalgebraa ofgwill be denoted by (ϕu)|a.

Denote by π the set of simple roots with respect to a fixed triangular decomposition

g=n⁺⊕h⊕n⁻

of g, and by ∆π (respectively ∆⁺_π, ∆⁻_π) the corresponding root system (respectively positive root system, negative root system). Ifπ⁰ is a subset ofπ, we denote by ∆_π⁰ the root subsystem of ∆_π generated byπ⁰ and we set ∆^±_π0 = ∆π⁰ ∩∆^±_π. For α∈ ∆π, denote by gα the α-root subspace of gand leth_α be the unique element of [g_α,g_−α] such that α(h_α) = 2. For eachα∈∆π, fixxα∈gαso that the family {xα, hβ ; α∈∆π, β∈π}is a Chevalley basis ofg. In particular, for non-colinear rootsαandβ, we have [x_α, x_β] =±(p+ 1)x_α+β ifβ−pαis the source of the α-string throughβ. We briefly recall a classical construction due to B. Kostant. It associates to a subset of π a system of strongly orthogonal positive roots in ∆_π. This construction is known to be very helpful to obtain regular forms on biparabolic subalgebras ofg. For a recent account about thecascade construction of Kostant, we refer to [22, §1.5] or [21, §40.5].

(6)

Forλinh^∗andα∈∆π, we shall writehλ, α^∨iforλ(h_α). Recall that two rootsαandβ in ∆πare said to be strongly orthogonalif neitherα+β nor α−β is in ∆_π. Let π⁰ be a subset of π. The cascadeKπ⁰ of π⁰ is defined by induction on the cardinality ofπ⁰ as follows:

(1) K(∅) =∅,

(2) Ifπ⁰₁,. . . ,π_r⁰ are the connected components ofπ⁰, thenKπ⁰ =Kπ₁⁰ ∪

· · · ∪Kπ⁰_r,

(3) If π⁰ is connected, then Kπ⁰ = {π⁰} ∪KT where T = {α ∈ π⁰ | hα, ε^∨_π0i= 0}andεπ⁰ is the highest positive root of ∆⁺_π0.

ForK∈Kπ⁰, set

ΓK ={α∈∆K | hα, ε^∨_Ki>0} and Γ⁰_K= ΓK\ {εK}. Notice that the subspace P

K∈ΓK

gαis a Heisenberg Lie algebra whose center isg_ε_K.

The cardinality kπ of Kπ only depends on g; it is independent of the choices of hand π. The values of kπ for the different types of simple Lie algebras are given in Table 1.1; in this table, for a real numberx, we denote by [x] the largest integer6x.

A`, `>1 B`, `>2 C`, `>3 D`, `>4 G2 F4 E6 E7 E8

`+ 1 2

` ` 2

` 2

2 4 4 7 8

Table 1.1. k_π for the simple Lie algebras.

For π⁰ a subset of π, we denote by Eπ⁰ the set of the highest rootsε_K whereKruns over the elements of the cascade ofπ⁰. By construction, the subsetEπ⁰ is a family of pairwise strongly orthogonal roots in ∆_π⁰. For the convenience of the reader, the setEπ, for each simple Lie algebra of type π, is described in the Tables 1.2 and 1.3. We denote byE_π⁰ the subspace ofh^∗ which is generated by the elements ofEπ⁰.

(7)

A_`, `>1:

α1 α2 α−1 α {ε_i = α_i + · · · + α_i+(`−2i+1), i6

`+ 1 2

}

B_`, `>2 :

α1 α2 α₋₁ α

>

{ε_i = α_i−1+ 2α_i+· · ·+ 2α_`, ieven, i6`} ∪ {ε_i= α_i, iodd, i6`}

C_`,`>3:

α1 α2 α₋₁ α

< {εi= 2α_i+· · ·+ 2α_`−1+ α_`, i6`−1} ∪ {ε_`=α_`}

D_`,`even,`>4:

α1 α2 α₋₂

α

α−1

{ε_i = α_i−1 + 2α_i +

· · · + 2α_`−2 + α_`−1 + α_`, ieven, i < `−1} ∪ {ε_i=α_i, iodd, i < `} ∪ {ε_`=α_`}

D_`,`odd,`>5:

α1 α2 α−2

α

α−1

{εi = α_i−1 + 2α_i +

· · · + 2α`−2 + α`−1 + α_`, ieven, i < `−1} ∪ {ε_i=α_i, iodd, i < `} ∪ {ε`−1=α_`−2+α_`−1+α_`} Table 1.2. Eπ for the classical Lie algebras.

1.3.Abiparabolicsubalgebra ofgis defined to be the intersection of two parabolic subalgebras whose sum isg. This class of algebras has first been studied in the case ofsl_n by Dergachev and Kirillov [2] under the name of seaweedalgebras.

For a subsetπ⁰ofπ, we denote byp⁺_π0 the standard parabolic subalgebra of gwhich is the subalgebra generated by b⁺ = h⊕n⁺ and by g−α, for α∈ π⁰. We denote by p⁻_π0 the “opposite parabolic subalgebra” generated by b⁻ = n⁻⊕h and by gα, for α ∈ π⁰. Set lπ⁰ = p⁺_π0 ∩p⁻_π0. Then lπ⁰ is a Levi factor of both parabolic subalgebrasp⁺_π0 and p⁻_π0 and we can write l_π⁰ = n⁺_π0 ⊕h⊕n⁻_π0 where n^±_π0 = n^±∩l_π⁰. Let m⁺_π0 (respectively m⁻_π0) be the nilradical of p⁺_π0 (respectively p⁻_π0). We denote by gπ⁰ the derived Lie algebra ofl_π⁰ and byz(l_π⁰) the center ofl_π⁰. The Cartan subalgebrah∩g_π⁰ ofgπ⁰ will be denoted byhπ⁰.

(8)

G2:

α1 α2

> {ε1= 23, ε2= 01}

F4: ^α¹ ^α² > ^α³ ^α⁴ {ε1= 2342, ε2= 0122, ε3= 0120, ε4= 0100}

E6: ^α¹ ^α³ ^α⁴ ^α⁵ ^α⁶

α2

{ε1= 12321 2

, ε₂= 11111 0

, ε3= 01110

0

, ε4= 00100 0

}

E₇: ^α¹ ^α³ ^α⁴ ^α⁵ ^α⁶ ^α⁷

α2

α1

{ε1= 234321 2

, ε2= 012221 1

, ε3= 012100

1

, ε4= 000001 0

, ε5= 000000

1

, ε6= 010000 0

, ε7= 000100

0 }

E8: ^α¹ ^α³ ^α⁴ ^α⁵ ^α⁶ ^α⁷ ^α⁸

α2

{ε1= 2465432 3

, ε2= 2343210 2

, ε3= 0122210

1

, ε4= 0121000 1

, ε5= 0000010

0

, ε6= 0000000 1

, ε₇= 0100000

0

, ε₈= 0001000 0

}

Table 1.3. Eπ for the exceptional Lie algebras.

Definition 1.3. — The subalgebraqπ₁,π₂ of ggiven as follows by the subsetsπ₁, π₂⊂π

qπ₁,π₂:=p⁺_π₁∩p⁻_π₂=n⁺_π₂⊕h⊕n⁻_π₁

is called thestandardbiparabolic subalgebra (associated toπ1andπ2). Its nilpotent radical isu_π₁_,π₂ := (n⁺_π

2∩m⁺_π

1)⊕(n⁻_π

1∩m⁻_π

2)andl_π₁_,π₂ :=l_π₁_∩π₂ is the standard Levi factor ofqπ₁,π₂.

Any biparabolic subalgebra is conjugate to a standard one, see [22, §2.3]

or [10, §2.5]. So, for our purpose, it will be enough to consider standard biparabolic subalgebras.

(9)

Remark 1.4. — The classification of quasi-reductive (bi)parabolic subalgebras of reductive Lie algebras can be deduced from the classification of quasi-reductive (bi)parabolic subalgebras of simple Lie algebras: A stabilizer of a linear form on g is the product of its components on each of the simple factors of gand of the center of g. As a consequence, we may assume thatgis simple without loss of generality.

Letπ₁, π₂be two subsets ofπ. The dual ofq_π₁_,π₂is identified toq_π₂_,π₁via the Killing form ofg. Fora= (aK)_K∈_K_π

2 ∈(C^∗)^k^π² andb= (bL)_L∈_K_π

1 ∈ (C^∗)^k^π¹, set

u(a, b) = X

K∈Kπ2

aKx_−ε_K+ X

L∈Kπ1

bLxε_L

It is an element ofuπ2,π1 and the linear form (ϕu)|qπ1,π2 is a regular element ofq^∗_π₁_,π₂ for any (a, b) running through a nonempty open subset of (C^∗)^k^π²^+k^π¹, cf. [22, Lemma 3.9].

We denote by E_π₁_,π₂ the subspace generated by the elements ε_K, for K∈Kπ₁∪Kπ₂. Thus, dimEπ₁,π₂ = kπ₁+ kπ₂−dim(Eπ₁∩Eπ₂). As it has been proved in [10, §7.16], we have

indqπ1,π2 = (rkg−dimEπ1,π2) + (kπ1+ kπ2−dimEπ1,π2) (1.1)

Remark 1.5. — By (1.1), the index of q_π₁_,π₂ is zero if and only if Eπ₁ ∩Eπ₂ = {0} and kπ₁ + kπ₂ = rkg. For example, in type E6, there are exactly fourteen standard parabolic subalgebrasp⁺_π0 with index zero.

The corresponding subsetsπ⁰⊂πof the simple roots are the following:

{α1, α5};{α3, α6};{α1, α4, α5};{α3, α4, α6};{α1, α5, α6};

{α₁, α₃, α₆};{α₁, α₃, α₅};{α₃, α₅, α₆};{α₁, α₃, α₄};{α₄, α₅, α₆};

{α1, α3, α4, α5};{α3, α4, α5, α6};{α1, α2, α3, α4};{α2, α4, α5, α6}.

This was already observed in the unpublished work [7] of A. Elashvili (with a small error).

In the sequel, we will often make use of the following element of u_π₂_,π₁ on our way to construct reductive forms:

u⁻_π₁_,π₂ = X

ε∈Eπ2,ε /∈∆⁺π1

x_−ε

Ifπ₂=π, we simply writeu⁻_π

1 foru⁻_π

1,π and, in the special case ofπ₁=∅ andπ₂=π, we writeu⁻ foru⁻_∅. LetB be the Borel subgroup ofGwhose Lie algebra isb⁺. We summarize in the following proposition useful results of Kostant concerning the linear form (ϕ_u−)|_b+. They can be found in [21, Proposition 40.6.3].

(10)

Proposition 1.6. — (i)The linear form(ϕ_u−)|b⁺ is of reductive type for b⁺. More precisely, the stabilizer of ϕ_u⁻ in b⁺ is the subspace

T

K∈Kπ

kerε_K ofhof dimensionrkg−k_π.

(ii) Letm be an ideal ofb⁺ contained inn⁺. TheB-orbit of(ϕ_u−)|_m in m^∗ is an open dense subset ofm^∗.

1.4.We end the section by reviewing what is known in the classical case.

First recall that the biparabolic subalgebras of simple Lie algebras of type A and C are always quasi-reductive as has been shown by D. Panyushev in [19].

The classification of quasi-reductive parabolic subalgebras of the orthogonal Lie algebras is given in the recent work [4] of Duflo, Khalgui and Torrasso. Since we will use this result repeatedly, we state it below.

Let E be a complex vector space of dimension N endowed with a non- degenerate symmetric bilinear form. Denote byso_N the Lie algebra of the corresponding orthogonal group. LetV={{0}=V0(V1(· · ·(Vs=V} be a flag of isotropic subspaces in E, with s > 1. Its stabilizer in soN

is a parabolic subalgebra of soN and any parabolic subalgebra of soN is obtained in this way. We denote byp_V the stabilizer ofVinsoN.

Theorem 1.7. — [4] Let V = {{0} = V0 ( V1 ( · · · ( Vs = V} be a flag of isotropic subspaces in E with s > 1. Denote by V⁰ the flag of isotropic subspaces in E which is equal to V\ {V} if dimV is odd and equal toN/2, and equal toVotherwise.

The Lie algebrap_V is quasi-reductive if and only if the sequenceV⁰ does not contain two consecutive subspaces of odd dimension.

Example 1.8. — Forg=D₆ there are twelve standard parabolic subalgebrasp =p⁺_π0 which are not quasi-reductive. The corresponding subsets π⁰⊂πof the simple roots are the following:

{α₂},{α₄},{α₁, α₄},{α₂, α₄},{α₂, α₅},{α₂, α₆}, {α1, α2, α4},{α2, α3, α4},{α2, α4, α5},{α2, α4, α6}, {α2, α5, α6},{α2, α4, α5, α6}.

Among these, the connectedπ⁰ are{α2},{α4},{α2, α3, α4}.

Thus it remains to determine the quasi-reductive parabolic subalgebras of the exceptional Lie algebras. This is our goal.

(11)

2. Methods of reduction

In this section, we develop methods of reduction to deduce the quasi- reductivity of a parabolic subalgebra from the quasi-reductivity of other subalgebras. We assume thatπ₂ =π. Nevertheless we keep the notations of biparabolic subalgebras where it is convenient.

2.1.The following theorem seems to be standard. As there is no proof to our knowledge, we give a short proof here:

Theorem 2.1(Transitivity). — Letπ⁰⁰, π⁰be subsets ofπwithπ⁰⁰⊂π⁰. Suppose thatKπ⁰ ⊂Kπ. Then,qπ⁰⁰,πis quasi-reductive if and only ifqπ⁰⁰,π⁰

is.

Proof. — Note that the assumption Kπ⁰ ⊂ Kπ implies indqπ⁰⁰,π⁰ = indq_π⁰⁰_,π+ (kπ−kπ⁰) by formula (1.1). Sinceu_π⁰_,π is an ideal ofb⁺ contained in n⁺, Proposition 1.6(ii) enables to choose w⁰ in lπ⁰ such that both (ϕ_w0+u⁻

π0)|_q_π₀₀_,π and (ϕ_w⁰)|_q_π₀₀_,π₀ are regular linear forms ofq_π⁰⁰_,π and qπ⁰⁰,π⁰ respectively. Then one can show thatqπ⁰⁰,π⁰(ϕw⁰) =qπ⁰⁰,π(ϕ_w0+u⁻

π0)⊕

P

K∈Kπ\Kπ0

Ch_ε_K. By Proposition 1.2, ifq_π⁰⁰_,π (respectivelyq_π⁰⁰_,π⁰) is quasi- reductive, then we can assume furthermore that (ϕ_w0+u⁻

π0)|q_π00,π (respectively (ϕw⁰)|q_π00,π0) has reductive type. Hence the equivalence of the theo-

rem follows.

Suppose that gis simple and let eπbe the subset of π defined byKπ = {π} ∪K

e^π

. Ifgis of exceptional type,π\πeonly consists of one simple root which we denote byαπ. Note thatαπ is the simple root which is connected to the lowest root in the extended Dynkin diagram.

As a consequence of Theorem 2.1, to describe all the quasi-reductive parabolic subalgebras ofg, forgof exceptional type, it suffices to consider the case of parabolic subalgebrasp⁺_π0 withαπ ∈π⁰. This will be an important reduction in the sequel.

Remark 2.2. — If g has type F4 (resp. E6, E7, E8), then g e^π

has type C3 (resp. A5, D6, E7). In particular, if g has type F4 or E6, then p⁺_π0 is quasi-reductive for any π⁰ which does not containαπ because in types A and C all (bi)parabolic subalgebras are quasi-reductive.

2.2.As a next step we now focus on a property that we call “additivity”

to relate the quasi-reductivity of different parabolic subalgebras (cf. Theo- rem 2.11). Throughout this paragraph,gis assumed to be simple.

(12)

Definition 2.3. — Let π⁰, π⁰⁰ be subsets of π. We say that π⁰ is not connected toπ⁰⁰ifα⁰ is orthogonal toα⁰⁰, for all(α⁰, α⁰⁰)in π⁰×π⁰⁰.

Notation 2.4. — For a positive root α, we denote byK_π⁺(α) the only elementLofKπ such thatα∈ΓL. Note that unlessα∈Eπ,K_π⁺(α) is the only elementLofKπ for whichεL−αis a positive root. ForK∈Kπ, we haveK_π⁺(εK) =K.

Remark 2.5. — It can be checked thatK_π⁺(α) =K_π⁺(β) forα,β simple if and only ifαandβare in the same orbit of−w₀wherew₀is the longest element of the Weyl group ofg. This suggests thatw0 should play a role in these questions, as may be guessed from a result of Kostant which says that Eπ is a basis of the space of fixed points of −w0 and from work of Joseph and collaborators ([10, 11]).

Definition 2.6. — We shall say that two subsets π⁰, π⁰⁰which are not connected to each othersatisfy the condition (∗)if:

(∗) K_π⁺(α⁰)6=K_π⁺(α⁰⁰) ∀(α⁰, α⁰⁰)∈π⁰×π⁰⁰.

Note that if k_π = rkg(that is if−w₀acts trivially onπ), the condition (∗) is always satisfied. Moreover, by using Table 1.3, a case-by-case discussion shows:

Lemma 2.7. — Assume thatgis simple of exceptional type and letπ⁰ be a connected subset of π containing α_π. Then, for any subset π⁰⁰ of π which is not connected toπ⁰, the two subsets π⁰, π⁰⁰ satisfy the condition (∗), unless g= E₆,π⁰ ={α₁, α₂, α₃, α₄} and π⁰⁰ ={α₆} or by symmetry π⁰={α2, α4, α5, α6} andπ⁰⁰={α1}.

Remark 2.8. — If g= E₆, with π⁰ ={α1, α₂, α₃, α₄} and π⁰⁰ ={α6}, then K_π⁺(α1) = K_π⁺(α6) = {{α1, α3, α4, α5, α6}}, so π⁰ and π⁰⁰ do not satisfy the condition (∗). As a matter of fact, the parabolic subalgebra p⁺_π0∪π⁰⁰ will appear as a very special case (see Remark 2.12).

Letπ⁰, π⁰⁰be two subsets ofπwhich are not connected to each other and assume thatπ⁰, π⁰⁰satisfy condition (∗). By Proposition 1.6(ii), we can let w⁰be in l_π⁰ such that (ϕ_w)|_p+

π0 is regular wherew=w⁰+u⁻_π0. Denote bys⁰ be the image ofp⁺_π0(ϕ_w) by the projection map fromp⁺_π0 to its derived Lie algebragπ⁰⊕m⁺_π0 with respect to the decompositionp⁺_π0=z(lπ⁰)⊕gπ⁰⊕m⁺_π0. Letk⁰ be the intersection of z(lπ⁰) with T

ε∈Eπ, ε6∈∆⁺

π0

kerε.

Lemma 2.9. — (i) indp⁺_π0 = dims⁰+ dimk⁰. (ii) [s⁰,p⁺_π0∪π⁰⁰]⊂p⁺_π0 andϕw([s⁰,p⁺_π0∪π⁰⁰]) ={0}.

(13)

Proof. — (i) We have dimp⁺_π0(ϕ_w) = indp⁺_π0. Since the image ofp⁺_π0(ϕ_w) by the projection map from p⁺_π0 to gπ⁰ ⊕m⁺_π0 is s⁰, it suffices to observe that the intersection ofz(l_π⁰) withp⁺_π0(ϕ_w) isk⁰. And this follows from the choice ofw.

(ii) Letxbe an element ofp⁺_π0(ϕw); writex=x₀+x⁰+x⁺withx₀∈z(lπ⁰), x⁰∈gπ⁰ andx⁺∈m⁺_π0. Since [x⁺, w⁰] lies inm⁺_π0, the fact thatx∈p⁺_π0(ϕw) means [x0, u⁻_π0] + [x⁰, w⁰] + [x⁰, u⁻_π0] + [x⁺, u⁻_π0]∈m⁺_π0. First, we have to show [x⁰+x⁺,p⁺_π0∪π⁰⁰]⊂p⁺_π0. As [x⁰,p⁺_π0∪π⁰⁰]⊂p⁺_π0 sinceπ⁰⁰, π⁰ are not connected, it suffices to prove thatx⁺ ∈m⁺_π0∪π⁰⁰. If not, there areγ∈∆⁺_π00,K ∈Kπ, andα⁰ ∈∆⁺_π0 such that

γ−ε_K+

π(γ)=−(α⁰+εK) , i.e.ε_K+

π(γ)=γ+ (α⁰+εK) Henceγ, α⁰ ∈Γ⁰_K+

π(γ)that isK_π⁺(α⁰) =K_π⁺(γ). But this contradicts condition (∗). Thus [x⁰+x⁺,p⁺_π0∪π⁰⁰]⊂p⁺_π0.

It remains to show: ϕ_w([x⁰+x⁺,p⁺_π0∪π⁰⁰]) = {0} that is [x⁰+x⁺, w] ∈ m⁺_π0∪π⁰⁰. If [x⁰+x⁺, w] 6∈m⁺_π0∪π⁰⁰, there must be γ ∈ ∆⁺_π \∆π⁰, K ∈Kπ, andα⁰⁰ ∈∆⁺_π00 such that γ−ε_K =α⁰⁰. In particular α⁰⁰ ∈ Γ⁰

K_π⁺(γ) that is K_π⁺(α⁰⁰) =K_π⁺(γ). On the other hand, [x, w]∈m⁺_π0 implies that there exist α⁰ ∈∆⁺_π0 andL∈Kπ, such that

γ−ε_K+

π(γ)=−(α⁰+εL) , i.e.ε_K+

π(γ)=γ+ (α⁰+εL) As before, we deduce that α⁰ ∈Γ⁰_K+

π(γ), i.e.K_π⁺(α⁰) =K_π⁺(γ) =K_π⁺(α⁰⁰)

and this contradicts condition (∗).

Corollary 2.10. — Letπ⁰, π⁰⁰ be two subsets ofπwhich are not connected to each other and satisfy condition(∗). Ifp⁺_π0∪π⁰⁰ is quasi-reductive thenp⁺_π0 andp⁺_π00 are both quasi-reductive.

Proof. — Suppose thatp⁺_π0∪π⁰⁰is quasi-reductive and that any one of the other two parabolic subalgebras is not quasi-reductive and show that this leads to a contradiction. By assumption we can choose ϕ ∈ (p⁺_π0∪π⁰⁰)^∗ of reductive type for p⁺_π0∪π⁰⁰ such that ϕ⁰ = ϕ_|

p+ π0

and ϕ⁰⁰ = ϕ_|

p+ π00

are p⁺_π0- regular andp⁺_π00-regular respectively. Suppose for instance that p⁺_π0 is not quasi-reductive. By Proposition 1.6(ii) we can suppose furthermore that ϕ⁰ = (ϕw)|_p+

π0 for somew=w⁰+u⁻_π0 withw⁰∈lπ⁰.

Since we assumed that p⁺_π0 is not quasi-reductive, (ϕw)|_p+

π0 contains a nonzero nilpotent element, x, which is so contained in the derived Lie algebra of p⁺_π0. Then, Lemma 2.9(ii) gives [x,p⁺_π0∪π⁰⁰] ⊂ p⁺_π0 and {0} = ϕ_w([x,p⁺_π0∪π⁰⁰]) = ϕ⁰([x,p⁺_π0∪π⁰⁰]) = ϕ([x,p⁺_π0∪π⁰⁰]). As a consequence, p⁺_π0∪π⁰⁰(ϕ) contains the nonzero nilpotent elementx. This contradicts the

(14)

choice ofϕ. The same line of arguments works if we assume thatp⁺_π00 is not

quasi-reductive.

Under certain conditions, the converse of Corollary 2.10 is also true as we show now. To begin with, let us express the index ofp⁺_π0∪π⁰⁰in terms of those ofp⁺_π0 andp⁺_π00. AsEπ⁰∪π⁰⁰,π=Eπ⁰,π+Eπ⁰,π, we get: dimEπ⁰∪π⁰⁰,π= dimE_π⁰_,π+ dimE_π⁰⁰_,π−dim(E_π⁰_,π∩E_π⁰⁰_,π). Hence, formula (1.1) implies indp⁺_π0∪π⁰⁰ = indp⁺_π0+ indp⁺_π00−(rkg+ k_π−2 dim(E_π⁰_,π∩E_π⁰⁰_,π)) . (2.1)

In case rkg= kπ, the intersection Eπ⁰,π∩Eπ⁰⁰,π is equal to Eπ and has dimension rkg. Hence, the index is additive in that case, as (2.1) shows.

Theorem 2.11 (Additivity). — Assume thatgis simple and of exceptional type and that kπ = rkg. Let π⁰, π⁰⁰ be two subsets of π which are not connected to each other. Then,p⁺_π0∪π⁰⁰ is quasi-reductive if and only if bothp⁺_π0 andp⁺_π00 are quasi-reductive.

Remark 2.12. — The conclusions of Theorem 2.11 is valid for classical simple Lie algebras, even without the hypothesis kπ = rkg. In types A or C this follows from the fact that all biparabolic subalgebras are quasi- reductive. Ifgis an orthogonal Lie algebra, this is a consequence of Theo- rem 1.7. However, for the exceptional Lie simple algebra E6, the only one for which kπ6= rkg, the conclusions of Theorem 2.11 may fail. Indeed, let us consider the following subsets ofπforgof type E6:π⁰={α1, α2, α3, α4} andπ⁰⁰={α₆}.. By Remark 1.5,p⁺_π0 is quasi-reductive as a Lie algebra of zero index. On the other hand, p⁺_π00 is quasi-reductive by the transitivity property, cf. Remark 2.2. But, it will be shown in Theorem 4.6 thatp⁺_π0∪π⁰⁰

is not quasi-reductive.

As a consequence of Lemma 2.7 and Corollary 2.10, even in type E6

where rkg 6= kπ, if p⁺_π0∪π⁰⁰ is quasi-reductive, then p⁺_π0 and p⁺_π00 are both quasi-reductive.

As a by-product of our classification, we will see that the above situation is the only case which prevents the additivity property to be true for all simple Lie algebras (see Remark 5.3).

Proof. — We argue by induction on the rank of g. By the transitivity property (Theorem 2.1), Remark 2.12 and the induction, we can assume that απ ∈ π⁰. Then, by Lemma 2.7 and Corollary 2.10, only remains to prove that if bothp⁺_π0 andp⁺_π00 are quasi-reductive, then so isp⁺_π0∪π⁰⁰.

Assume that both p⁺_π0 and p⁺_π00 are quasi-reductive. By Proposition 1.2, we can find a linear regular formϕin (p⁺_π0∪π⁰⁰)^∗ such that ϕ⁰ =ϕ_|

p+ π0

and

(15)

ϕ⁰⁰=ϕ_|

p+ π00

are regular and of reductive type forp⁺_π0 andp⁺_π00 respectively.

By Proposition 1.6(ii), we can assume thatϕ= (ϕ_w+u−)|_p+

π0 ∪π00, wherew= h+w⁰+w⁰⁰, withw⁰ ∈n⁺_π0,w⁰⁰∈n⁺_π00andh∈h. Hence,ϕ⁰= (ϕ_h+w⁰_+u⁻)|_p+

π0

andϕ⁰⁰= (ϕ_h+w00+u⁻)|_p+ π00.

Use the notations of Lemma 2.9. By Lemma 2.9(ii), s⁰ is contained in p⁺_π0∪π⁰⁰(ϕ). Show now thatk⁰is zero. Lethbe an elementk⁰. Sinceh∈k⁰, we haveε(h) = 0 for anyε∈Eπwhich is not in ∆⁺_π0. On the other hand, for any ε∈Eπ∩∆⁺_π0, we have ε(h) = 0 sincehlies in the center oflπ⁰. Hence, our assumption rkg= kπimpliesh= 0. As a consequence of Lemma 2.9(i), we deduce that indp⁺_π0 = dims⁰. Similarly, ifs⁰⁰ denotes the image ofp⁺_π00(ϕ⁰⁰) under the projection fromp⁺_π00 togπ⁰⁰⊕m⁺_π00, Lemma 2.9(ii) tells us thats⁰⁰ is contained inp⁺_π0∪π⁰⁰(ϕ) and that indp⁺_π00= dims⁰⁰.

To summarize, our discussion shows thats⁰+s⁰⁰is contained inp⁺_π0∪π⁰⁰(ϕ) and that these two subspaces have the same dimension by equation (2.1). So s⁰+s⁰⁰=p⁺_π0∪π⁰⁰(ϕ). But by assumption,s⁰+s⁰⁰only consists of semisimple elements. From that we deduce that ϕ is of reductive type for p⁺_π0∪π⁰⁰,

whence the theorem.

3. Some classes of quasi-reductive biparabolic subalgebras In this section we show that, under certain conditions on the interlace- ment of the two cascades ofπ1andπ2, we can deduce thatqπ₁,π₂ is quasi- reductive (Theorem 3.6). We assume in this section thatgis simple.

3.1.We start by introducing the necessary notations. Recall that for a positive root α, K_π⁺(α) stands for the only element L of Kπ such that α∈Γ_L, cf. Notation 2.4. To any positive root α∈∆⁺_π we now associate the subsetK⁻π(α) of the cascadeKπ of allLsuch that the highest rootεL

can be added toα:

K⁻π(α) = {L∈Kπ|ε_L+α∈∆⁺_π}.

Observe that the setK⁻π(α) may be empty or contain more than one element.

Examples 3.1. — (1) IfK is in the cascadeKπ thenK⁻π(ε_K) is empty.

(2) In type E7, forα=α4+α5+α6, the set K⁻π(α) has more than one element:ε₄+α, ε₅+α, ε₆+αare all positive roots.

We need also the following notation:

∆e⁺_π ={α∈∆⁺_π, α= 1

2(ε_K−ε_K⁰) ; K, K⁰∈Kπ}.