On the variety of lagrangian subalgebras, II

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ON THE VARIETY OF LAGRANGIAN SUBALGEBRAS, II

B

Y

S

AM

EVENS

AND

J

IANG

-H

UA

LU

ABSTRACT. – Motivated by Drinfeld’s theorem on Poisson homogeneous spaces, we study the varietyL of Lagrangian subalgebras ofg⊕gfor a complex semi-simple Lie algebrag. LetGbe the adjoint group ofg. We show that the(G×G)-orbit closures inLare smooth spherical varieties. We also classify the irreducible components ofLand show that they are smooth. Using some methods of M. Yakimov, we give a new description and proof of Karolinsky’s classification of the diagonalG-orbits inL, which, as a special case, recovers the Belavin–Drinfeld classification of quasi-triangularr-matrices ong. Furthermore, Lhas a canonical Poisson structure, and we compute its rank at each point and describe its symplectic leaf decomposition in terms of intersections of orbits of two subgroups ofG×G.

RÉSUMÉ. – Motivé par le théorème de Drinfeld sur les espaces de Poisson homogènes, nous étudions la variétéLdes sous-algèbres de Lie lagrangiennes deg⊕gpourg, une algèbre de Lie complexe semi- simple. SoitGle groupe adjoint deg. Nous montrons que les adhérences des(G×G)-orbites dansLsont les variétés sphériques et lisses. Aussi, nous classifions les composantes irréductibles deLet nous montrons qu’elles sont lisses. Nous employons des méthodes de M. Yakimov pour donner une nouvelle description et une nouvelle preuve de la classification de Karolinsky des orbites diagonales deGdansL, qui, comme cas spécial, donne la classification de Belavin–Drinfeld desr-matrices quasi-triangulaires deg. En outre,L possède une structure de Poisson canonique, et nous calculons son rang à chaque point et nous décrivons sa décomposition en feuilles symplectiques en terme des intersections des orbites des deux sous-groupes de G×G.

Contents

1 Introduction . . . 348

2 The varietyLof Lagrangian subalgebras ofg⊕g. . . 351

2.1 Lagrangian subspaces . . . 351

2.2 Isometries . . . 352

2.3 Karolinsky’s classification . . . 353

2.4 Partition ofL. . . 353

2.5 (G×G)-orbits inL . . . 355

2.6 (B×B⁻)-orbits inL. . . 355

2.7 The De Concini–Procesi compactificationsZd(G)ofG. . . 356

2.8 Closures of(G×G)-orbits inL. . . 357

2.9 The geometry of the strataL^ε(S, T, d). . . 358

2.10 The geometry of the closure ofL^ε(S, T, d) . . . 359

2.11 Irreducible components ofL. . . 359

3 Classification ofGΔ-orbits inL. . . 360

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3.1 Some results on Weyl groups and generalized BD-triples . . . 360

3.2 A double coset theorem . . . 361

3.3 GΔ-orbits inL. . . 364

3.4 Normalizer subalgebras ofgΔatl∈ L. . . 365

3.5 Intersections ofgΔwithl∈ L . . . 367

3.6 Examples of smoothGΔ-orbit closures inL. . . 368

4 The Poisson structureΠ0onL. . . 370

4.1 Lagrangian splittings ofg⊕g . . . 370

4.2 The rank of the Poisson structureΠ0 . . . 371

4.3 The action ofHΔon the set of symplectic leaves of the Poisson structureΠ0 . . . 374

5 Lagrangian subalgebras ofg⊕h. . . 377

Acknowledgement . . . 378

References . . . 378

1. Introduction

Let d be a 2n-dimensional Lie algebra overk=Ror C, together with a symmetric, non- degenerate, and ad-invariant bilinear form,. A Lie subalgebralofdis said to beLagrangian iflis maximal isotropic with respect to,, i.e., ifdimkl=nand ifx, y= 0for allx, y∈l.

By aLagrangian splittingofdwe mean a direct sum decompositiond=l₁+l₂, wherel₁and l₂ are two Lagrangian subalgebras ofd. Denote byL(d)the set of all Lagrangian subalgebras ofd. It is an algebraic subvariety of the GrassmannianGr(n,d)ofn-dimensional subspaces ofd, and every connected Lie groupD with Lie algebradacts onL(d)via the adjoint action ofD ond. We proved in [9] that each Lagrangian splittingd=l₁+l₂gives rise to a Poisson structure Πl1,l2 onL(d), makingL(d)into a Poisson variety. Moreover, ifL1andL2 are the connected subgroups ofD with Lie algebrasl₁ andl₂ respectively, all theL1 andL2-orbits inL(d)are Poisson submanifolds ofΠl1,l2.

The above construction in [9] was motivated by the work of Drinfeld [7] on Poisson homogeneous spaces. Indeed, a Lagrangian splittingd=l₁+l₂ of d gives rise to the Manin triple(d,l₁,l₂), which in turn defines Poisson structuresπ1andπ2onL1 andL2respectively, making them into Poisson Lie groups [19]. A Poisson space (M, π) is said to be (L₁, π₁)- homogeneous if L₁ acts on M transitively and if the action map L₁×M →M is Poisson.

In [7], Drinfeld constructed anL₁-equivariant mapM→ L(d)for every(L₁, π₁)-homogeneous Poisson space(M, π)and proved that(L₁, π₁)-homogeneous Poisson spaces correspond toL₁- orbits inL(d)in this way. The Poisson structureΠl1,l2onL(d)is constructed in such a way that the Drinfeld mapM → L(d)is a Poisson map. In many cases, the Drinfeld mapM → L(d)is a local diffeomorphism onto its image. Thus we can think of L₁-orbits inL(d)as models for (L₁, π₁)-homogeneous Poisson spaces. For this reason, it is interesting to study the geometry of the varietyL(d), theL₁andL₂-orbits inL(d), and the Poisson structuresΠl1,l2 onL(d).

There are many examples of Lie algebrasdwith symmetric, non-degenerate, and ad-invariant bilinear forms. The geometry ofL(d)is different from case to case. Moreover, there can be many Lagrangian splittings for a givend, resulting in many Poisson structures onL(d).

Example 1.1. – Letgbe a complex semi-simple Lie algebra and regard it as a real Lie algebra, and let,be the imaginary part of the Killing form ofg. The geometry ofL(g)was studied in [9]. In particular, we determined the irreducible components ofL(g)and classified theG-orbits inL(g), whereGis the adjoint group ofg. Letg=k+a+nbe an Iwasawa decomposition ofg.

Then bothkanda+nare Lagrangian, sog=k+ (a+n)is a Lagrangian splitting, resulting in a Poisson structure onL(g)which we denote byπ0. Many interesting Poisson manifolds appear as GorK-orbits insideL(g), whereKis the connected subgroup ofGwith Lie algebrak. Among

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such Poisson manifolds are the flag manifolds ofGand the compact symmetric spaces associated to real forms ofG. Detailed studies of the Poisson geometry of these Poisson structures and some applications to Lie theory have been given in [22,23,8,11]. For example, a flag manifoldX of Gcan be identified with a certainK-orbit inL(g). The resulting Poisson structureπ₀ onX is called theBruhat–Poisson structurebecause its symplectic leaves are Bruhat cells inX. In [22]

and [8], we established connections between the Poisson geometry ofπ₀and the harmonic forms onXconstructed by Kostant [20] in 1963, and we gave a Poisson geometric interpretation of the Kostant–Kumar approach [21] to Schubert calculus onX.

Example 1.2. – Letgbe anyn-dimensional Lie algebra, and letdbe the semi-direct product of gand its dual spaceg^∗. The symmetric bilinear form x+ξ, y+η=x, η+y, ξfor x, y∈gandξ, η∈g^∗ is non-degenerate and ad-invariant. Whengis semi-simple, Lagrangian subalgebras of d are not easy to classify (except for low dimensional cases), for, as a sub- problem, one needs to classify all abelian subalgebras ofg. See [19,16], and their references for more detail. The description of the geometry ofL(d)in this case is an open problem.

In this paper, we consider the complexification of Example 1.1. Namely, letgbe a complex semi-simple Lie algebra andd=g⊕gthe direct sum Lie algebra with the bilinear form

(x1, x2),(y1, y2)

=x1, y1 − x2, y2, x1, x2, y1, y2∈g,

where,is a fixed symmetric, non-degenerate, and ad-invariant bilinear form ong. The variety of Lagrangian subalgebras ofdwith respect to,will be denoted byL.

The classification of Lagrangian subalgebras of d has been given by Karolinsky [15], and Lagrangian splittings ofg⊕ghave been classified by Delorme [6]. In this paper, we establish the first few steps in the study of the Poisson structures onLdefined by Lagrangian splittings of g⊕g. Namely, we will first describe the geometry ofLin the following terms:

(1) the(G×G)-orbits inLand their closures, whereGis the adjoint group ofg;

(2) the irreducible components ofL;

We then look at the Poisson structureΠ0 onL defined by the so-calledstandardLagrangian splittingd=g_Δ+g^∗_st, whereg_Δ={(x, x): x∈g}andg^∗_st⊂b⊕b⁻, wherebandb⁻ are two opposite Borel subalgebras ofg. Let GΔ={(g, g): g∈G}, and letB andB⁻ be the Borel subgroups ofGwith Lie algebrasbandb⁻respectively. We will compute the rank ofΠ0inL and study the symplectic leaf decomposition ofLwith respect toΠ0in terms of the intersections ofG_Δand(B×B⁻)-orbits inL.

We regard the Poisson structureΠ₀as the most important Poisson structure onL, especially since it is more closely related to applications to other areas of mathematics. The study of the symplectic leaf decomposition of the Poisson structureΠ_l₁_,l₂defined by an arbitrary Lagrangian splittingg⊕g=l₁+l₂will be carried out in [25] using results of [24]. This analysis is much more technically involved, so we think it is worthwhile to present the important special case of Π0separately. In particular, in computing the rank of Π0 inL, we need to classifyGΔ-orbits inLand compute the normalizer subalgebra ofg_Δ for everyl∈ L. The classification ofGΔ- orbits inLfollows directly from a special case of Theorem 2.2 in [24]. However, the inductive procedure used in the proof of Theorem 2.2 in [24] is rather involved. To make the paper self- contained and more comprehensible, we present a shorter inductive proof for the classification ofG_Δ-orbits inL. Our proof and the one in [24] are both adapted from [33]. The description of the normalizer subalgebra ofg_Δatl∈ Lcan be derived from Theorem 2.5 in [24]. Again for the purpose of completeness and comprehensibility, we give a simple and more direct computation of these subalgebras.

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We point out that E. Karolinsky has in [15] given a classification of GΔ-orbits in L in different terms. Our classification is more in line with that of Lagrangian splittings in [6], and in particular, the Belavin–Drinfeld theorem [3] on Lagrangian splittings of the formg⊕g=g_Δ+l follows easily from our classification, but does not seem to be easily derived from Karolinsky’s classification. A special case of our classification is the classification of G_Δ-orbits on the wonderful compactification ofG, and this special case was first determined by Lusztig in [26]

and [27], where it is used to develop a theory of character sheaves for the compactifications. Our arguments are somewhat different from those in [26] and [27]. It would be very interesting to find connections between Poisson geometry and character sheaves.

We now give more details of the main results in this paper:

Following O. Schiffmann [28], we define ageneralized Belavin–Drinfeld triple(generalized BD-triple) to be a triple (S, T, d), whereS andT are two subsets of the setΓ of vertices of the Dynkin diagram ofg, andd:S→T is an isometry with respect to,. For a generalized BD-triple (S, T, d), let PS and P_T⁻ be respectively the standard parabolic subgroups ofGof typeS and opposite typeT (see Notation 2.11) with Levi decompositions PS=MSNS and P_T⁻=MTN_T⁻. LetGS andGT be the quotients ofMSandMT by their respective centers, and letχ_S:M_S→G_S andχ_T:M_T →G_T be the natural projections. Letγ_d:G_S→G_T be a group isomorphism induced byd, and define the subgroupR_{S,T ,d}ofP_S×P_T⁻by

R_{S,T ,d}=

(m_S, m_T)∈M_S×M_T:γ_d

χ_S(m_S)

=χ_T(m_T)

(N_S×N_T⁻).

In Section 2, we establish the following facts on (G×G)-orbits and their closures in L (Proposition 2.17, Corollary 2.22, and Proposition 2.25):

(1) Every(G×G)-orbit inLis isomorphic to(G×G)/R_{S,T ,d}for a generalized BD-triple (S, T, d), so the(G×G)-orbit types inLcorrespond bijectively to generalized BD-triples forG;Every(G×G)-orbit inLis a(G×G)-spherical homogeneous space.

(2) For a generalized BD-triple(S, T, d), the closure of a(G×G)-orbit of type(S, T, d)is a fiber bundle over the flag manifoldG/PS×G/P_T⁻whose fiber is isomorphic to a De Concini–Procesi compactification of G_S.

We also study in Section 2 the irreducible components of L. We prove (Corollary 2.27, Theorems 2.29 and 2.32):

(1) The irreducible components of Lare roughly(see Theorem 2.32for detail)labeled by quadruples(S, T, d, ε), where(S, T, d)are generalized BD-triples andε∈ {0,1};

(2) The irreducible component corresponding to(S, T, d, ε)is a fiber bundle over the flag manifoldG/P_S×G/P_T⁻with fiber isomorphic to the product of a De Concini–Procesi compactification of G_S and a homogeneous space of a special orthogonal group. In particular, all the irreducible components ofLare smooth;

(3) Lhas two connected components.

In Section 3, we give the classification ofGΔ-orbits inL, and we compute the normalizer subalgebra ofg_Δat everyl∈ L. In Section 4, we compute the rank of the Poisson structureΠ0. In particular, we prove (Theorems 4.5 and 4.14):

Every non-empty intersection of aGΔ-orbitO and a(B×B⁻)-orbitO inLis a regular Poisson variety with respect toΠ0;The Cartan subgroupHΔ={(h, h): h∈H}ofGΔ, where H=B∩B⁻, acts transitively on the set of symplectic leaves inO ∩ O.

An interesting Poisson subvariety of (L,Π₀) is the De Concini–Procesi compactification Z1(G)ofGas the closure of the(G×G)-orbit throughg_Δ∈ L. Conjugacy classes inGand their closures inZ1(G)are all Poisson subvarieties of(Z1(G),Π0). In particular,Π0restricted to a conjugacy class C in G is non-degenerate precisely on the intersection of C with the open Bruhat cellB⁻B (see Corollary 4.7). It will be particularly interesting to compare the

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Poisson structureΠ0 on the unipotent variety inGwith the Kirillov–Kostant structure on the nilpotent cone ing^∗. Another interesting Poisson subvariety of(L,Π0)is the De Concini–Procesi compactification Xσ of a complex symmetric space G/G^σ for an involutive automorphism σof G(Proposition 3.21). Intersections of GΔ-orbits and(B×B⁻)-orbits inside the closed (G×G)-orbits inLare related to double Bruhat cells inG(see Example 4.9), and Kogan and Zelevinsky [18] have constructed toric charts on some of the symplectic leaves in these closed orbits. It would be interesting to see how their methods can be applied to other symplectic leaves ofΠ₀.

2. The varietyLof Lagrangian subalgebras ofg⊕g

Throughout this paper,gwill be ann-dimensional complex semi-simple Lie algebra, and, will be a fixed symmetric and non-degenerate ad-invariant bilinear form ong. Equip the direct product Lie algebrag⊕gwith the symmetric non-degenerate ad-invariant bilinear form

(x1, x2),(y1, y2)

=x1, y1 − x2, y2, x1, x2, y1, y2∈g.

(2.1)

A Lagrangian subalgebra ofg⊕gis by definition ann-dimensional complex Lie subalgebra of g⊕gthat is isotropic with respect to,.

NOTATION 2.1. – We will denote byLthe variety of all Lagrangian subalgebras ofg⊕gand byLspace(g⊕g)the variety of alln-dimensional isotropicsubspacesofg⊕g.

LetGbe the adjoint group ofg. In this section, we will use the classification of Lagrangian subalgebras ofg⊕gby Karolinsky [15] to study the(G×G)-orbits and their closures inL, and we will determine the irreducible components ofL.

2.1. Lagrangian subspaces

Let U be a finite-dimensional complex vector space with a symmetric and non-degenerate bilinear form,. A subspaceV ofU is said to be Lagrangian ifV is maximal isotropic with respect to,. IfdimU= 2nor2n+ 1, Witt’s theorem says that the dimension of a Lagrangian subspace ofU isn. Denote byLspace(U)the set of Lagrangian subspaces ofU. It is a closed algebraic subvariety ofGr(n, U), the Grassmannian ofn-dimensional subspaces ofU.

PROPOSITION 2.2. – (Arbarello et al. [1, pp. 102–103].) Let dimU = 2n (respectively 2n+ 1) with n >0. Then Lspace(U) is a smooth algebraic subvariety of Gr(n, U) with two(respectively one) connected components, each of which is isomorphic to SO(2n,C)/P (respectivelySO(2n+ 1,C)/P)whereP has Levi factor isomorphic toGL(n,C). Moreover, Lspace(U) has complex dimension ⁿ⁽ⁿ₂⁻¹⁾ (respectively ⁿ⁽ⁿ⁺¹⁾₂ ). When dimU = 2n, two Lagrangian subspaces V1 and V2 are in the same connected component of Lspace(U)if and only ifdim(V₁)−dim(V₁∩V₂)is even.

NOTATION 2.3. – ForU=g⊕gwith the bilinear form,in (2.1), letL⁰be the intersection of L with the connected component of Lspace(g⊕g) containing the diagonal of g⊕g. The intersection ofLwith the other connected component ofLspace(g⊕g)will be denoted byL¹.

Lethbe a Cartan subalgebra ofg, and letnandn⁻ be the opposite nilpotent subalgebras of gcorresponding to a choice of positive roots for(g,h). For a Lagrangian subspaceV of h⊕h with respect to,, letl_V =V +{(x, y): x∈n, y∈n⁻}. Thenl_V ∈ L. It is easy to see from Proposition 2.2 thatl_V₁ andl_V₂ are in the same connected component of Lspace(g⊕g)if and

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only ifV1 andV2 are in the same connected component ofLspace(h⊕h). In particular,L¹ is non-empty.

2.2. Isometries

We collect some results on automorphisms that will be used in later sections.

NOTATION 2.4. – Throughout this paper, we fix a Cartan subalgebrahand a choiceΣ⁺of positive roots in the setΣof all roots ofgrelative toh, and letg=h+

α∈Σg_αbe the root decomposition. Let Γ be the set of simple roots inΣ⁺. For α∈Σ, let H_α∈h be such that Hα, H=α(H)for allH ∈h. Forα∈Σ⁺, fix root vectorsE_α∈g_αandE_−α∈g_−αsuch thatEα, E₋α= 1. For a subsetS ofΓ, let[S]be the set of roots in the linear span ofS, and letGSbe the adjoint group of the semi-simple Lie algebrag_Sgiven by

g_S= span_C{Hα: α∈S}+

α∈[S]

g_α⊂g.

For S, T ⊂Γ, we are interested in Lie algebra isomorphisms g_S →g_T that preserve the restrictions of,tog_S andg_T. We will simply refer to this property as preserving,.

DEFINITION 2.5. – LetS, T⊂Γ. By an isometry fromStoTwe mean a bijectiond:S→T such thatdα, dβ=α, βfor allα, β∈S, whereα, β=H_α, H_β. LetI(S, T)be the set of all isometries fromStoT. Following [28], a triple(S, T, d), whereS, T ⊂Γandd∈I(S, T), will also be called ageneralized Belavin–Drinfeld(generalized BD-)triple forG.

LEMMA 2.6. – For anyS, T⊂Γandd∈I(S, T), there is a unique isomorphismγd:g_S→g_T such that

γd(Eα) =E_d(α), γd(Hα) =H_d(α), ∀α∈S.

(2.2)

Moreover, γ_d preserves,, and for every Lie algebra isomorphismμ:g_S →g_T preserving ,, there is a uniqued∈I(S, T)and a uniqueg∈G_Ssuch thatμ=γ_dAd_g.

Proof. –Existence and uniqueness of γd is by Theorem 2.108 in [17]. For α∈Σ⁺, let λα, μα∈C be such that γd(Eα) =λαEdα andγd(E₋α) =μαE₋dα. By applying γd to the identity [Eα, E₋α] =Hα we get λαμα= 1 for every α∈Σ⁺. It follows that γd preserves ,. Suppose that μ:g_S→g_T is a Lie algebra isomorphism preserving ,. Letd1 be any isomorphism from the Dynkin diagram ofg_Sto the Dynkin diagram ofg_T. Letγd1:g_S→g_T be defined as in (2.2). Thenν:=γ_d⁻¹₁ μis an automorphism ofg_S. Recall that there is a short exact sequence

1→GS→AutgS→AutS→1,

whereAutgS is the group of automorphisms ofg_S, andAutS is the group of automorphisms of the Dynkin diagram ofg_S. Letd2∈AutS be the image ofνunder the mapAutgS→AutSand writeν=γd2Adg for someg∈GS. Thusμ=γd1γd2Adg=γd1d2Adg. SinceμandAdg are isometries of,, so isγd1d2. Thusd:=d1d2∈I(S, T)is an isometry, andμ=γdAdg.

Uniqueness ofdandgfollows from the fact that ifg0∈GSpreserves a Cartan subalgebra and acts as the identity on all simple root spaces, theng0is the identity element. 2

DEFINITION 2.7. – A,-preserving Lie algebra isomorphismμ:g_S→g_T is said to be of type dfor d∈I(S, T) ifd is the unique element in I(S, T)such that μ=γdAdg for some g∈GS.

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2.3. Karolinsky’s classification

Karolinsky [15] has classified the Lagrangian subalgebras ofg⊕gwith respect to the bilinear form,given in (2.1). We recall his results now.

NOTATION 2.8. – For a parabolic subalgebrapofg, letnbe its nilradical andm:=p/nits Levi factor. Letm= [m,m] +zbe the decomposition ofm into the direct sum of its derived algebra[m,m]and its centerz. Recall that[m,m]is semi-simple and that the restrictions of the bilinear form,tom andz are both non-degenerate. Ifp is another parabolic subalgebra, denote its nilradical, Levi factor, and center of Levi factor, etc. byn, m, andz, etc. When speaking of Lagrangian subspaces ofz⊕z, we mean with respect to the restriction of,to z⊕z.

DEFINITION 2.9. – A quadruple (p,p, μ, V)is called admissible if pandp are parabolic subalgebras ofg,μ: [m,m]→[m,m]is a Lie algebra isomorphism preserving,, andV is a Lagrangian subspace ofz⊕z.

If(p,p, μ, V)is admissible, set l(p,p, μ, V) :=

(x, x): x∈p, x∈p, μ(x_[m,m]) =x_[m,m], (xz, x_z)∈V

⊂g⊕g, where forx∈p,x_[m,m]∈[m,m]andxz∈zare respectively the[m,m]- andz-components of x+n∈p/n= [m,m] +z. We use similar notation forx∈p.

THEOREM 2.10. – (See [15].) l(p,p, μ, V) is a Lagrangian subalgebra if (p,p, μ, V) is admissible, and every Lagrangian subalgebra of g⊕g is of the form l(p,p, μ, V) for some admissible quadruple.

2.4. Partition ofL

NOTATION 2.11. – Recall the fixed choiceΣ⁺of positive roots from Notation 2.4. Set n=

α∈Σ⁺

g_α, n⁻=

α∈Σ⁺

g_−α.

A parabolic subalgebrapofgis said to bestandardif it contains the Borel subalgebrab:=h+n.

Recall also that, forS⊂Γ,[S]denotes the set of roots in the linear span ofS. Let m_S=h+

α∈[S]g_α, n_S=

α∈Σ⁺−[S]

g_α, n⁻_S =

α∈Σ⁺−[S]

g₋_α

andp_S=m_S+n_S,p⁻_S =m_S+n⁻_S. We will refer top_S as the standard parabolic subalgebra of gdefined bySandp⁻_S theoppositeofp_S. A parabolic subalgebra ofpofgis said to be oftype Sifpis conjugate top_S and of theopposite-typeSifpis conjugate top⁻_S. Letg_S= [mS,m_S] as in Notation 2.4 and

h_S=h∩g_S= span_C

Hα: α∈[S]

, z_S=

x∈h: α(x) = 0, ∀α∈S .

Thenp_S =z_S+g_S +n_S andp⁻_S =z_S+g_S +n⁻_S. The connected subgroups of Gwith Lie algebras p_S,p⁻_S,m_S,n_S and n⁻_S will be respectively denoted by PS, P_S⁻, MS, NS and N_S⁻. Correspondingly we havePS=MSNSandP_S⁻=MSN_S⁻, whereMS∩NS={e}=MS∩N_S⁻.

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Recall thatGS denotes the adjoint group ofg_S. LetχS:PS →GS be the composition of the projection from PS =MSNS toMS along NS and the projection MS→GS. The similarly defined projection fromP_S⁻toGSwill also be denoted byχS.

Returning to the notation in 2.8, we have

LEMMA–DEFINITION 2.12. – Let (p,p, μ) be a triple, where p and p are parabolic subalgebras of g, and μ: [m,m]→[m,m] is a Lie algebra isomorphism preserving ,. Assume thatpis of typeSandpis of opposite-typeT. Letg1, g2∈Gbe such thatAdg1p=p_S andAdg2p=p⁻_T. LetAdg1 andAdg2 be the induced Lie algebra isomorphisms

Adg1: [m,m]→g_S, Adg2: [m,m]→g_T.

Ifμ:= Ad_g₂◦μ◦(Ad_g₁)⁻¹:g_S→g_T is of typed∈I(S, T)as in Definition2.7, we say that (p,p, μ)is of type(S, T, d). The type of(p,p, μ)is independent of the choice ofg1andg2.

Proof. –If h1 andh2 in G are such that Adh1p=p_S and Adh2p =p⁻_T, then there exist pS∈PSandp⁻_T ∈P_T⁻such thath1=pSg1andh2=p⁻_Tg2. Thus

μ:= Ad_h₂◦μ◦(Ad_h₁)⁻¹= Ad_p−

T ◦μ◦(Ad_p_S)⁻¹.

The action ofAd_p_S ong_S is by definition the adjoint action of χ_S(p_S)∈G_S ong_S, and the same holds for the action ofAd_p−

T ong_T. Thus by Definition 2.7, the two mapsμ andμhave the same type. 2

We are now ready to partitionL. Recall the definitions ofL⁰andL¹in Notation 2.3.

DEFINITION 2.13. – LetS, T ⊂Γ, d∈I(S, T), andε∈ {0,1}. DefineL^ε(S, T, d)to be the set of all Lagrangian subalgebrasl(p,p, μ, V)such that

(1) l(p,p, μ, V)∈ L^ε;

(2) (p,p, μ)is of type(S, T, d).

We say thatl∈ Lis oftype(ε, S, T, d)ifl∈ L^ε(S, T, d).

It is clear that

L=

ε∈{0,1}

S,T⊂Γ,d∈I(S,T)

L^ε(S, T, d) (2.3)

is a disjoint union, and eachL^ε(S, T, d)is invariant underG×G. Set n_S⊕n⁻_T =

(x, y): x∈n_S, y∈n⁻_T

⊂g⊕g,

and for eachV ∈ Lspace(zS⊕z_T),set

l_{S,T ,d,V} =V + (n_S⊕n⁻_T) +

x, γ_d(x)

: x∈g_S

∈ L.

(2.4)

Note thatdimz_S= dimzT becauseh_S∼=h_T andh=z_S+h_S=z_T +h_T are direct sums.

LEMMA–DEFINITION 2.14. – ForV₁, V₂∈ Lspace(z_S⊕z_T),l_{S,T ,d,V}₁andl_{S,T ,d,V}₂are in the same connected component ofLspace(g⊕g)if and only ifV₁andV₂are in the same connected component ofLspace(z_S⊕z_T). Forε={0,1}, let

L^ε_space(zS⊕z_T) =

V ∈ Lspace(zS⊕z_T): l_{S,T ,d,V} ∈ L^ε .

e ◦

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Proof. –The statement follows from Proposition 2.2 and the fact

dim(l_{S,T ,d,V}₁)−dim(l_{S,T ,d,V}₁∩l_{S,T ,d,V}₂) = dim(V₁)−dim(V₁∩V₂). 2 PROPOSITION 2.15. – For any generalized BD-triple(S, T, d)andε∈ {0,1},

L^ε(S, T, d) =

V∈L^εspace(zS⊕zT)

(G×G)·l_{S,T ,d,V} (disjoint union).

(2.5)

Proof. –By Definition 2.13, every(G×G)-orbit inL^ε(S, T, d)passes through anl_{S,T ,d,V} for some V ∈ L^ε_space(zS ⊕ z_T). If V1, V ∈ L^ε_space(zS ⊕ z_T) are such that l_{S,T ,d,V}₁ = Ad_(g₁_,g₂₎l_{S,T ,d,V}, then(g1, g2)normalizesn_S⊕n⁻_T, so(g1, g2)∈PS×P_T⁻, and it follows that V1=V. 2

2.5. (G×G)-orbits inL

The following theorem follows immediately from Proposition 2.15 and the decomposition of Lin (2.3).

THEOREM 2.16. – Every (G×G)-orbit in L passes through an l_{S,T ,d,V} for a unique quadruple(S, T, d, V), whereS, T⊂Γ, d∈I(S, T)andV ∈ Lspace(z_S⊕z_T).

For eachS, T⊂Γandd∈I(S, T), define the group (see Notation 2.11) RS,T ,d:=

(pS, p⁻_T)∈PS×P_T⁻: γd

χS(pS)

=χT(p⁻_T)

⊂PS×P_T⁻. (2.6)

PROPOSITION 2.17. – LetS, T⊂Γ, d∈I(S, T), andV ∈ Lspace(zS⊕z_T).

(1) The (G×G)-orbit inL throughl_{S,T ,d,V} is isomorphic to (G×G)/RS,T ,d and it has dimensionn−z, wheren= dimgandz= dimz_S.

(2) (G×G)·l_{S,T ,d,V} fibers overG/PS×G/P_T⁻with fiber isomorphic toGS.

Proof. –It is routine to check that the stabilizer of l_{S,T ,d,V} is R_{S,T ,d}, and the dimensional formula follows. The fiber in (2) may be identified with(P_S ×P_T⁻)/R_{S,T ,d}, which may be identified withG_Svia the map

(P_S×P_T⁻)/R_{S,T ,d}→G_S: (p_S, p⁻_T)→γ_d⁻¹

χ_T(p⁻_T)

χ_S(p_S)₋1

. 2

Remark 2.18. – It follows that the number of orbit types forG×GinLis exactly the number of generalized BD-triples forG.

LEMMA 2.19. – R_{S,T ,d}is connected.

Proof. –The projection map p:RS,T ,d→PS, (pS, p⁻_T)→pS is surjective and has fiber N_T⁻×ZT, where ZT is the center of the group MT. Clearly N_T⁻ is connected, and ZT is connected by Proposition 8.1.4 of [4]. Since PS is connected, it follows that RS,T ,d is also connected. 2

2.6. (B×B⁻)-orbits inL

LetB andB⁻ be the Borel subgroups ofGwith Lie algebrasb=h+nandb⁻=h+n⁻ respectively. By Proposition 2.17, to classify (B×B⁻)-orbits in L, it suffices to consider (B×B⁻)-orbits in(G×G)/RS,T ,dfor any generalized BD-triple(S, T, d).

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Let W be the Weyl group of Σ. For S⊂Γ, letWS be the subgroup of W generated by elements inS, and letW^S be the set of minimal length representatives of elements in the cosets inW/WS. Fix a representativew˙ inGfor eachw∈W. The following assertion can be proved in the same way as Lemma 1.3 in [31]. It also follows directly from Proposition 8.1 of [24].

PROPOSITION 2.20. – Let(S, T, d)be an generalized BD-triple forG. Then every(B×B⁻)- orbit in (G×G)/RS,T ,d is of the form (B×B⁻)( ˙w,v)R˙ S,T ,d for a unique pair (w, v)∈ W×W^T.

COROLLARY 2.21. – Every (B×B⁻)-orbit in L goes through exactly one point in L of the formAd_{( ˙}_w,_v)_˙ l_{S,T ,d,V}, where(S, T, d)is a generalized BD-triple,V ∈ Lspace(zS⊕z_T), and (w, v)∈W×W^T.

Since each(G×G)-orbit inLhas finitely many(B×B⁻)-orbits, at least one of them is open.

COROLLARY 2.22. – All(G×G)-orbits inLare(G×G)-spherical homogeneous spaces.

2.7. The De Concini–Procesi compactificationsZ_d(G)ofG

In this section, we will consider the closure inL of some special(G×G)-orbits. Namely, whenS=T= Γandd∈I(Γ,Γ), we have the graphl_γ_dofγ_d:g→gas a point inL:

l_γ_d=

x, γd(x)

: x∈g . (2.7)

The(G×G)-orbit inLthroughl_γ_dcan be identified withGby the map Id:G→(G×G)·l_γ_d: g→

x, γdAdg(x)

: x∈g . (2.8)

The identificationI_dis(G×G)-equivariant if we equipGwith the action ofG×Ggiven by (g1, g2)·g=γ_d⁻¹(g2)gg₁⁻¹.

(2.9)

Since an orbit of an algebraic group on a variety is locally closed (Section 8.3 in [14]), the orbit(G×G)·l_γ_dhas the same closure in the Zariski topology and in the classical topology. The closure(G×G)·l_γ_d, called aDe Concini–Procesi compactification ofG, is a smooth projective variety of dimensionn= dimG(see [5,§6]). We denote this closure byZd(G).

It is known [5] thatG×Ghas finitely many orbits inZd(G)indexed by subsets ofΓ. Indeed, for eachS⊂Γ, letl_S,d∈ Lbe given by

l_S,d=n_S⊕n⁻_d(S)+

x, γd(x)

: x∈m_S . (2.10)

Chooseλ∈hsuch thatα(λ) = 0forα∈Sandα(λ)>0forα∈Γ−S, and lete^λ:C^∗→H be the one parameter subgroup ofH corresponding toλ. Then it is easy to see that

t→lim+∞Ad_(eλ(t),e)l_γ_d=l_S,d∈Gr(n,g⊕g).

Thusl_S,d∈Z_d(G). It is easy to see thatl_γ_d∈ L^ε(Γ,Γ, d)forε= (dimh−dimh^γ^d) mod 2. Thus l_S,d∈ L^ε(S, d(S), d|S)for the same value ofε.

THEOREM 2.23. – (See [5].)For everyd∈I(Γ,Γ),Z_d(G) =

S⊂Γ(G×G)·l_S,d.

e ◦

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2.8. Closures of(G×G)-orbits inL

Let(S, T, d, V)be a quadruple where(S, T, d)is a generalized BD-triple andV ∈ Lspace(z_S⊕ z_T). Forl_{S,T ,d,V} given in (2.4), we will now study the closure of the (G×G)-orbit through l_{S,T ,d,V} inGr(n,g⊕g). To this end, letGr(m,g_S⊕g_T)be the Grassmannian ofm-dimensional subspaces ing_S⊕g_T, wherem= dimg_S. For the Lie algebra isomorphismγ_d:g_S→g_T given in (2.2), let

l_γ_d=

x, γd(x)

: x∈g_S

∈Gr(m,g_S⊕g_T).

NOTATION 2.24. – LetZ_d(G_S)be the closure of(G_S×G_T)·l_γ_dinGr(m,g_S⊕g_T). Under the identification

G_S→(G_S×G_T)·l_γ_d:g→

(x, γ_dAd_gx): x∈g_S (2.11) ,

Z_d(G_S)can be regarded as a(P_S×P_T⁻)-equivariant compactification ofG_S, whereP_S×P_T⁻ act onG_S by

(p_S, p⁻_T)·g_S=γ_d⁻¹

χ_T(p⁻_T) g_S

χ_S(p_S)₋1

, (p_S, p⁻_T)∈P_S×P_T⁻, g_S∈G_S (2.12)

(see Notation 2.11), and onGr(m,g_S⊕g_T)via the group homomorphismχ_S×χ_T:P_S×P_T⁻→ G_S×G_T.

PROPOSITION 2.25. – For every generalized BD-triple (S, T, d) and every V ∈ Lspace(zS⊕z_T),

(1) the closure(G×G)·l_{S,T ,d,V} inGr(n,g⊕g)is a smooth subvariety ofGr(n,g⊕g)of dimensionn−z, wheren= dimgandz= dimz_S, and the map

a: (G×G)×_(P_S_×P⁻

T)Zd(GS)→(G×G)·l_{S,T ,d,V}: (g1, g2),l

→Ad_(g₁_,g₂₎

V + (nS⊕n⁻_T) +l is a(G×G)-equivariant isomorphism;

(2) (G×G)·l_{S,T ,d,V} is the finite disjoint union (G×G)·l_{S,T ,d,V} =

S1⊂S

(G×G)·l_S₁_,d(S₁_),d₁_,V₁,

where forS1⊂S, we setd1=d|S1, andV1=V+{(x, γd(x)):x∈h_S∩z_S₁} ⊂z_S₁⊕z_T₁. Proof. –SinceG/PS×G/P_T⁻ is complete, the image of ais closed. Since a[(e, e),l_γ_d] = l_{S,T ,d,V} andais(G×G)-equivariant, ahas dense image. Henceais onto. (2) follows easily from the fact thatais onto and the description of orbits inZ_d(G_S)in Theorem 2.23. Further, by this description of orbits, for l∈Z_d(G_S), the nilradical of V + (n_S⊕n⁻_T) +l contains n_S ⊕n⁻_T. It follows that ifa[(g₁, g₂),l]∈a[(e, e), Z_d(G_S)], then(g₁, g₂)is in the normalizer of the nilradical ofV + (nS ⊕n⁻_T) +l. This normalizer, and hence also(g1, g2), is contained inPS×P_T⁻. Now supposea[(g1, g2),l] =a[(x1, x2),l₁]forx1, x2∈Gandl₁∈Zd(GS). Then a[(x⁻¹₁ g1, x⁻¹₂ g2),l] =a[(e, e),l₁], so by the above(x⁻¹₁ g1, x⁻¹₂ g2)∈PS×P_T⁻. It follows easily thatais injective.

To show that the differential a_∗ of a is injective everywhere, it suffices to show a_∗ is injective at all points[(e, e),l]by(G×G)-equivariance. ForX ∈g⊕g, letξX be the vector field on (G×G)×_(P_S_×_P⁻

T)Z_d(G_S)tangent to the (G×G)-action. Recall that the tangent