The Variety of Lagrangian Subalgebras of Real Semisimple Lie Algebras

THE VARIETY OF LAGRANGIAN SUBALGEBRAS OF REAL SEMISIMPLE LIE ALGEBRAS

A Dissertation

Submitted to the Graduate School of the University of Notre Dame in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy

Oleksandra Lyapina

Samuel R. Evens, Director

Graduate Program in Mathematics Notre Dame, Indiana June 2009 THE VARIETY OF LAGRANGIAN SUBALGEBRAS OF REAL SEMISIMPLE LIE ALGEBRAS

Abstract by Oleksandra Lyapina

The purpose of our work is to generalize a result of Evens and Lu [9] from the complex case to the real one. In [9], Evens and Lu considered the variety L of complex Lagrangian subalgebras of g × g, where g is a complex semisimple Lie algebra. They described irreducible components, and showed that all irreducible components are smooth. In particular, some irreducible components are De Concini-Procesi compactiﬁcations of the adjoint group G associated to g, and its geometry at inﬁnity plays a role in understanding L and in applications in Lie theory [10], [11].

In this work we consider an analogous problem, where g0 is a real semisimple Lie algebra. We describe the irreducible components of Λ, the real algebraic variety of Lagrangian subalgebras of g0 × g0.

Let g be the complexiﬁcation of g0. Denote by σ the complex conjugation on g with respect to g0. Let G be the adjoint group of g. Then σ can be lifted to an antiholomorphic involutive automorphism σ : G → G, which we denote by σ too. The set of σ-ﬁxed points Gσ = {g ∈ G : σ(g) = g} is a real algebraic group

σ with Lie algebra g0. We study G using a result of Steinberg [20] and the Atlas project software [23]. Our results also give a description of a compactiﬁcation of Oleksandra Lyapina Gσ. Although our methods are similar to the methods of [9], the real structure introduces some additional diﬃculties. In memory of my mother Neonila Lyapina

ii CONTENTS

TABLES ...... iv

ACKNOWLEDGMENTS ...... v

SYMBOLS ...... vi

CHAPTER 1: INTRODUCTION ...... 1

CHAPTER 2: THE VARIETY OF LAGRANGIAN SUBALGEBRAS . .6 2.1 Evens and Lu’s classiﬁcation of (G × G)-orbits in L ...... 7

CHAPTER 3: PRELIMINARIES ON REAL SEMISIMPLE LIE ALGEBRAS 10

CHAPTER 4: AUTOMORPHISMS OF A REAL SEMISIMPLE LIE AL- GEBRA ...... 19 4.1 The group of quasi-inner automorphisms ...... 19 4.2 QInt g0-orbits through σ-stable parabolic subalgebras ...... 42 4.3 The group of outer automorphisms ...... 43

CHAPTER 5: (Gσ × Gσ)-ORBITS IN Λ ...... 47 5.1 σ-ﬁxed points in L ...... 47 5.2 (Gσ × Gσ)-orbits in Λ ...... 50 σ σ 5.3 Lagrangian subspaces in zS × zT ...... 52 5.4 Closures of (Gσ × Gσ)-orbits in Λ ...... 53 5.4.1 The variety of Lagrangian subalgebras in g0 × g0, where g0 is a compact real form ...... 59 5.4.2 Closures of (Gσ × Gσ)-orbits in Λ, continued...... 60 5.5 Irreducible components ...... 61

BIBLIOGRAPHY ...... 66

iii TABLES

4.1 FUNDAMENTAL GROUPS OF SIMPLE LIE GROUPS OF AD- JOINT TYPE ...... 25 4.2 THE CONNECTED COMPONENTS OF SIMPLE REAL ALGE- BRAIC GROUPS ...... 29

iv ACKNOWLEDGMENTS

I would like to express my inﬁnite gratitude to my advisor, Professor Samuel Evens, for his guidance, encouragement and inspiration provided throughout my graduate studies at Notre Dame. I would like to thank to my undergraduate advisor Professor Eugene Karolin- sky. I wish to thank Eugene for many virtues I have learned from him. I would like to express my appreciation to the members of the doctoral com- mittee, J. Arlo Caine, William Dwyer, Michael Gekhtman, for reviewing the dissertation and helpful discussions. I also wish to thank all of my friends and colleagues for their help and support. I am especially grateful to my father, Yurij Lyapin, for his love and support.

v SYMBOLS

g a complex semisimple Lie algebra g0 a real form of g

σ the complex conjugation on g with respect to g0

τ a Cartan involution on g0, a compact real form on g η a split real form on g compatible with σ

l a Lagrangian subalgebra

L the variety of complex Lagrangian subalgebras

Λ the variety of real Lagrangian subalgebras

vi CHAPTER 1

INTRODUCTION

Let d be a 2n-dimensional Lie algebra over R or C equipped with a nondegenerate symmetric ad-invariant bilinear form h·, ·i (of signature (n, n) in the real case). Let D be a connected group with Lie D = d.

Deﬁnition 1.0.1. A Lie subalgebra l ⊂ d is Lagrangian if l is maximal isotropic, i.e., hx, yi = 0 for all x, y ∈ l, and dim l = n.

The notion of Lagrangian subalgebra was introduced by Drinfeld in [7]. The main motivation for studying Lagrangian subalgebras is through the connection with the theory of Poisson Lie groups. Let us explain this connection in more detail.

Deﬁnition 1.0.2. A Manin pair is a pair (d, g), where g is a Lagrangian subalgebra of d.

Now let us deﬁne the notion of Lie quasi-bialgebra following [6].

Deﬁnition 1.0.3. Let g be a Lie algebra, δ a g ∧ g-valued 1-cocycle of g, and ϕ ∈ V3 g. A triple (g, δ, ϕ) is called a Lie quasi-bialgebra if

1 Alt(δ ⊗ id)δ(x) = ad ϕ for any x ∈ g, 2 x

1 Alt(δ ⊗ id ⊗ id)ϕ = 0, where

adx(a ⊗ b ⊗ c) = [x ⊗ 1 ⊗ 1 + 1 ⊗ x ⊗ 1 + 1 ⊗ 1 ⊗ x, a ⊗ b ⊗ c] and Alt is the alternation map.

Furthermore, if ϕ = 0 we say that (g, δ) is a Lie bialgebra. A Manin pair (d, g) deﬁnes a Lie quasi-bialgebra structure on g uniquely up to twisting t ∈ V2 g [1]. Namely, (g, δ0, ϕ0) is obtained by twisting (by t ∈ V2 g) from (g, δ, ϕ) if

0 δ (x) = δ(x) + adx t for all x ∈ g,

1 ϕ0 = ϕ + Alt(δ ⊗ id)t − CYB(t), 2 where CYB(t) = [t12, t13] + [t12, t23] + [t13, t23]

(see [6]). Twisting is an equivalence relation.

Deﬁnition 1.0.4. A Manin quasi-triple is (d, g, h), where (d, g) is a Manin pair and h is an isotropic linear subspace of d such that d = g ⊕ h.

A Manin quasi-triple (d, g, h) deﬁnes uniquely a Lie quasi-bialgebra structure on g. This structure can be locally integrated to a quasi-Poisson structure πg,h on the Lie subgroup G of D with Lie G = g. Recall that g is a Lie bialgebra if and only if h is also a Lagrangian subalgebra of d. In the latter case (d, g, h) is called a Manin triple, and (G, πg,h) is a Poisson Lie group. In the case d is a complex or real reductive Lie algebra, all Manin triples were classiﬁed in [5].

2 Now denote by L the variety of all Lagrangian subalgebras in d. Let M be a G-homogeneous space, i.e., the action of G on M is transitive. It follows from [7] and [14] that a (quasi-)Poisson (G, πg,h)-homogeneous structure on M is equivalent to a G-equivariant map P : M → L, m 7→ lm such that lm ∩ g = gm, where gm is the Lie algebra of the stabilizer subgroup of G at m.

Thus in order to describe the set of (quasi-)Poisson (G, πg,h)-homogeneous spaces up to local isomorphism it is enough to describe G-conjugacy classes of Lagrangian subalgebras of d. Now we consider L as an algebraic subvariety of the Grassmannian Gr(n, d). The Lie group G acts on L by the adjoint action. Evens and Lu [8] associated to each Manin triple (d, g, h) a (quasi-)Poisson structure Πg,h on L, making L into a (quasi-)Poisson variety. Moreover, each G-orbit in L is a (quasi-)Poisson submanifold and consequently a (G, πg,h)-homogeneous space. The (quasi-)Poisson structure Πg,h on L is constructed in such a way that for any (G, πg,h)-homogeneous space (M, π), the map P : M → L is a (quasi-)Poisson map onto the G-orbit of lm for any m ∈ M. In many cases, the map P : M → L is a local diﬀeomorphism onto its image. Therefore, we can think of G-orbits in L as models for (G, πg,h)- homogeneous (quasi-)Poisson spaces. There are many examples of Lie algebras with a symmetric nondegenerate and ad-invariant bilinear form. The geometry on L depends on d and the bilinear form. Moreover, there are many Lagrangian splittings for a given d, resulting in diﬀerent Poisson structures on L. It is also important to consider D-orbits in L. The D-orbits and their closures have an interesting geometric structure, which is crucial in understanding the geometry of L.

3 Now let g be a semisimple Lie algebra (real or complex) and d := g × g with the bilinear form h·, ·i given by

1 h(x , x ), (y , y )i = (hx , y i − hx , y i ), x , x , y , y ∈ g, 1 2 1 2 2 1 1 g 2 2 g 1 2 1 2

where h·, ·ig is the Killing form on g. The case of a complex semisimple Lie algebra g was considered in [13] and [8]. In [13], Karolinsky classified the Lagrangian subalgebras in g × g. Further, in [9], Evens and Lu studied the geometry of the variety L. In this paper we study the real algebraic variety of Lagrangian subalgebras in d = g × g, where g is a real semisimple Lie algebra. In future work, we plan to study the Poisson geometry of L defined by a Manin triple (d, g, h), where g is embedded diagonally into d. It would also be of interest to study quasi-Poisson geometry on L in the case when there is no Lagrangian subalgebra complementary to g. The paper is organized as follows. In Chapter 2 we recall Evens and Lu’s classification of Lagrangian subalgebras in the complex case. In Chapter 3 we review some facts from the theory of real semisimple Lie groups, which we use in this paper. Most of the results are well known ([2], [16], [11]). In Chapter 4 we study the group of automorphisms of a real semisimple Lie algebra. We describe the group Gσ of quasi-inner automorphisms for a real simple Lie algebra, which is a real analog of the group of inner automorphisms in the complex case. The concept was introduced and studied in [15], but our approach is different. We also prove that the group of automorphisms of a real semisimple Lie algebra is a semidirect product of the group of quasi-inner automorphisms and the group of automorphisms of the corresponding Satake diagram. In Chapter 5, we study

4 (Gσ ×Gσ)-orbits in the real algebraic variety L of Lagrangian subalgebras in g×g. We also describe the closures of the orbits and irreducible components in L.

5 CHAPTER 2

THE VARIETY OF LAGRANGIAN SUBALGEBRAS

Let g0 be a real semisimple Lie algebra equipped with an invariant non-

degenerate symmetric bilinear form h·, ·ig0 .

We consider D(g0) := g0 × g0 with the scalar product

1 h(x , x ), (y , y )i := (hx , y i − hx , y i ) , (2.1) 1 2 1 2 2 1 1 g0 2 2 g0

where x1, x2, y1, y2 ∈ g0.

Let g be the complexiﬁcation of g0, and let G be the adjoint group of g. Let L be the variety of Lagrangian subalgebras of g × g. The structure of L was studied in [9, 13].

Let σ be the complex conjugation on g with respect to g0. Abusing notation, we will also denote by σ the corresponding complex conjugation of G and the complex conjugation of g × g given by σ(x, y) = (σ(x), σ(y)). We set Gσ to be the real algebraic subgroup of σ-ﬁxed points in G. The variety of Lagrangian subalgebras L is stable under σ. Let Lσ denote the real subvariety of σ-ﬁxed points in L. We denote by Λ the real algebraic variety of Lagrangian subalgebras in g0 × g0. We have the following simple proposition.

Proposition 2.0.5. The real variety Λ can be identiﬁed with the variety of σ-ﬁxed points in L via the map a : Lσ → Λ, l 7→ lσ.

6 2.1 Evens and Lu’s classiﬁcation of (G × G)-orbits in L

In [9] Evens and Lu classiﬁed the (G × G)-orbits in the variety L. Before recalling the result, let us introduce some standard notation. Let h be a Cartan subalgebra in g. Denote by ∆+ a choice of positive roots in the set of all roots ∆ of g relative to h, and by Π the set of simple roots in ∆+. For a subset S of Π, let [S] be the set of roots in the linear span of S. Let

X X − X mS = h + gα, nS = gα, nS = g−α. (2.2) α∈[S] α∈∆+\[S] α∈∆+\[S]

The standard parabolic subalgebra of the type S is pS = mS +nS, and the standard

− − − parabolic subalgebra of the opposite-type S is pS = mS + nS . Note that pS is conjugate to the standard parabolic subalgebra of type −w0(S), where w0 is the longest element of the Weyl group. Further, set

gS = [mS, mS], hS = h ∩ gS, and zS = {x ∈ h : α(x) = 0, ∀α ∈ S}. (2.3)

− We denote the connected subgroups of G with Lie algebras pS, pS , zS, mS, nS, − − − and nS by PS,PS ,ZS,MS,NS, and NS respectively. The corresponding group decompositions are

− − − PS = MSNS,PS = MSNS , and MS ∩ NS = {e} = MS ∩ NS .

Deﬁne GS = MS/ZS. The adjoint action of MS on mS leaves gS invariant and induces a natural projection MS → GS, so there are well deﬁned projections

− χS : PS → GS and χS : PS → GS. (2.4)

7 Let S and T be two subsets of Π. A bijection d : S → T is called an isometry if d induces an isomorphism of subdiagrams of the Dynkin diagram with vertices S and T. Denote by I(S,T ) the set of all isometries from S to T. A triple (S, T, d), where d ∈ I(S,T ), is called a generalized Belavin-Drinfeld triple for G. Let us choose a set of Chevalley generators {hα, eα, fα : α ∈ Π} of g. Then {hα, eα, fα : α ∈ S} is a set of Chevalley generators of gS and {hα, eα, fα : α ∈ T } is a set of Chevalley generators of gT .

Deﬁnition 2.1.1. An isometry d : S → T deﬁnes the Dynkin isomorphism γd : gS → gT given on the generators by γd(hα) = hd(α), γd(eα) = ed(α), γd(fα) = fd(α).

Denote by Lspace(zS ×zT ) the variety of Lagrangian subspaces in zS ×zT , deﬁned with respect to nondegenerate restriction of h·, ·i to zS × zT .

For a generalized Belavin-Drinfeld triple (S, T, d) and each V ∈ Lspace(zS ×zT ), set

− lS,T,d,V := V + (nS × nT ) + {(x, γd(x)) : x ∈ gS} ∈ L.

The stabilizer of lS,T,d,V in G × G is

− − − − RS,T,d := {(pS, pT ) ∈ PS × PT : γd(χS(pS)) = χT (pT )} ⊂ PS × PT . (2.5)

Finally, consider the coordinate projections pi : g × g → g, pi(x1, x2) = xi, i =

1, 2. The images of lS,T,d,V under the projections are the corresponding parabolic

− subalgebras, i.e., p1(lS,T,d,V ) = pS, and p2(lS,T,d,V ) = pT .

Theorem 2.1.2. [9] 1) Every (G × G)-orbit in L passes through an lS,T,d,V for a unique quadruple (S, T, d, V ), where (S, T, d) is a generalized Belavin-Drinfeld triple and V ∈ Lspace(zS × zT ).

8 2) The (G × G)-orbit in L through lS,T,d,V is isomorphic to (G × G)/RS,T,d, and it has the complex dimension n − z, where n = dim g and z = dim zS.

− 3) (G × G) · lS,T,d,V ﬁbers over G/PS × G/PT with the ﬁber isomorphic to GS.

9 CHAPTER 3

PRELIMINARIES ON REAL SEMISIMPLE LIE ALGEBRAS

In this chapter we will ﬁx a Cartan subalgebra and a set of simple roots which we will use throughout the paper. Let

g0 = k0 ⊕ a0 ⊕ n0 (3.1)

be an Iwasawa decomposition of g0. The Iwasawa decomposition deﬁnes uniquely

τ −τ a Cartan involution τ of g0 such that k0 = g0 and p0 := (Int k0) · a0 = g0 [16, Theorem 6.51]. We also denote by τ the unique anti-linear extension τ : g → g,

τ so that g = k0 ⊕ ip0 is a compact real form of g.

Let t0 be a maximal abelian subalgebra of the centralizer zk0 (a0) of a0 in k0.

Then h0 = t0 ⊕ a0 is a maximal noncompact Cartan subalgebra of g0, i.e., the

R-diagonalizable subspace a0 has the maximal possible dimension.

Lemma 3.0.3. [16] The maximal noncompact Cartan subalgebras of g0 are conjugate under the action of Int g0.

Let Σ be the set of roots of g0 with respect to a0. Choose the set of positive

+ ∗ P roots Σ ⊂ a0 so that n0 = λ∈Σ+ gλ. Let h be the complexiﬁcation of h0. Clearly,

σ h is a σ-stable Cartan subalgebra of g and h = h0. Denote by ∆ the root system of g with respect to h.

10 Set σt(γ)(x) := γ(σ(x)) for any γ ∈ h∗, x ∈ h. This deﬁnes an anti-linear transformation σt : h∗ → h∗. Let us choose a σ-compatible set of positive roots

+ + t + ∆ ⊂ ∆, i.e., for every α ∈ ∆ such that α|a0 6= 0 we have σ (α) ∈ ∆ [11,

+ Deﬁnition 3.4]. Let Π ⊂ ∆ be the set of simple roots. Set Π0 = {α ∈ Π: α|a0 =

0} and Π1 = Π\Π0. Also let ∆0 be the set of roots which are in the span of Π0 and ∆1 = ∆ \ ∆0. We will refer to Π0 as the set of simple compact roots, and to

∆0 as the set of compact roots. In the sequel we assume that h is a Cartan subalgebra of g such that hσ is a

+ maximal noncompact Cartan subalgebra of g0, and ∆ is a σ-compatible set of positive roots.

P Deﬁnition 3.0.4. The Borel subalgebra b = h⊕ α∈∆+ gα is called a σ-compatible Borel subalgebra.

Notice that a σ-compatible Borel subalgebra is not σ-stable in general because

+ t + if α ∈ ∆0 , then σ (α) ∈ −∆ . But it has the following important property.

Proposition 3.0.5. [22] The group Int g0 acts transitively on the set of the σ- compatible Borel subalgebras.

We also consider h(R) = {x ∈ h : α(x) ∈ R for all α ∈ ∆} = a0 + it0. One can choose a basis {hα : α ∈ Π} of h(R) such that α(hα) = 2. Then this basis can be extended to a set of Chevalley generators {hα, eα, fα : α ∈ Π} of g such that the split real form gsplit generated by {hα, eα, fα : α ∈ Π} over R is compatible with σ, i.e., σ(gsplit) = gsplit. Denote by η the corresponding real structure, i.e.,

η g = gsplit. The compatibility implies ση = ησ.

t + The involutive automorphism ση : g → g preserves h. Moreover, (ση) (∆1 ) =

+ t + + ∆1 and (ση) (∆0 ) = −∆0 . Let W be the Weyl group of g with respect to h.

11 Let w0 be the longest element of the subgroup of W generated by the reﬂections

0 corresponding to the roots in the set ∆0. Then there is a representative n ∈ NG(h) of w0 such that n0ση permutes the simple root vectors, i.e., n0ση(h) = h and

0 0 t 0 n ση{eα : α ∈ Π} = {eα : α ∈ Π}. Thus (n ση) = c ∈ Aut Π and n ση = γc is the Dynkin automorphism corresponding to c ∈ Aut Π and the Chevalley generators

{hα, eα, fα : α ∈ Π}. Thus we deduce

0 σ = n γcη, (3.2)

0 where n , c and η are as above. It is easy to check that c(Πi) = Πi, i = 0, 1, and

0 c|Π0 = −w . Let us deﬁne ω := c|Π1 ∈ Aut Π1.

t Now for any α ∈ Π1 we have σ (α) = β+δ, for some δ ∈ ∆0, and β = ω(α) ∈ Π1

[11]. Notice that for any α, β ∈ Π1 we have α|a0 = β|a0 if and only if α = β or α = ω(β) [21]. These data allows us to construct the Satake diagram [21] of g0 obtained from the Dynkin diagram of g in the following way: all the vertices corresponding to roots of Π0 are black, the vertices corresponding to roots of Π1 are white, and the 2-element orbits of ω are joined by arrows. Notice that a Satake diagram on the set of simple roots Π is uniquely deﬁned by a subset Π0 of Π and an involution ω :Π1 → Π1, where Π1 = Π \ Π0.

Recall that the longest element w0 ∈ W takes a set of simple roots Π into the opposite set of simple roots −Π, and it is the unique element in W with such a

2 property. Clearly, w0 = id.

Deﬁnition 3.0.6. Let w0 be the longest element in the Weyl group. Then ν =

−w0 ∈ Aut Π is called the canonical involution.

0 Corollary 3.0.7. c|Π0 = −w ∈ Aut Π0 is the canonical involution of Π0.

12 Clearly, ν takes each irreducible component of the system Π into itself and induces a canonical involution on the irreducible components. It is therefore suf- ﬁcient to consider the case of a simple Lie algebra g.

Theorem 3.0.8. [12] For all simple noncommutative Lie algebras g other than sln(C), so4n+2(C), and E6, the canonical involution ν of the system of simple roots is the identity transformation. For g = sln(C)(n ≥ 3), so4n+2(C)(n ≥ 1), and

E6, the involution ν coincides with the only nontrivial automorphism of the system Π.

Corollary 3.0.9. The canonical involution is in the center of Aut Π for any Π.

Note that not any subset of Π can be a set of complex roots of a real simple

Lie algebra. For a set Π0 ⊂ Π of simple compact roots the canonical involution −w0 can be extended to an automorphism c ∈ Aut Π. It follows immediately from the arguments leading to (3.2).

Proposition 3.0.10. The canonical involution ν ∈ Aut Π preserves the set of compact roots, i.e., ν(Π0) = Π0.

Proof. The canonical involution of a subset of compact roots Π0 ⊂ Π can be extended to an automorphism c ∈ Aut Π. In the case ν = id, the statement is obvious. Now let us assume that ν 6= id. Then c = ν or c = id. If c = ν the statement is obvious since c(Π0) = Π0.

Finally, ν 6= id and c = id appears for g0 = sll(H)(l ≥ 2), g0 = sup,l+1−p with

0 < l − 2p ≤ 2, and g0 = sop,2l−p with l odd and l − p even. It is straightforward to verify that ν(Π0) = Π0 in each of these cases.

13 Example 1. Let us consider

    XY  g sl   gl 0 = 3(H) =   : X,Y ∈ 3(C), Re tr X = 0 ,  −Y X  which is a real form of g = sl6(C). Let h be the diagonal Cartan subalgebra of g

∗ and W be the Weyl group of g with respect to h. Deﬁne εi ∈ h by

εi(diag(x1, . . . , x6)) = xi.

We consider the following set of simple roots

Π = {α1 = ε1 − ε4, α2 = ε4 − ε3, α3 = ε3 − ε6, α4 = ε6 − ε5, α5 = ε5 − ε2}.

It is clear that h ∩ g0 is a Cartan subalgebra of g0 with the R-diagonalizable subspace

a0 = {diag(x1, x2, x3, x1, x2, x3): xi ∈ R, x1 + x2 + x3 = 0}.

Since dim a0 = 2 equals to the real rank of g0, we conclude that h∩g0 is a maximal

noncompact Cartan subalgebra. We have αi|a0 = 0 for i = 1, 3, 5, and α2, α4 are nonzero on a0. Moreover, α2|a0 6= α4|a0 . Thus the corresponding Satake diagram is

s c s c s

It is deﬁned by Π0 = {α1, α3, α5} and ω = id. Let ∆0 be the set of roots in the linear span of Π0. A compatible split real form is gsplit = sl6(R). Denote by η the corresponding complex conjugation on g. Let w0 be the longest element of

14 the subgroup of W generated by the reﬂections corresponding to the roots in the   0 −E3 0 0   set ∆0. Then σ = n η with n = Ad   , where E3 ∈ Mat3(C) is the E3 0 identity matrix.

The following proposition is obvious.

+ Proposition 3.0.11. There is a Chevalley basis {hα, eα, fα : α ∈ ∆ } on g with the following property. For any α ∈ Π0 we have

σ(eα) = −fα, σ(hα) = −hα, σ(fα) = −eα.

+ For any α ∈ Π1 there is δ ∈ ∆0 such that

σ(eα) = eω(α)+δ, σ(hα) = hω(α)+δ, σ(fα) = fω(α)+δ.

Deﬁnition 3.0.12. A subdiagram of a Satake diagram is a subdiagram of the underlying Dynkin diagram with the set of vertices S, such that all black vertices are in S and if α ∈ S is white, then ω(α) ∈ S.

Proposition 3.0.13. Let ν ∈ Aut Π be a canonical involution and let S ⊂ Π be a set of vertices of a subdiagram of a Satake diagram on Π. Then ν(S) is also a set of vertices of a subdiagram of the Satake diagram.

Proof. By Theorem 3.0.8, ν is either the identity or coincides with the only nontrivial automorphism of the system Π. If ν = id then the statement is obvious. Now let us assume that ν is the only nontrivial automorphism of the system Π.

By Proposition 3.0.10, we have ν(Π0) = Π0 and ν(Π1) = Π1. Thus Π0 ⊂ ν(S)∩S, i.e., all black vertices are in ν(S).

15 Now let us check that if α ∈ ν(S) ∩ Π1, then ω(α) ∈ ν(S). We have ω = c|Π1 for some c ∈ Aut Π, where c is either the identity or c = ν. If c = id, then ω =

id : Π1 → Π1 and the condition is satisﬁed. Now let c = ν. Then ω = c|Π1 = ν|Π1 . Finally, we have

ω(ν(S) ∩ Π1) = ω(ν(S ∩ Π1)) = S ∩ Π1 = ω(S ∩ Π1) = ν(S ∩ Π1) = ν(S) ∩ Π1.

To explain why it is important to consider subdiagrams of a Satake diagram let us recall the deﬁnition of a real parabolic subalgebra.

Deﬁnition 3.0.14. A subalgebra p0 of a real semisimple Lie algebra g0 is said to be parabolic if its complexiﬁcation p is a parabolic subalgebra of the complex Lie algebra g = g0 + ig0.

Proposition 3.0.15. Let S ⊂ Π. The standard parabolic subalgebra pS of type S is σ-stable if and only if S is a subdiagram of the Satake diagram.

Proof. Indeed, S is a subdiagram of the Satake diagram, it is equivalent to Π0 ⊂ S and c(S) = S. Then

0 0 0 0 σ(pS) = n γcη(pS) = n γc(pS) = n (pc(S)) = n (pS). (3.3)

0 0 + + Finally, n maps the nil-radical nS ⊂ pS into itself since w (∆ \ [S]) = ∆ \ [S].

0 Thus n0 ∈ PS and Ad n (pS) = pS.

Now let us prove similar proposition for the standard parabolic subalgebra of the opposite-type S.

16 − Proposition 3.0.16. Let S ⊂ Π. The standard parabolic subalgebra pS of opposite-type S is σ-stable if and only if S is a subdiagram of the Satake diagram.

− Proof. Notice that pS is Int g-conjugate to pν(S), which is σ-stable by Proposition 3.0.13 and the arguments above. The same computations as in (3.3) show that

− pS is σ-stable.

Corollary 3.0.17. Let S be a set of vertices of a subdiagram of the Satake dia-

− gram. Then zS, gS, nS, nS are σ-stable.

Proof. Since S is a set of vertices of a subdiagram of the Satake diagram, pS = zS + gS +nS is σ-stable by Proposition 3.0.15 and clearly σ preserves the decomposition.

Thus zS, gS, nS are σ-stable.

− − Now it is left to prove that nS is σ-stable. Similarly, let us consider pS = − zS + gS + nS , which is also σ-stable by Proposition 3.0.16. Thus σ preserves the − nil-radical nS .

Proposition 3.0.18. [2], [16] Any σ-stable parabolic subalgebra of g is Int g0- conjugate to a unique standard parabolic subalgebra pS, where S is a set of vertices of a subdiagram of the Satake diagram.

Deﬁnition 3.0.19. We call d ∈ Aut Π an automorphism of a Satake diagram if

d preserves the Satake diagram structure, i.e., d(Π0) = Π0 and d|Π1 ◦ ω = ω ◦ d|Π1 .

Denote the group of automorphisms of a Satake diagram by Aut(Π, σ).

For any d ∈ Aut Π consider the Dynkin automorphism γd corresponding to the

+ Chevalley basis {hα, eα, fα : α ∈ ∆ } chosen as in Proposition 3.0.11.

Proposition 3.0.20. The automorphism γd commutes with σ if and only if d ∈ Aut(Π, σ).

17 + + Proof. We have d(∆ ) = ∆ . Let us recall that Π = Π0 ∪ Π1. From (3.2) we get

t t + σ |Π0 = −id|Π0 , and for any α ∈ Π1, σ (α) = ω(α) + δ for some δ ∈ ∆0 .

t t Assume that γdσ = σγd, then (γdσ) = (σγd) .

t t Let α ∈ Π0. Then (σγd) (α) = d(σ (α)) = −d(α). On the other hand

t t t (γdσ) (α) = σ (d(α)). Thus σ (d(α)) = −d(α), which is true if and only if d(α) ∈ Π0.

Further, for any α ∈ Π1 we have

t t (σγd) (α) = d(σ (α)) = d(ω(α) + δ) = d(ω(α)) + d(δ).

t t t Also (γdσ) (α) = σ (d(α)), so σ (d(α)) = d(ω(α)) + d(δ) ∈ ∆1. Since d(α) ∈ Π1, it follows that ωd(α) = dω(α).

Conversely, if d ∈ Aut(Π, σ), then γd clearly commutes with σ.

18 CHAPTER 4

AUTOMORPHISMS OF A REAL SEMISIMPLE LIE ALGEBRA

In this chapter we study automorphisms of real semisimple Lie algebras. Let

σ Aut g0 be the group of all automorphisms on g0. Clearly, Aut g0 = (Aut g) are the automorphisms of g commuting with the complex conjugation σ.

4.1 The group of quasi-inner automorphisms

The conjugation σ on g induces a real form on Int g. Denote the corresponding real structure by σ also. The corresponding algebraic real form

σ QInt g0 = (Int g) = Int g ∩ Aut g0

is called the group of quasi-inner automorphisms of g0 [15]. Clearly, the identity component of QInt g0 is Int g0. The group Int g0 is a linear algebraic group over

R if and only if Int g0 = QInt g0. In this section we will describe the connected

σ components of (Int g) = QInt g0. We begin with simple statements which reduce the problem to the case of a simple Lie algebra g0 such that the complexiﬁcation g is also a simple Lie algebra.

Proposition 4.1.1. Let g0 be a simple real Lie algebra such that g = g0 + ig0 is not a simple Lie algebra. Then

19 1) g0 admits a complex structure, i.e., g0 = dR for a complex simple Lie algebra d.

2) QInt g0 = Int g0 ' Int d.

Proof. The ﬁrst statement is well known, for a proof see, for example, [12]. Hence we have g0 = dR, where d is a complex simple Lie algebra [12]. Now we consider another Lie algebra d over C obtained from d by reversing the sign of the complex structure. It is clear that Int d ' Int d, and an isomorphism Int d → Int d is an antilinear automorphism of Int d. It is known [12] that g ' d ⊕ d. The transformation σ : d ⊕ d → d ⊕ d deﬁned by σ(x, y) = (y, x) is a real structure on d ⊕ d and the map (x, x) 7→ x is an

σ isomorphism of (d ⊕ d) onto g0. The group of complex inner automorphisms on g is isomorphic to Int d × Int d.

σ It is easy to check that (Int d×Int d) = (Int d)diag, where (Int d)diag is the diagonal

σ subgroup of Int d × Int d. Finally, we have QInt g0 = (Int g) = (Int d)diag ' Int d.

In particular, QInt g0 is connected and is the complex Lie group Int d considered as a group over R.

PN Further, let g0 be a semisimple real Lie algebra, and g0 = k=1(gk)0 is the decomposition into simple components. Assume that (gk)0 for k = 1,...,L admit

R complex structures, i.e., (gk)0 = dk for a simple complex Lie algebra dk and the complexiﬁcation of (gk)0 is isomorphic to dk ⊕ dk. Assume also that (gk)0 is a real form of a simple complex Lie algebra gk, for each k = L + 1,...,N.

QN Proposition 4.1.2. π0(QInt g0) = k=L+1 π0(QInt(gk)0).

Proof. The complexiﬁcation g of g0 uniquely decomposes into the direct sum of

20 σ-invariant subalgebras, i.e., we have

L N M M g = (dk ⊕ dk) ⊕ gk, k=1 k=L+1

where dk and gk are simple Lie algebras. Now we have

L N Y Y Int g = Int(dk ⊕ dk) × Int gk. k=1 k=L+1

Thus the σ-stable points are

L N σ Y σ Y σ (Int g) = (Int(dk ⊕ dk)) × (Int gk) . k=1 k=L+1

By Proposition 4.1.1 we have

σ (Int(dk ⊕ dk)) ' Int dk

for k = 1,...,L. The result follows since π0(Int dk) = 1 for k = 1,...,L.

From now on, g0 is a real form of a simple complex Lie algebra g.

Recall that τ is the Cartan involution of g0 deﬁned in the beginning of Chapter 3, i.e., τ is a compact real form on g compatible with σ. Then θ = τσ is an involutive automorphism on g.

Theorem 4.1.3. [12] The mapping σ 7→ θ deﬁnes a bijection from the set of isomorphism classes of real forms on g onto the set of classes of conjugate involutive automorphisms of g.

First, we reduce the problem of finding connected components in QInt g0 to that of finding the group of θ-fixed points in Int g, which is studied in [20].

21 θ Proposition 4.1.4. π0(QInt g0) = π0((Int g) ).

Proof. The conjugation τ induces a Cartan involution on the complex algebraic group Int g considered as a real group. The corresponding Cartan decomposition is Int g = KP, where K is a maximal compact subgroup of Int g and P = exp p (here p is the Cartan subspace of g with respect to τ). The conjugations τ and

σ σ σ σ σ commute, so QInt g0 = K P = K exp p is a Cartan decomposition. Since

σ σ exp(p ) is connected, we have π0(QInt g0) = π0(K ). Let us consider the involutive automorphism θ = τσ : g → g. Notice that θ(K) = K and θ|K = σ|K , thus

σ θ θ θ θ θ π0(K ) = π0(K ). Finally, π0(K ) = π0(K P ) = π0((Int g) ).

Corollary 4.1.5. Let σ be a compact real form. Then QInt g0 = Int g0.

Proof. Indeed, σ is a compact real form compatible with σ, thus θ = id. Hence

θ π0((Int g) ) = π0(Int g) = π0(QInt g0) is trivial.

In the sequel, we will consider only noncompact real forms. We collect some of Steinberg’s results [20] on automorphisms of complex semisimple algebraic groups that will be used later. First, let us introduce some notation.

Let G be a semisimple algebraic group over C and θ a semisimple automorphism of G. Let π : G˜ → G be the universal covering, F = Ker π, i.e., the

˜ −1 fundamental group of G. For a θ-stable subgroup S ⊂ G, we set Sθ := {s · θ(s ): s ∈ S}.

Theorem 4.1.6. [20] Let G˜ be a simply connected semisimple complex Lie group. Let θ : G˜ → G˜ be a semisimple automorphism. Then G˜θ is connected.

Theorem 4.1.7. [20] Let θ be a semisimple automorphism of G. Then there exists a unique semisimple automorphism θ˜ : G˜ → G˜ such that πθ˜ = θπ.

22 By abuse of notation, we will use the same letter θ for θ˜.

Theorem 4.1.8. [20] Let G be a semisimple complex algebraic group and let G˜ be the simply connected cover of G. Let θ be a semisimple automorphism of

θ θ θ θ G. Denote by G0 the identity component of G . Then G /G0 is isomorphic to ˜ (Gθ ∩ F )/Fθ ⊂ F/Fθ.

Now let g be a complex semisimple Lie algebra, b ⊂ g a Borel subalgebra, and h ⊂ b a Cartan subalgebra.

Theorem 4.1.9. [21] Any involution θ ∈ Aut g is Int g-conjugate to Ad t0γd, where γd is the Dynkin automorphism corresponding to d ∈ Aut Π, and t0 ∈ exp(hγd ).

Now we consider G := Int g and let Ge := Intgg be the simply connected cover of G. Let T ⊂ Ge be a maximal torus. In this case, F ⊂ T is the center of Ge. Notice that there is an isomorphism Aut g ' Aut G ' Aut Ge. In particular, Int g ' Int G ' Int Ge.

Corollary 4.1.10. F/Fθ depends only on the class of θ in Aut G/ Int G.

Proof. Let ∆ be the set of roots with respect to T . Let Π ⊂ ∆ be a subset of simple roots. Recall that Aut Π ' Aut G/ Int G. For any θ ∈ Aut G denote by [θ] the class of θ in Aut G/ Int G. Let [θ] = d ∈ Aut Π. Now F is the center, thus for any g ∈ G and f ∈ F we have Int g(f) = gfg−1 = 1. By Theorem 4.1.9, there is

−1 γd g ∈ G such that Int g ◦θ ◦Int g = Int t0 ◦γd, for some t0 ∈ T . Since γd(F ) = F , we have

−1 −1 −1 Fθ = {f · θ(f ): f ∈ F } = {f · (Int g ◦ Int t0 ◦ γd ◦ Int g)(f ): f ∈ F }

23 −1 = {f · γd(f ): f ∈ F }.

Now we assume that θ ∈ Aut G is of the form described in Theorem 4.1.9. Let W be the Weyl group of g with respect to h. Then an inner automorphism acts trivially on W . Further, we set W θ = W d to be the subgroup of θ-ﬁxed points in

θ −1 d W . Finally, for θ = Ad t0γd we denote by Wt0 := {t0 · w(t0 )|w ∈ W }. Note that θ θ if θ is an inner automorphism then W = W , and we refer to Wt0 simply by Wt0 .

d Lemma 4.1.11. [20] Let θ = Adt0γd for some d ∈ AutΠ and t0 ∈ T . Then

˜ θ Gθ ∩ F = (Wt0 · Tθ) ∩ F.

Proposition 4.1.12. Let θ be an involutive automorphism of G and assume that the order of F is odd. Then Gθ is connected.

θ θ 0 −1 Proof. Suppose there is g ∈ G \ G0. Consider g ∈ π (g) ∈ Ge. Then

θ(g0) = fg0 (4.1) for some f ∈ F . By Theorem 4.1.6 Geθ is connected, thus f 6= e. Acting by θ on (4.1) we get

g0 = θ(fg0) = θ(f)θ(g0) = θ(f)fg0.

Thus θ(f) = f −1 and f 2 = fθ(f)−1 = e. Hence f generates a subgroup of F of order 2. And the latter is impossible because the order of F is odd.

It is known that the center of G˜ is the fundamental group of G = Int g. It is

24 well known that this group is equal to the fundamental group of the root system

∆ of g, i.e., π1(G) = π(∆) [12]. In the table below the fundamental groups π(∆) are listed for all indecompos- able reduced root systems ∆.

TABLE 4.1

FUNDAMENTAL GROUPS OF SIMPLE LIE GROUPS OF ADJOINT TYPE

Dl, Dl, ∆ Al Bl Cl E6 E7 E8 F4 G2 l odd l even

π(∆) Zl+1 Z2 Z2 Z4 Z2 × Z2 Z3 Z2 {e} {e} {e}

We are interested in the case when θ is an involutive automorphism.

Corollary 4.1.13. If the complexiﬁcation g = g0 + ig0 is of type A2l, E6, E8,

F4 or G2, then the group of quasi-inner automorphisms of g0 is connected, i.e.,

QInt g0 = Int g0.

θ Proof. By Proposition 4.1.4, π0(QInt g0) = π0(G ), where θ is an involutive au-

θ θ θ tomorphism. Further, by Theorem 4.1.8, we have π0(G ) = G /G0 ⊂ F/Fθ. In the cases E8, F4, and G2, the fundamental group is trivial, thus the statement is obvious. For the types A2l and E6 the fundamental group is of odd order. Thus by Corollary 4.1.12 Gθ is connected.

For a noncompact real semisimple Lie algebra g0 with the complexiﬁcation being one of the types A2l−1, Bl, Cl, Dl or E7 we compute π0(QInt g0) case by case.

25 Let the complexiﬁcation g of g0 be a simple Lie algebra of type Ln. By Theorem 4.1.3 there is a one-to-one correspondence between isomorphism classes of real forms of g and the set of conjugation classes of involutive automorphisms. Further, there is a bijection between the conjugacy classes in Aut g of automorphisms of

(k) order 2 and isomorphism classes of Kac diagrams of types Ln with k = 1 for an inner automorphism θ and k = 2 for an outer one. Let us describe the bijection in more details. Recall that G is the adjoint group of g. Let T be the the Cartan subgroup of

G and h = Lie T . Any involutive automorphism θ is in γdG for some d ∈ Aut Π

2 (here γd is a Dynkin automorphism deﬁned in (3.0.20)). The condition θ = id implies d2 = id ∈ Aut Π, i.e., the order of d is either 1 or 2. Let q be the order of d.

d d Up to conjugation in Int g we may assume that θ ∈ γdT , where T the sub-

d group of γd-stable points in T . We deﬁne the quasitorus S as the direct product

d d d of T and the cyclic group generated by γd. The group χ(S ) of characters of S

d is the direct product of χ(T ) and χ(hγdi) ' Zq. For q = 2 we will denote the

d d elements of Zq as 1 and −1. For each character λ ∈ χ(S ) we denote by λ ∈ χ(T ) the restriction of λ to T d. Then any character λ ∈ χ(Sd) is a pair (λ, k), where

d λ ∈ χ(T ) and k ∈ Zq. The action of Sd on g deﬁnes the decomposition into the direct sum of the weight subspaces X g = gλ. λ∈χ(Sd)

d A nontrivial character α ∈ χ(S ) for which gα 6= 0 is called a d-root of g.A d-root α is called real if α 6= 0, and imaginary otherwise. In the case d is a trivial automorphism we obtain the usual root decomposition whose all roots are real.

26 d d Let ∆ be the system of all d-roots and ∆re be the system of all real d-roots. d Now let us denote by ∆ the set of restrictions to T d of real d-roots.

d Proposition 4.1.14. [12] ∆ is a root system (not necessarily reduced).

d A system of simple roots Πd of ∆ is called a system of simple d-roots of g.

γd Now let g = g1 ⊕g−1, where g1 = g , be the Zq grading deﬁned by the Dynkin automorphism γd. Notice that for q = 1 we have g−1 = 0. It is known [12] that gγd is a simple Lie algebra and Πd is a system of simple roots of gγd . Let

d d Π = {β1, . . . , βl(d)}. Let us extend Π to an admissible system of vectors in the following way. In the case q = 1, let δ be the highest root of g. For q = 2, i.e., d is a nontrivial

γd automorphism on Π, the representation of g in g−1 is irreducible [12]. Let δ be

γd the highest weight of the representation of g in g−1. In both cases, we have

l(d) X 0 0 0 δ = njβj, nj ∈ Z, nj > 0. j=1

γd Let us identify the roots βj (j = 1, . . . , l(d)) of the algebra g with the d-roots

(βj, 1) of the algebra g. Now we add to them the d-root β0 = (−δ, 1) for d trivial,

d and β0 = (−δ, −1) in the case of a nontrivial d. The system Πe = {β0, β1, . . . , βl(d)} of d-roots is called the extended system of simple d-roots of the algebra g [12]. For any λ = (λ, k), µ = (µ, l) ∈ ∆d we deﬁne a scalar product by hλ, µi := hλ, µi. Then Πe d is an admissible system of vectors. There is a unique (up to a scalar multiple) linear relation

l(d) X njβj = 0, j=0

27 0 d where n0 = q, nj = qnj (j = 1, . . . , l(d)). The metric properties of Πe are described by the so called aﬃne Dynkin diagram [12]. Any element x ∈ hd can be identiﬁed with its barycentric coordinates

1 x = − δ(x), x = β (x), . . . , x = β (x), 0 q 1 1 l(d) l(d) satisfying the condition

l(d) X njxj = 1. (4.2) j=0

d d Now we have θ = γd exp 2πix, where x ∈ h = Lie T with the barycentric

2 coordinates x0, x1, . . . , xl(d). The additional condition θ = id implies that

p x = j , p ∈ (j = 0, 1, . . . , l(d)), j q j Z and (4.2) can be rewritten in the form

l(d) X njpj = q. (4.3) j=0

According to [12], up to conjugation in Aut g, we may assume that pi ≥ 0. Thus an involutive automorphism θ ∈ Aut g is uniquely, up to conjugation in Aut g, defined by a connected affine Dynkin diagram with nonnegative integral labels pj satisfying (4.3). A classification of affine Dynkin diagrams is given in

[21]. Now since nj and pj are nonnegative integers, condition (4.3) implies that there are two possibilities:

1) pj = 0 for all j except some j = s; ns = 2 and ps = 1;

28 2) pj = 0 for all j except some j = s, t (s 6= t); ns = nt = 1 and ps = pt = 1. In the second case we may assume that t = 0 if the aﬃne Dynkin diagram is considered up to isomorphism. The Kac diagram of an involution θ is the corresponding aﬃne Dynkin diagram such that the vertices with pj 6= 0 are black.

Theorem 4.1.15. Let g0 be a noncompact real Lie algebra with the complexiﬁca- tion g = g0 + ig0 of type A2l−1, Bl, Cl, Dl or E7. Then the connected components group π0(QInt g0) is given by the following table.

TABLE 4.2

THE CONNECTED COMPONENTS OF SIMPLE REAL ALGEBRAIC GROUPS

(k) g Ln Kac diagram of θ g0 π0(QInt g0)

0 ¨ H ¨¨ HH ¨¨ s HH sl ( ), ¨ H su , {e}, p < l 2l C (1) ¨ ...... H p,2l−p A2l−1 l ≥ 2 1 2 p 2l − 1 1 ≤ p ≤ l , p = l c c s c Z2

0 A(2) , H 2l−1 H ... 1 ¨ <<< sl2l(R) Z2 l ≥ 3 c¨ 2 3 l − 1 l c c c s c 0 HH ¨ ... <<< sll(H) Z2 1s¨ 2 3 l − 1 l c c c c c

29 TABLE 4.2 Continued

(k) g Ln Kac diagram of θ g0 π0(QInt g0)

(1) su1,1 ' sl2(C) A1 Z2 sl ( ) c c 2 R

0 so ( ), H so , 2l+1 C (1) H ...... 2p,2(l−p)+1 Bl 1 ¨ >>> Z2 l ≥ 3 c¨ 2 p l − 1 l 2 ≤ p ≤ l c s c c c 0 HH ¨ ... >>> so2,2l−1 Z2 1s¨ 2 3 l − 1 l c c c c s

{e}, p < l/2 sp2l(C), (1) spp,2l−p, C >>> ...... <<< 2, p = l/2, l 0 1 p l − 1 l l Z l ≥ 2 1 ≤ p ≤ [ 2 ] c c s c c l even

... >>> <<< sp2l(R) Z2 1 1 2 l − 1 l s c c c s

0 l − 1 , p < l/2 H ¨ Z2 so2l(C), (1) H ¨ so2p,2(l−p), D ¨ ...... H 2 × 2, p = l/2, l 1c¨ 2 p l − 2 Hlc l Z Z l ≥ 4 1 ≤ p ≤ [ 2 ] c s c l even c c

0 l − 1 HH ¨¨ ¨ ... H so2,2l−2 Z2 1s¨ 2 3 l − 2 Hlc c c c 0s l −c 1 H ¨ {e}, l odd H ... ¨ ∗ 1 ¨ H l ul (H) s¨ 2 3 l − 2 H c , l even c c c Z2 c s so , (2) ...... 2p+1,2(l−p)−1 l ≥ 3 Dl <<< >>> {e} 0 1 p l − 2 l − 1 0 ≤ p ≤ l c c s c c 2

30 TABLE 4.2 Continued

(k) g Ln Kac diagram of θ g0 π0(QInt g0)

(1) 1 026354 E7 E7 E V Z2 c c cc cc c 7 s 1 026354 E VI {e} c c cc cs c 7 c 1 026354 E VII Z2 s c cc cc s 7 c

The theorem follows from Propositions 4.1.17, 4.1.22, 4.1.18, 4.1.23, 4.1.24. Let us ﬁrst state a preliminary lemma and introduce some more notation.

Let S be a subgroup of the maximal torus T ⊂ Intgg. Take θ = Ad t0γd as in Theorem 4.1.9. The following lemma is obvious.

Lemma 4.1.16. i) θ(S) = γd(S).

ii) If θ is inner, then Tθ = {e}. In particular, Fθ = {e}.

In the sequel, En ∈ GL(n, C) denotes the identity matrix and

Ip,n−p = diag(1,..., 1, −1,..., −1) ∈ GL(n, C) is the diagonal matrix with the ﬁrst p eigenvalues 1 and the other eigenvalues −1.

Proposition 4.1.17. Let g0 be a noncompact real Lie algebra with the complex- iﬁcation g = g0 + ig0 of type A2l−1. Then QInt g0 is connected for g0 = sup,2l−p,

31 1 ≤ p < l, and π0(QInt g0) = Z2 otherwise.

Proof. We have G = PSL(2l, C), Ge = SL(2l, C). The maximal torus T ⊂ SL(2l, C) is the subgroup of the diagonal matrices. The center F ⊂ SL(2l, C)

iπ is the cyclic group of order 2l generated by e l E2l. The Weyl group is the permutation group S2l. Let us ﬁrst consider the class of conjugate involutive automorphisms represented by θp = Ad Ip,2l−p for 1 6 p 6 l. The corresponding real forms are sup,2l−p. iπ Note that Ad Ip,2l−p = Ad e l Ip,2l−p, so θp is an inner automorphism for any p. By Lemma 4.1.16 θp acts trivially on the maximal torus and d = id. Hence

θp Tθ = Fθ = {e} and W = W . It follows from Proposition 4.1.4 and Theorem 4.1.8 and 4.1.11 that

π0(QInt sup,2l−p) = [(WIp,2l−p · Tθ) ∩ F ]/Fθ = WIp,2l−p ∩ F,

−1 where WIp,2l−p = {Ip,2l−p · w(Ip,2l−p): w ∈ W }. If p < l, then for any w ∈ W the −1 matrix Ip,2l−p · w(Ip,2l−p) has some eigenvalues equals 1. Thus for p < l we have −1 −1 Ip,2l−p · w(Ip,2l−p) ∈ F if and only if Ip,2l−p · w(Ip,2l−p) = E2l.

Now if p = l, take wl ∈ W which sends the ith fundamental weight to the

−1 (l + i)th fundamental weight for 1 ≤ i ≤ l. Then Ip,2l−p · wl(Ip,2l−p) = −E2l ∈ F .

Thus QInt sup,2l−p has two connected components for p = l, and is connected otherwise. If l > 1, then there are two conjugacy classes of outer involutions. The ﬁrst one is represented by θ(x) = −xt for x ∈ g and it corresponds to the split real

t form sl2l(R). The second class is represented by θ(x) = − Ad Il,l(x ) for x ∈ g, and the real form is sll(H).

iπ In both cases, let us take g = e 2l diag(1,..., 1, −1) ∈ SL(2l, C), then g ·

32 −1 iπ/l θ(g ) = e E2l = z is a generator of the center of SL(2l, C). It is easy to check

2 that Fθ is generated by z . Thus SL(2l, C)θ ∩ F = F/Fθ ' Z2.

Hence the groups QInt sl2l(R) and QInt sll(H) have two connected components.

Proposition 4.1.18. Let g0 be a noncompact real Lie algebra with the complexi-

ﬁcation g = g0 + ig0 of type Cl, l > 1. Then QInt g0 is connected for g0 = spp,2l−p with 1 ≤ p < l, and π0(QInt g0) = Z2 otherwise.

Proof. We have Ge = Sp(2l, C), with the center F = {±E2l}' Z2. The subgroup

−1 −1 ∗ T = {diag(z1, . . . , zl, z1 , . . . zl ), zi ∈ C }

is a maximal torus in Sp(2l, C). The Weyl group is a semi-direct product of Sl,

l permuting the fundamental weights, by the group (Z2) , acting by zi 7→ (±)zi.

Let us consider conjugation classes of involutions represented by θp = Ad tp,

l −1 −1 1 6 p < 2 , where tp = diag(z1, . . . , zl, z1 , . . . , zl ), with zi = −1, for i = 1, . . . , p, and zp+1 = ... = zl = 1. The corresponding real forms of sp2l(C) are spp,2l−p.

Clearly, θp is an inner automorphism for any p. Thus

θp −1 (Wtp · Tθp ) ∩ F/Fθp = Wtp ∩ F = {tp · w(tp )|w ∈ W } ∩ F.

It is easy to check that the intersection above is nontrivial if and only if l is even

l and p = 2 . l Hence QInt spp,2l−p with 1 ≤ p < is connected, and QInt sp l l (l is even) has 2 2 , 2 two connected components.

The real form sp2l(R) corresponds to the second conjugacy class of involutions in Cl. Let tl = diag(i, . . . , i, −i, . . . , −i) ∈ Sp(2l, C). An involution of the second

33 type is θl = Ad tl. Let us take wl ∈ W, which multiplies all the fundamental

−1 weights by −1, then tl · wl(tl ) = −E2l ∈ F .

Thus the group of quasi-inner automorphisms of sp2l(R) has two connected components.

Now we consider the types Bl and Dl. The simply connected group Spin(n, C) has type Bl for n = 2l + 1 and type Dl for n = 2l. The spin group does not have a simple matrix realization. To apply the results of Steinberg we will introduce coordinates on a maximal torus T ⊂ Spin(n, C) so that the center F ⊂ T is apparent, and the Weyl group action is easy to compute.

Recall that Spin(n, C) is the double cover of the special orthogonal group

SO(n, C). (For n = 2l + 1 the group SO(n, C) is the adjoint group of type Bl; for n = 2l the group SO(n, C) is of type Dl, and it has a nontrivial center isomorphic

n to Z2.) For our purposes it is convenient to choose a basis in C so that the matrix of the SO(n, C)-invariant form is

    0 El 0   0 El     for n = 2l,  E 0 0  for n = 2l + 1.    l  El 0   0 0 1

In this realization

0 −1 −1 ∗ T = {diag(x1, . . . , xl, x1 , . . . , xl ): xi ∈ C } is a maximal torus of SO(2l, C), and

0 −1 −1 ∗ T = {diag(x1, . . . , xl, x1 , . . . , xl , 1) : xi ∈ C }

34 is a maximal torus of SO(2l + 1, C)

Proposition 4.1.19. 1) A maximal torus T ⊂ Spin(n, C) can be realized as

∗ n+1 2 T1 := {(z1, . . . , zn, q) ∈ (C ) : q = z1 . . . zn}.

In this realization

0 2) Let T be the diagonal maximal torus of SO(n, C). The projection p : T1 → T 0 is given by the following formulas.

For n = 2l + 1 (type Bl)

−1 −1 p(z1, . . . , zl, q) = (z1, . . . , zl, z1 , . . . , zl , 1).

For n = 2l (type Dl)

−1 −1 p(z1, . . . , zl, q) = (z1, . . . , zl, z1 , . . . , zl ).

Proof. Let h be the tangent algebra of T . Consider the homomorphism E : h → T deﬁned by

E(x) = exp(2πix).

Consider the root and the weight lattices Q ⊂ P ⊂ h(R)∗ and their dual lattices

∨ ∨ Q ⊂ P ⊂ h(R). Since T is a maximal torus of a simply connected group E is surjective with kernel Q∨. Therefore, we want to ﬁnd a surjective group

∨ ∼ homomorphism E1 : h → T1 with kernel Q . This will prove that T1 = T .

Let consider the usual linear coordinates on h ⊂ son(C) ⊂ gln(C), so that for

35 n = 2l + 1

h = {x = (x1, . . . , xl, −x1,..., −xl, 0) : xi ∈ C}, and for n = 2l

h = {x = (x1, . . . , xl, −x1,..., −xl): xi ∈ C}.

Identify h ' Cl by using only the ﬁrst l coordinates. We have the embedding Q∨ → h. In the coordinates as above

∨ Q = {(x1, . . . , xl): xi ∈ Z, x1 + ... + xl ∈ 2Z}.

Let us deﬁne E1 : h → T1 by the formula

2πix1 2πixl πi(x1+···+xl) E1(x1, . . . , xl) = (e , . . . , e , e ). (4.4)

∨ It is obvious that E1 is surjective. Let us check that E1 has kernel exactly Q . Note that x = (x1, . . . , xn) ∈ Ker(E1) if and only if xi ∈ Z for all i and x1 +...+xl ∈ 2Z,

∨ hence Ker(E1) = Q .

To compute the Weyl group action, recall that the Weyl group is the semidirect product of the group Sl, acting by permutation of the fundamental weights zi, by

l l l the group Θ of sign changes, i.e., acting by zi → (±)izi. In the Bl case, Θ ' (Z2) .

l l−1 In the case of Dl, the group Θ ' (Z2) changes signs in pairs of coordinates, Q i.e., zi → (±)izi with i(±)i = 1.

Proposition 4.1.20. 1) If w ∈ Sl, then

w(z1, . . . , zl, q) = (zw−1(1), . . . , zw−1(l), q),

36 i.e., w acts by permuting the z coordinates and ﬁxes q.

2) Let wi act by changing the sign of the kth coordinate only, then

−1 wk(z1, . . . , zl, q) = (z1, . . . , zk−1, zk , zk+1, . . . , zl, q/zk).

For the type Bl automorphisms wk, k = 1, . . . , l, generate Θl.

For the type Dl all products wjwk ∈ Θl generate Θl.

Proof. Let us recall that there is the homomorphism E1 : h → T1 deﬁned by (4.4).

Let w ∈ Sl. Then

2πix −1 2πix −1 πi(x1+...+xl) w(E1(x1, . . . , xl)) = E1(w(x1, . . . , xl)) = (e w (1) , . . . , e w (l) , e ),

since x1 + ... + xl is unchanged by the action of a permutation.

Further, let wk change the sign of kth coordinate , then

wk(E1(x1, . . . , xl)) = E1(wk(x1, . . . , xl)) = E1(x1,..., −xk, . . . , xl)

= (e2πix1 , . . . , e−2πixk , . . . , e2πixn , eπi(x1+...+xl−2xk)).

Proposition 4.1.21. 1) If n = 2l + 1 is odd, then the center F ' Z2 is generated by

(z1, . . . , zl, q) = (1,..., 1, −1).

2.1) If n = 2l and l is odd, then the center F ' Z4 is generated by

(z1, . . . , zl, q) = (−1,..., −1, i).

37 2.2) If n = 2l with l is even, then the center F ' Z2 × Z2 is generated by

g1 = (z1, . . . , zl, q) = (−1,..., −1, −1) and

g2 = (z1, . . . , zl, q) = (1,..., 1, −1).

Proof. Recall that there is an identiﬁcation F = P ∨/Q∨. The homomorphism

∨ ∨ ∨ ∨ given by (4.4) establishes an isomorphism E : P /Q → F . Further, let α1 , . . . , αl l be the generators of the character group of T . Deﬁne isomorphisms ψ :(C∗) → T

n ∨ ∨ and dψ : C → h by ψ(z1, . . . , zl) = (α1 (z1), . . . , αl (zl)) and dψ is the diﬀerential of ψ.

Now let us consider the mape ˜l : Cl → (C∗)l given by

l 2πix1 2πixl e˜ (x1, . . . , xl) = (e , . . . , e ).

Then the diagram (C∗)l → T ↑ ↑

Cl → h commutes, where the horizontal maps are the isomorphisms ψ :(C∗)l → T and dψ : Cl → h deﬁned below, and the vertical maps aree ˜l and E.

Now let us consider the Bl case. The center F is a cyclic group of order 2. It is generated by g = ψ(1,..., 1, −1). Then g = E(x), where x = (0,..., 0, 1). Hence the center F ⊂ T1 is generated by E1(x) = (1,..., 1, −1).

Next we consider the Dl case with l = 2m + 1. Then F is cyclic of order 4, and it is generated by g = ψ(−1, 1, −1, 1,..., −1, i, −i), where −1 and 1 alternate

38 through the ﬁrst 2m − 1 entries. Then

1 1 1 1 1 g = E(x), x = , − ,..., , , , 2 2 2 2 2 where the signs alternate through the first 2m − 1 entries and we are identifying h with the coordinates for diagonal matrices in so2l. Thus, the center F ⊂ T1 is generated by E1(x) = (−1,..., −1, i). Now let l = 2m be even. Then F is a Klein 4-group, and is generated by g1 = ψ(−1, 1, −1, 1,..., −1, 1, 1, −1) (signs alternate through the first 2m − 2 entries), and by g2 = ψ(1,..., 1, −1, −1) (entry is 1 through the first 2m − 2 items). Then g1 = E(x1), where

1 1 1 1 1 1 x = , − ,..., , − , , 1 2 2 2 2 2 2

and g2 = E(x2) where

x2 = (0,..., 0, 1, 0).

Thus, the center F ⊂ T1 is generated by E1(x1) = (−1,..., −1, −1) and E1(x2) = (1,..., 1, −1).

Proposition 4.1.22. Let g0 be a noncompact real Lie algebra with the complexi-

ﬁcation g = g0 + ig0 of type Bl, l > 1. Then π0(QInt g0) = Z2.

Proof. From the classiﬁcation of the Kac diagrams it follows that all representa- tives of conjugation classes of nontrivial involutions are θp = exp Ad tp, 1 6 p 6 l, p/2 where tp = (−1,..., −1, 1,..., 1, (−1) ) ∈ T1 with the ﬁrst p coordinates −1.

Let us apply Lemma 4.1.11. We take wp ∈ W which changes the sign of pth

p/2 fundamental weight, then wptp = (−1,..., −1, 1,..., 1, −(−1) ) (only the last

39 −1 coordinate is changed). And tpwp(tp ) = (1,..., 1, −1), which is the generator of the center.

Thus the group of quasi-inner automorphisms of so2p,2(l−p)+1, p = 1, . . . , l, has two connected components.

Proposition 4.1.23. Let g0 be a noncompact real Lie algebra with the complexi-

ﬁcation g = g0 + ig0 of type Dl, l > 1. Then

l ∗ 1) π0(QInt g0) = Z2 for g0 = so2p,2(l−p), 1 ≤ p < 2 and for g0 = ul (H) with l even.

2) π0(QInt g0) = Z2 × Z2 for g0 = sol,l, with l even.

3) QInt g0 is connected otherwise.

l Proof. The ﬁrst class of involutions is given by θp = Ad tp, 1 ≤ p ≤ [ 2 ], where p/2 tp = (−1,..., −1, 1,..., 1, (−1) ) ∈ T1 has the ﬁrst p coordinates equal −1. The corresponding real Lie algebras are so2p,2(l−p).

Let us take wpwp+1 ∈ W which changes the signs of pth and (p + 1)st fun-

p/2 damental weights, then wpwp+1(tp) = (−1,..., −1, 1,..., 1, −(−1) ), and tp ·

−1 l wpwp+1(tp ) = (1,..., 1, −1) ∈ F. It is clear, that if p 6= 2 , then it is the only nonunit element of the center which lies in the intersection (Wtp · Tθp ) ∩ F.

l −1 Let l be even and p = 2 . As in the case of any p, we have t l w l w l +1(t l ) = 2 2 2 2

(1,..., 1, −1) ∈ F. Now consider w = w1 · ... · wl ∈ W, which multiplies by −1 all

−1 pairs of coordinates. Then t l w(t l ) = (−1,..., −1, 1) ∈ F, which is the second 2 2 generator of the center.

l Thus, for p = 1,..., 2 , the group of inner automorphisms of the real Lie algebra so2p,2(l−p) has two connected components. And in the case of an even l and the group of the inner automorphisms of sol,l has four connected components.

The second class of conjugate involutive automorphisms is represented by θl =

40 l/2 ∗ Ad tl, where tl = (i, . . . , i, i ) ∈ T1. The corresponding real form is ul (H). −1 If l is an odd number, then {tlw(tl ): w ∈ W } intersects with F at identity.

∗ Thus the group of quasi-inner automorphisms of ul (H) with l odd is connected.

Now let l be an even number. Consider w = w1·...·wl ∈ W , which acts by mul-

−1 l tiplication of all fundamental weights by −1. Then tlw(tl ) = (−1,..., −1, i ) ∈ F and w is the only element of the Weyl group with the property. Thus the group of

∗ quasi-inner automorphisms of ul (H) with l even has two connected components. Finally, let us consider the case of the conjugacy classes of outer involutions.

2 Let d ∈ Aut Dl be nontrivial with d = id and γd the corresponding Dynkin automorphism. We may assume that γd = wl, which is deﬁned in Proposition

l 4.1.20. Consider θp = Ad tpγd, where 1 6 p 6 2 and tp is as above. The involutions l correspond to the real Lie algebras so2p+1,2(l−p)−1, 1 6 p 6 2 .

It is easy to check that Fθp ' Z2 is generated by (1,..., 1, −1) for any p. It is a straightforward computation to check that

d [Wtp · Tθp ] ∩ F ⊂ Fθp

l for any 1 6 p 6 2 . Thus the group of quasi-inner automorphisms of the Lie algebra

l so , 0 p 2p+1,2(l−p)−1 6 6 2 is connected.

Proposition 4.1.24. Let g0 be a noncompact real Lie algebra with the complex- iﬁcation g = g0 + ig0 of type E7. Then π0(QInt g0) = {e} for g0 = EVI, and

π0(QInt g0) = Z2 for g0 = EV,EVII.

41 To calculate π0(QInt g0) when g = g0 + ig0 is of type E7, we use the ATLAS software [23].

4.2 QInt g0-orbits through σ-stable parabolic subalgebras

Notice that for a σ-stable parabolic subgroup PS we have the obvious inclusion

σ σ σ Int g0/(PS ∩ Int g0) ⊂ G /PS ⊂ (G/PS) . (4.5)

The following lemma proves that the inclusions are equalities.

Lemma 4.2.1. Let PS be a σ-stable parabolic subgroup. Then

σ σ σ (G/PS) = G /PS = Int g0/(PS ∩ Int g0).

σ In particular, (G/PS) is a compact connected real algebraic variety.

σ Proof. First, let us recall that QInt g0 = G , and Int g0 ⊂ QInt g0 is the connected component of the identity.

By Proposition 3.0.18 all σ-stable parabolic subgroups of type S are Int g0-

σ σ conjugate to PS. Thus (G/PS) = Int g0/(PS ∩ Int g0). Since G = QInt g0, we

σ σ σ σ have PS = PS ∩ QInt g0. Hence G /PS = QInt g0/(PS ∩ QInt g0) ' (G/PS) .

σ σ σ σ Now G /PS is connected because Int g0 is connected. Finally, G /PS is a

σ compact variety, because it is a ﬁxed set (G/PS) of an involution on the compact variety G/PS.

42 4.3 The group of outer automorphisms

In this section we follow the notation introduced in Chapter 3. Let h0 be

+ the maximal noncompact Cartan subalgebra, and h = h0 + ih0. Let ∆ be a σ- compatible set of positive roots on g, and Π ⊂ ∆+ be the set of simple roots. We

+ choose the Chevalley basis {hα, eα, fα : α ∈ ∆ } as in Proposition 3.0.11. Then

0 σ = n γcη (see (3.2) for more details). Following the steps in [21, Chapter 4.4], consider

t Aut(g0, h0, Π) = {θ ∈ Aut g0 : θ(h0) = h0, θ (Π) = Π}.

t Since θ ∈ Aut g0, it follows from Proposition 3.0.20 that θ ∈ Aut(Π, σ). Assigning

t −1 to an automorphism θ ∈ Aut(g0, h0, Π) the automorphism (θ |Π) ∈ Aut(Π, σ) we get a homomorphism ξ : Aut(g0, h0, Π) → Aut(Π, σ).

Lemma 4.3.1. ξ is surjective.

+ Proof. Fix a Chevalley basis {hα, eα, fα : α ∈ ∆ } of g as in Proposition 3.0.11.

For any d ∈ Aut(Π, σ) there exists a unique automorphism θd ∈ Aut g such that

θd(hα) = hd−1(α), θd(eα) = ed−1(α), θd(fα) = fd−1(α) for any α ∈ Π.

−1 Since d ∈ Aut(Π, σ), we have d ∈ Aut(Π, σ). By Proposition 3.0.20 θd = γd−1 commutes with σ. It is easy to check that θd(h0) = h0. Thus the map

ζ : d 7→ θd (4.6)

is a homomorphism of Aut(Π, σ) into Aut(g0, h0, Π) such that ξζ = idAut(Π,σ).

43 Let Aut(Π\, σ) := Imζ ⊂ Aut(g0, h0, Π), where ζ is deﬁned by (4.6). We see that ξ isomorphically maps Aut(Π\, σ) onto Aut(Π, σ). It is clear that

Aut(g0, h0, Π) = Ker ξ o Aut(Π\, σ).

We denote by H = exp ad h the maximal torus of Int g. It is easy to check that

σ Ker ξ = Aut(g0, h0, Π) ∩ Int g = QInt g0 ∩ H = H ,

σ thus Aut(g0, h0, Π) = H o Aut(Π\, σ). Let b = h + P g . Notice that b is a σ-compatible Borel subalgebra of g, α∈∆+ α σ i.e., b = b ∩ g0 = t0 + a0 + n0 has the maximal possible dimension. Then for any

θ ∈ Aut g0 the image θ(b) is also a σ-compatible Borel subalgebra. By Proposition

3.0.5, there is g ∈ Int g0 such that gθ preserves the Iwasawa decomposition, and stabilizes h and b. Hence

σ Aut g0 = Int g0 · Aut(g0, h0, Π) = Int g0 · H o Aut(Π\, σ).

Now we extend ξ to a homomorphism of the whole group Aut g0 by letting

ξ(Int g0) = id .

The extended homomorphism will be denoted by the same letter ξ. Then

σ Ker ξ = Int g0 · H .

Now QInt g0 ⊂ Aut g0. Every g ∈ QInt g0 induces the identity automorphism on

44 σ Π, so QInt g0 ⊂ Ker ξ. Clearly, Int g0 · H ⊂ QInt g0. And since

σ Ker ξ = Int g0 · H ⊂ QInt g0 ⊂ Ker ξ we have the equality

σ Int g0 · H = QInt g0 = Ker ξ. (4.7)

Finally, we have the following result.

Theorem 4.3.2. Aut g0 = QInt g0 o Aut(Π\, σ). In particular,

Aut g0/ QInt g0 ' Aut(Π, σ).

Proposition 4.3.3. Let S and T be sets of vertices of subdiagrams of the Satake diagram and let gS and gT be the corresponding semisimple subalgebras of g. Let d ∈ I(S,T ) be such that

d(Π0) = Π0 and ω ◦ (d|S∩Π1 ) = (d|S∩Π1 ) ◦ ω. (4.8)

Then the isomorphism γd : gS → gT deﬁned by

γd(eα) = edα, γd(hα) = hdα, γd(fα) = fdα

commutes with σ. Moreover, for every Lie algebra isomorphism µ : gS → gT

σ commuting with σ there is a unique d ∈ I(S,T ) satisfying (4.8) and g ∈ QInt(gT ) such that µ = gγd.

Proof. Arguments analogous to the ones in the proof of Proposition 3.0.20 show

45 that γd commutes with σ.

Further, let us take an arbitarary de∈ I(S,T ) satisfying identities (4.8). Then

0 −1 0 we consider µ = γ µ : gS → gS. It is clear that µ commutes with σ. By Theorem de 0 σ 0 4.3.2, we have µ = gγd0 for some g ∈ QInt(gS) and d ∈ I(S,S) satisfying (4.8).

0 −1 σ Then µ = γ µ = γ (g)γ 0 , where γ (g) = γ ◦ g ◦ γ ∈ QInt(gT ) . Now let us de de dde de de de set d = dde 0 ∈ I(S,T ), then d clearly satisﬁes (4.8).

46 CHAPTER 5

(Gσ × Gσ)-ORBITS IN Λ

In this chapter we will use the classiﬁcation of (G × G)-orbits in L described in Section 2.1 and the one-to-one correspondence between σ-ﬁxed points in L and points in Λ to describe the (Gσ × Gσ)-orbits in Λ. Throughout the chapter

σ G = Int g and G = QInt g0.

5.1 σ-ﬁxed points in L

0 By (3.2) we may assume that σ = n γcη for a certain set of Chevalley genera-

0 tors. Recall that c ∈ Aut Π is deﬁned by c|Π0 = −w , the canonical involution on

Π0, and c|Π1 = ω, where ω ∈ Aut Π1 is deﬁned by the Satake diagram structure.

Proposition 5.1.1. Let l ∈ (G × G) · lS,T,d,V . Then σ(l) ∈ (G × G) · lS0,T 0,d0,V 0 ,

0 0 0 −1 0 where S = c(S),T = c(T ), d = cdc ,V = γcη(V ).

Proof. Since l ∈ (G × G) · lS,T,d,V , there is (g1, g2) ∈ G × G such that

− l = (g1, g2) · lS,T,d,V = (g1, g2) · (V + (nS × nT ) + {(x, γd(x)) : x ∈ gS}).

Further, σ(l) = (σ(g1), σ(g2)) · σ(lS,T,d,V ). Now

− σ(lS,T,d,V ) = σ(V + (nS × nT ) + {(x, γd(x)) : x ∈ gS}) =

47 − σ(V ) + σ(nS × nT ) + σ({(x, γd(x)) : x ∈ gS}).

0 0 To describe the subsets S ,T , let us consider the projection of σ(lS,T,d,V ) onto the ﬁrst and the second components. We have p1(σ(lS,T,d,V )) = σ(pS). Now

− − recall that the Chevalley generators are η-stable, so η(pS) = pS and η(pT ) = pT . − − Further, it follows from the deﬁnition that γc(pS) = pc(S), and γc(pT ) = pc(T ). Thus

0 0 0 σ(pS) = n γcη(pS) = n γc(pS) = n · pc(S).

− Similarly, p2(σ(lS,T,d,V )) = σ(pT ). Then

− 0 − 0 − 0 − σ(pT ) = n γcη(zT + gT + nT ) = n · (γc(zT ) + gc(T ) + γc(nT )) = n · pc(T ).

0 0 Finally, σ({(x, γd(x)) : x ∈ gS}) = (n , n ) ·{(x, γcdc−1 (x)) : x ∈ gc(S)}. Therefore, we have

0 0 −1 σ(l) = (σ(g1), σ(g2)) · σ(lS,T,d,V ) = (σ(g1), σ(g2))(n , n ) · lc(S),c(T ),cdc ,γcη(V ).

σ Lemma 5.1.2. If (G × G) · lS,T,d,V ∩ L 6= ∅, then σ(lS,T,d,V ) ∈ (G × G) · lS,T,d,V .

Proof. If σ(lS,T,d,V ) ∈/ (G×G)·lS,T,d,V , then σ((G×G)·lS,T,d,V ) = (G×G)·lS0,T 0,d0,V 0

0 0 0 0 for a diﬀerent quadruple (S ,T , d ,V ). Suppose there is l = (g1, g2) · lS,T,d,V ∈

(G×G)·lS,T,d,V such that σ(l) = l. Then we have σ(l) = (σ(g1), σ(g2))·σ(lS,T,d,V ).

−1 Hence σ(lS,T,d,V ) = (σ(g1), σ(g2)) · (g1, g2) · lS,T,d,V ∈ (G × G) · lS,T,d,V . But by assumption σ((G × G) · lS,T,d,V ) = (G × G) · lS0,T 0,d0,V 0 . Since the quadruple (S, T, d, V ) describing a (G × G)-orbit is unique, we have a contradiction.

Remark 1. The condition σ(lS,T,d,V ) ∈ (G × G) · lS,T,d,V is necessary but not

48 suﬃcient. Indeed, let us return to the Example 1. In this case σ = n0η with   0 −E3 0   n = Ad   . We ﬁx the Chevalley generators and the order of simple E3 0 roots as in Chapter 3. Let us consider any generalized Belavin-Drinfeld triple

(S, T, d). Then zS ×zT is η-stable. Let V ∈ Lspace(zS ×zT ) be such that η(V ) = V .

0 The Lagrangian subalgebra lS,T,d,V has the property σ(lS,T,d,V ) = n · lS,T,d,V ∈

(G × G) · lS,T,d,V .

Now let S = T = {α1} and d = id. Let V ∈ Lspace(zα1 × zα1 ) be such that η(V ) = V . Consider the coordinate projections pi : g × g → g, pi(x1, x2) =

xi, i = 1, 2, which are clearly commuting with σ. The images of lα1,α1,d,V under the projections are the corresponding parabolic subalgebras, i.e., p1(lα1,α1,id,V ) = pα1 ,

− and p2(lα1,α1,id,V ) = pα1 .

Suppose there are (g1, g2) ∈ (G × G) such that σ((g1, g2) · lα1,α1,id,V ) = (g1, g2) · lα1,α1,id,V . Applying the projection on the ﬁrst coordinate we get σ(g1 · pα1 ) = g1 ·pα1 . From the classiﬁcation of σ-stable parabolic subalgebras in Theorem 3.0.18 it follows that α1 is a subdiagram of the Satake diagram. We get a contradiction.

σ − σ Notice that in this case (G/PS) = ∅ and (G/PT ) = ∅.

σ Lemma 5.1.3. If (G × G) · lS,T,d,V ∩ L 6= ∅, then S and T are subdiagrams of the Satake diagram.

Proof. Assume that

σ (G × G) · lS,T,d,V ∩ L 6= ∅. (5.1)

Let us consider the projections pi, i = 1, 2 of (G × G) · lS,T,d,V on the ﬁrst and the second coordinate. We have p1((G×G)·lS,T,d,V ) = G·pS and p2((G×G)·lS,T,d,V ) =

− σ G · pT . The projection maps commute with σ, so (5.1) implies (G · pS) 6= ∅

49 − σ σ and (G · pT ) = (G · p−w0(T )) 6= ∅. Recall that by Proposition 3.0.13, T is a subdiagram of the Satake diagram if and only if −w0(T ) is so. Finally, notice that

σ by Theorem 3.0.18, we have (G/PS) 6= ∅ if and only if S is a subdiagram of the

− σ Satake diagram. Similarly, (G/PT ) 6= ∅ if and only if T is a subdiagram of the Satake diagram.

Deﬁnition 5.1.4. A generalized Belavin-Drinfeld triple (S, T, d) is said to be σ-compatible if the Dynkin diagrams with vertices S and T are subdiagrams of the Satake diagram, and the isometry d : S → T preserves the Satake diagram structure.

5.2 (Gσ × Gσ)-orbits in Λ

Now we are ready to prove in the real case a theorem analogous to Theorem 2.1.2.

Theorem 5.2.1. 1) The (Gσ × Gσ)-orbits in the variety Λ are parametrized by quadruples (S, T, d, V ), where (S, T, d) is a σ-compatible generalized Belavin-

σ σ Drinfeld triple and V is a real Lagrangian subspace in zS × zT . 2) The (Gσ×Gσ)-orbit in Λ through lσ is isomorphic to (Gσ×Gσ)/Rσ S,T,d,V (C) S,T,d and has the real dimension n − z, where n = dimC g and z = dimC zS. σ σ σ σ − σ 3) (G ×G )·l ﬁbers over (G/PS) ×(G/P ) with the ﬁber isomorphic S,T,d,V (C) T σ to (GS) .

Proof. Let (S, T, d) be a σ-compatible generalized Belavin-Drinfeld triple. Then

− by Corollary 3.0.17 the subalgebras gS, gT , nS, nT , zS and zT are σ-stable. Let V

σ σ be a real Lagrangian subspace in zS × zT . Consider

− lS,T,d,V (C) = V (C) + (nS × nT ) + {(x, γd(x)) : x ∈ gS}.

50 Since d preserves the Satake diagram structure, γd : gS → gT commutes with σ (see

σ σ σ Proposition 3.0.20). Hence σ(lS,T,d,V ( )) = lS,T,d,V ( ) and (G ×G )·l ⊂ Λ. C C S,T,d,V (C) Let us prove the reverse inclusion. Let l ∈ Lσ. We consider the coordinate projections pi : g × g → g, pi(x1, x2) = xi, i = 1, 2. The images of l under the projections are parabolic subalgebras p1 and p2. Since the projection maps commute with σ, we deduce that p1 and p2 are σ-stable parabolic subalgebras of g. From Theorem 3.0.18 it follows that there are gi ∈ Int g0, i = 1, 2 such

− that Ad g1(p1) = pS and Ad g2(p2) = pT , where S and T are vertices of Satake − subdiagrams, and pS and pT are σ-stable standard parabolic subalgebras. Further,

0 − it is proven in [9] that up to Int g0 × Int g0 we can replace l by V + (nS × nT ) +

0 {(x, γ(x)) : x ∈ gS} for some V ∈ Lspace(zS ×zT ) and an isomorphism γ : gS → gT preserving the scalar product. Further, σ preserves the reductive part of l, i.e.,

0 0 σ(V + {(x, γ(x)) : x ∈ gS}) = V + {(x, γ(x)) : x ∈ gS}.

0 0 Thus σ(V ) = V and σ({(x, γ(x)) : x ∈ gS}) = {(x, γ(x)) : x ∈ gS}. From

σ({(x, γ(x)) : x ∈ gS}) = {(x, γ(x)) : x ∈ gS} it follows that γ commutes with σ.

σ By Proposition 4.3.3 we have γ = gγd for some g ∈ QInt(gT ) and an isometry d : S → T preserving the Satake diagram structure. Hence (S, T, d) is a σ- compatible generalized Belavin-Drinfeld triple.

From the equality σ(V 0) = V 0 it follows that V 0 = V (C) for some Lagrangian

σ σ subspace V ⊂ zS × zT . This completes the proof of 1). Further, for a σ-compatible generalized Belavin-Drinfeld triple (S, T, d) the subgroup RS,T,d ⊂ G × G, deﬁned by (2.5), is σ-stable. Let lS,T,d,V be σ-stable.

The stabilizer subgroup of lS,T,d,V in (G × G) is RS,T,d by Theorem 2.1.2. Thus (Gσ × Gσ) · lσ ' (Gσ × Gσ)/Rσ . S,T,d,V (C) S,T,d

51 − Finally, the complex orbit (G × G) · lS,T,d,V (C) fibers over G/PS × G/PT with − fiber (PS × PT )/RS,T,d. The fiber is identified with GS via the map GS → (PS × − PT )/RS,T,d given by

g 7→ (g−1, e), (5.2)

Since (S, T, d) is a σ-compatible generalized Belavin-Drinfeld triple, the ﬁbration

σ and the identiﬁcation map (5.2) commutes with σ. By Lemma 4.2.1 (G/PS) =

σ σ − σ σ − σ G /PS and (G/PT ) = G /(PT ) , which proves 3).

σ σ 5.3 Lagrangian subspaces in zS × zT

Let S be a set of vertices of a subdiagram of the Satake diagram. Assume that the Cartan subalgebra and the set of positive roots are chosen as in the beginning of Section 5. Consider zS deﬁned by (2.3). Then by Proposition 3.0.17

σ zS is σ-stable. Moreover, Π0 ⊂ S implies zS ⊂ a0. Indeed,

σ σ σ zS = {x ∈ h : α(x) = 0 for all α ∈ S} ⊂ {x ∈ h : α(x) = 0 for all α ∈ Π0} = a0.

σ Thus the Killing form is positive deﬁnite on zS. Now let (S, T, d) be a σ-compatible generalized Belavin-Drinfeld triple. Then S and T are sets of vertices of a subdiagram of the Satake diagram. Consider

σ σ σ σ σ (zS × zT ) = zS × zT . The scalar product h·, ·i on zS × zT is nondegenerate

σ σ symmetric of signature (z, z), where z = dimR zS = dimR zT . Further, h·, ·i is σ σ positive deﬁnite on zS × {0} and negative deﬁnite on {0} × zT .

σ σ σ σ Denote by Λ(zS × zT ) the variety of Lagrangian subspaces of zS × zT with

σ σ respect to h·, ·i. Let Gr(z, zS × zT ) be the Grassmannian of z-dimensional real

52 σ σ σ σ σ σ subspaces of zS × zT , so Λ(zS × zT ) ⊂ Gr(z, zS × zT ).

σ σ σ Proposition 5.3.1. [8] Λ(zS × zT ) is a smooth algebraic subvariety of Gr(z, zS ×

σ zT ) diﬀeomorphic to the real orthogonal group O(z). There are two connected

σ σ components in Λ(zS × zT ).

0 σ σ 1 σ σ Denote the connected components by Λ (zS × zT ) and Λ (zS × zT ).

5.4 Closures of (Gσ × Gσ)-orbits in Λ

Deﬁnition 5.4.1. We will call a subvariety of a real algebraic variety closed if it is a set of real points of a closed complex subvariety.

To describe the closures of the (Gσ ×Gσ)-orbits in Λ, we will follow the scheme of arguments in [9]. Evens and Lu have described the closures of (G × G)-orbits in L. We recall the result now. Let (S, T, d) be a generalized Belavin-Drinfeld triple. Let s = dimC gS. Consider lγd = {(x, γd(x)) : x ∈ gS} as an element of the

Grassmannian Gr(s, gS × gT ) of s-dimensional subspaces of gS × gT . Let Zd(GS)

be the closure of (GS × GT ) · lγd in Gr(s, gS × gT ). Under the identiﬁcation

GS → (GS × GT ) · lγd , g 7→ {(x, γd(Adg x)) : x ∈ gS}, (5.3)

Zd(GS) can be regarded as the De Concini-Procesi compactiﬁcation of GS. Hence

Zd(GS) is a smooth complex projective variety of dimension s [4].

− Let the group PS ×PT act on Gr(m, gS ×gT ) through the group homomorphism

53 − − χS × χT : PS × PT → GS × GT , and let PS × PT act on GS by

− −1 − −1 − − (pS, pT ) · gS = γd (χT (pT ))gS(χS(pS)) , (pS, pT ) ∈ PS × PT , gS ∈ GS. (5.4)

− Then the embedding in (5.3) is PS × PT -equivariant. In particular, Zd(GS) is a − − PS ×PT -equivariant compactiﬁcation of GS for the action of PS ×PT on GS given in (5.4).

It is known [4] that GS × GT has ﬁnitely many orbits in Zd(GS) and they are

indexed by subsets of S. Namely, for any S1 ⊂ S consider lS1,d given by

l = n × n− + {(x, γ (x)) : x ∈ m }, S1,d S1 d(S1) d S1 where X X n = g ⊂ g , n− = g ⊂ g . S1 α S d(S1) −α T + + α∈[S] \[S1] α∈[T ] \[d(S1)]

Theorem 5.4.2. [4] For every d ∈ I(S,T ), we have the partition into locally closed GS-orbits

[ Zd(GS) = (GS × GT ) · lS1,d. (5.5)

S1⊂S

Now let (S, T, d) be a σ-compatible generalized Belavin-Drinfeld triple.

Theorem 5.4.3. Zd(GS) is σ-stable with the set of σ-ﬁxed points

σ [ σ σ σ Zd (GS) = (GS × GT ) · lS1,d. Π0⊂S1⊂S, ω(S1\Π0)=S1\Π0

σ σ σ Moreover, Zd (GS) is a smooth and compact subvariety of Gr(s, gS × gT ) of the

54 real dimension s.

Proof. Notice that σ is a real form on gS and gT and on their adjoint groups GS and GT . Since d ∈ I(S,T ) preserves the Satake diagram structure, by repeating the steps in the proof of Proposition 3.0.20 one can check that γd : gS → gT

σ σ commutes with σ. Thus σ(lγd ) = lγd . Moreover, γd(GS) = GT .

From σ(lγd ) = lγd it follows that the orbit (GS × GT ) · lγd is σ-stable. Since

(GS × GT ) · lγd is dense in Zd(GS) and σ is continuous, it follows that Zd(GS) is

σ σ σ also σ-stable. Thus Zd (GS) is a smooth and compact subvariety of Gr(s, gS × gT ) of real dimension s [19, 1.14].

Consider the isomorphism γed : gS × gT → gS × gS given by γed(x, y) = −1 (x, γd (y)). Then γed commutes with σ since γd does so. Let us associate to a

Lagrangian subalgebra l in gS × gT the Lagrangian subalgebra γed(l) ⊂ gS × gS.

Clearly, γed is an isomorphism between the varieties of Lagrangian subalgebras in gS × gT and of those in gS × gS. Let us apply Theorem 5.2.1 to the variety of La-

σ σ grangian subalgebras in gS ×gS. From the theorem it follows that for a subset S1 of S the set of σ-ﬁxed Lagrangian subalgebras in γ ((G ×G )·l ) = (G ×G )·l ed S T S1,d S S S1,id is non-empty if and only if Π0 ⊂ S1 ⊂ S, and ω(S1 \ Π0) = S1 \ Π0. Then ((G × G ) · l )σ = (Gσ × Gσ ) · lσ . Now we apply γ −1 to the results above S S S1,id S S S1,id ed to conclude that

σ [ σ σ σ Zd (GS) = (GS × GT ) · lS1,d. Π0⊂S1⊂S, ω(S1\Π0)=S1\Π0

The following result is a real analogue of Proposition 2.27 in [9].

55 Theorem 5.4.4. Let (S, T, d) be a σ-compatible generalized Belavin-Drinfeld tri-

σ σ ple and take V ∈ Λ(zS × zT ). Then

σ σ σ 1) The closure (G × G ) · l in Gr(n, g0 × g0) is a smooth subvariety S,T,d,V (C) of Λ of the real dimension n − z, where n = dimC g and z = dimC zS, and the map

σ σ σ σ σ σ a :(G × G ) × σ − σ Zd (GS) → (G × G ) · l (PS ×(PT ) ) S,T,d,V (C)

σ − σ [(g1, g2), l] 7→ Ad(g1,g2)(V + (nS × (nT ) ) + l) is a (Gσ × Gσ)-equivariant isomorphism. 2) We have a ﬁnite disjoint union

[ (Gσ × Gσ) · lσ = (Gσ × Gσ) · lσ , S,T,d,V (C) S1,d(S1),d1,V1(C) Π0⊂S1⊂S, ω(S1\Π0)=S1\Π0

h zσ where d1 = d|S1 , and V1 = V + {(x, γd(x)) : x ∈ S ∩ S1 }.

Proof. Consider the closure (G × G) · lS,T,d,V (C) in Gr(n, g × g). It follows from [9] that (G × G) · lS,T,d,V (C) is a smooth complex subvariety in L of complex dimension n − z, where n = dimC g and z = dimC zS. Moreover,

[ (G × G) · lS,T,d,V (C) = (G × G) · lS1,d(S1),d1,V1(C), S1⊂S

h zσ where d1 = d|S1 , and V1 = V + {(x, γd(x)) : x ∈ S ∩ S1 }. Since (S, T, d) is a σ-compatible generalized Belavin-Drinfeld triple and V ∈

σ σ Λ(zS × zT ), we have σ(lS,T,d,V (C)) = lS,T,d,V (C), so the orbit (G × G) · lS,T,d,V (C) is also σ-stable. Thus the closure (G × G) · lS,T,d,V (C) is σ-stable.

56 Further, from Theorem 5.2.1 it follows that

[ ((G × G) · l )σ = (Gσ ×Gσ)·lσ , (5.6) S,T,d,V (C) S1,d(S1),d1,V1(C) Π0⊂S1⊂S, ω(S1\Π0)=S1\Π0

h zσ where d1 = d|S1 , and V1 = V + {(x, γd(x)) : x ∈ S ∩ S1 }. σ σ σ σ Now ((G × G) · lS,T,d,V ( )) is the closure of (G ×G )·l in the analytic C S,T,d,V (C) topology. Indeed, for any S1 ⊂ S such that Π0 ⊂ S1, and ω(S1 \ Π0) = S1 \ Π0, we consider h ∈ a0 = {x ∈ h0 : α(x) = 0 for all α ∈ Π0} such that α(h) = 0 for all α ∈ S1 and α(h) > 0 for all α ∈ Π \ S1. The choice of h is possible because the conditions on Π \ S1 ⊂ Π1. Consider the one parameter subgroup

∗ exp(th): R → Int g0. Then

lim Ad(exp(th), e)lS,T,d,V ( ) = lS ,d(S ),d ,V ( ). t→+∞ C 1 1 1 1 C

We have

σ σ σ σ σ σ σ (G × G ) · l ⊂ ((G × G) · lS,T,d,V ( )) ⊂ (G × G ) · l . S,T,d,V (C) C S,T,d,V (C)

σ σ σ σ It follows that ((G × G) · lS,T,d,V ( )) is dense in (G × G ) · l . Hence C S,T,d,V (C)

σ σ σ σ (G × G ) · l = ((G × G) · lS,T,d,V ( )) . S,T,d,V (C) C

This proves 2). According to [9, Proposition 2.7] there is the (G×G)-equivariant isomorphism

a :(G × G) × − Zd(GS) → (G × G) · lS,T,d,V ( ) (PS ×PT ) C

57 − [(g1, g2), l] 7→ Ad(g1,g2)(V (C) + (nS × nT ) + l).

− Notice that PS,PT and Zd(GS) are σ-stable. It is easy to check that a commutes with σ. Thus a establishes a (Gσ × Gσ)-equivariant isomorphism between the sets of σ-ﬁxed points

σ σ a : ((G × G) × − Zd(GS)) → ((G × G) · lS,T,d,V ( )) . (PS ×PT ) C

Thus to prove 1) it is enough to check that

σ σ σ σ ((G × G) × − Zd(GS)) = (G × G ) × σ − σ Z (GS). (PS ×PT ) (PS ×(PT ) ) d

σ σ σ σ Clearly, (G × G ) × σ − σ Z (GS) ⊂ ((G × G) × − Zd(GS)) . Since (PS ×(PT ) ) d (PS ×PT ) σ σ ((G × G) × − GS) is nonempty, it is dense in ((G × G) × − Zd(GS)) . (PS ×PT ) (PS ×PT ) It follows from Lemma 4.2.1 that

σ σ σ σ ((G × G) × − GS) = (G × G ) × σ − σ G . (PS ×PT ) (PS ×(PT ) ) S

σ σ σ σ σ σ σ Now (G × G ) × σ − σ G ⊂ (G × G ) × σ − σ Z (GS), thus (G × (PS ×(PT ) ) S (PS ×(PT ) ) d σ σ σ G ) × σ − σ Z (GS) is dense in ((G × G) × − Zd(GS)) . (PS ×(PT ) ) d (PS ×PT ) − Since PS and PT are σ-stable, it follows from Lemma 4.2.1 that (G/PS ×

− σ σ σ σ G/P ) is connected and compact. Hence (G × G ) × σ − σ Z (GS) is a T (PS ×(PT ) ) d σ closed subset of ((G × G) × − Zd(GS)) . (PS ×PT )

Consider a triple (Π0, Π0, d), where d ∈ Aut(Π0). By Theorem 5.2.1, every

σ σ (G × G )-orbit corresponding to the triple (Π0, Π0, d) goes through a unique

58 Lagrangian subalgebra of the form

lσ = V + (nσ × (n− )σ) + {(x, γ (x)) : x ∈ gσ }, Π0,Π0,d,V (C) Π0 Π0 d Π0

zσ zσ for some V ∈ Λ( Π0 × Π0 ). By Theorem 5.4.4

(Gσ × Gσ) · lσ = (Gσ × Gσ) · lσ , Π0,Π0,d,V (C) Π0,Π0,d,V (C)

zσ zσ i.e., the orbit is closed for any V ∈ Λ( Π0 × Π0 ). Further, from Theorem 5.4.4 it follows that (Gσ × Gσ) · lσ is (Gσ × Gσ)-equivariantly isomorphic to Π0,Π0,d,V (C) σ σ σ σ − σ (G × G ) ×(P ×(P ) ) GΠ0 . Notice that σ|GΠ is a compact real form. Π0 Π0 0

5.4.1 The variety of Lagrangian subalgebras in g0 × g0, where g0 is a compact real form

Let τ be a compact real form on a semisimple complex Lie algebra g. Let Π be the set of simple roots of g and G the adjoint group. It follows from Corollary 4.1.5 that Gτ is connected and compact. Further, all τ-compatible generalized

Belavin-Drinfeld triples are of the form (Π, Π, d), where d ∈ Aut Π. Denote by Λτ

τ τ τ τ the variety of Lagrangian subalgebras in g × g . Thus any (G × G )-orbit in Λτ

is uniquely deﬁned by d ∈ Aut Π and is the orbit through lγd .

τ τ τ The (G × G )-orbit in Λτ through lγd can be identiﬁed with G by the map

τ τ τ τ G → (G × G ) · lγd , g 7→ {(x, Adg γd(x)) : x ∈ g }.

The following proposition is immediate.

τ τ τ Proposition 5.4.5. 1) Any (G × G )-orbit in Λτ is isomorphic to G . In par-

59 ticular, the orbits are closed and connected.

τ 2) Λτ is isomorphic to Aut g and has | Aut Π| connected components.

5.4.2 Closures of (Gσ × Gσ)-orbits in Λ, continued.

Corollary 5.4.6. The closed orbits in Λ correspond to the quadruples of the form

(Π0, Π0, d, V ), where (Π0, Π0, d) is a σ-compatible generalized Belavin-Drinfeld triple and V ∈ Λ(zσ × zσ ). Further, (Gσ × Gσ) · lσ is connected. Π0 Π0 Π0,Π0,d,V (C)

Proof. The closedness of (Gσ × Gσ) · lσ follows form Theorem 5.4.4. Π0,Π0,d,V (C) Now (Gσ × Gσ) · l fibers over (Gσ/P σ × Gσ/(P − )σ) with the fibers Π0,Π0,d,V Π0 Π0 σ σ σ isomorphic to GΠ0 . Further, GΠ0 is a compact real form of GΠ0 , so GΠ0 is connected by Corollary 4.1.5. Since the fiber and the base are connected, the orbit is connected.

Now in the case of a noncompact real form we have to following result.

Proposition 5.4.7. Let (S, T, d) be a σ-compatible generalized Belavin-Drinfeld triple and V ∈ Λ(zσ × zσ ). Then (Gσ × Gσ) · lσ is connected. There are S T S,T,d,V (C)

2 · | Aut Π0| connected components in Λ.

Proof. The orbit (Gσ × Gσ) · lσ has a unique closed orbit S,T,d,V (C)

(Gσ × Gσ) · lσ , where d = d| , Π0,Π0,d1,V1(C) 1 Π0 in its closure. It follows from Corollary 5.4.6 that (Gσ × Gσ) · lσ is Π0,Π0,d1,V1(C) connected. Further, every point in Λ is in the same connected component as lσ Π0,Π0,d1,V1(C) zσ zσ zσ zσ for some d1 ∈ Aut Π0 and V1 ∈ Λ( Π0 × Π0 ). The variety Λ( Π0 × Π0 ) has two connected components. Thus Λ has 2 · | Aut Π0| connected components.

60 Let G be a arbitrary complex semisimple Lie group, dimC G = n. Let σ be a real form on G. Now let us apply Theorem 5.4.4 and Proposition 5.4.7 to the σ-compatible generalized Belavin-Drinfeld triple (Π, Π, d) and the corresponding

σ σ σ σ-stable Lagrangian subalgebra lΠ,Π,d,0. The closure of the orbit (G ×G )·lΠ,Π,d,0 is (Gσ × Gσ)-equivariant isomorphic to the De Concini-Procesi compactiﬁcation

σ σ Zd (G) of G in Gr(n, g0 × g0).

σ Corollary 5.4.8. Zd (G) is a smooth and connected subvariety of Gr(n, g0 × g0)

σ of dimension n = dimC G = dimR G .

5.5 Irreducible components

Deﬁnition 5.5.1. We will call a real algebraic variety irreducible if it is a set of real points of a complex irreducible subvariety.

Let (S, T, d) be a generalized Belavin-Drinfeld triple. The G × G-orbits in L of type (S, T, d) are

[ L (S, T, d) = (G × G) · lS,T,d,V V ∈L (zS ×zT )

− − (see 2.9 [9]). The group PS × PT acts on GS by (5.4). Also PS × PT acts trivially

on L (zS × zT ). In the complex case the following theorem is true.

Theorem 5.5.2. [9] For every S,T ⊂ Π, d ∈ I(S,T ), and ∈ {0, 1}, we have 1) L(S, T, d) is a smooth and connected subvariety of Gr(n, g×g) of dimension

z(z−3) n + 2 , where n = dimC g and z = dimC zS. 2) The map

a :(G × G) × − (GS × L (zS × zT )) → (G × G) · lS,T,d,V (PS ×PT )

61 − [(g1, g2), (g, V )] 7→ Ad(g1,g2)(V + (nS × nT ) + {(x, γd Ad g(x)) : x ∈ gS}) is a (G × G)-equivariant isomorphism.

Similarly in the real case, for a σ-compatible generalized Belavin-Drinfeld triple (S, T, d), the Gσ × Gσ-orbits in Λ of type (S, T, d) are

[ Λ(S, T, d) = (Gσ × Gσ) · lσ , S,T,d,V (C) σ σ V ∈Λ (zS ×zT )

where ∈ {0, 1} and the union is disjoint. The bundle structure (G×G)× − (PS ×PT ) (GS × L (zS × zT )) is σ-compatible and the map a commutes with σ. This allows us to describe the geometry of Λ(S, T, d) by analogy with the complex case.

Theorem 5.5.3. Let (S, T, d) be a σ-compatible generalized Belavin-Drinfeld triple. Then

1) Λ (S, T, d) is a smooth subvariety of Gr(n, g0 × g0) of real dimension n +

z(z−3) σ 2 , where n = dimC g = dimR g0 and z = dimC zS = dimR zS. 2) The map

σ σ σ σ σ a :(G × G ) × σ − σ (G × Λ (z × z )) → Λ (S, T, d) (5.7) (PS ×(PT ) ) S S T

σ − σ σ [(g1, g2), (g, V )] 7→ Ad(g1,g2)(V + (nS × (nT ) ) + {(x, γd Ad g(x)) : x ∈ (gS) }) is a (Gσ × Gσ)-equivariant isomorphism.

σ Notice that GS is not connected in general, thus Λ (S, T, d) is not necessary

σ connected. From (5.7) it follows that π0(Λ (S, T, d)) = π0(GS). But in the same way as in the case of a single orbit, the closure of Λ(S, T, d) is connected. The

following theorem describes the geometry of the closure of Λ (S, T, d) in Gr(n, g0 ×

62 g0). The closure is taken in the Zariski topology, and we will show that it coincides with the closure in the analytic topology.

Theorem 5.5.4. 1) The closure Λ(S, T, d) is a smooth and connected algebraic variety and the map

σ σ σ σ σ a :(G × G ) × σ − σ (Z (GS) × Λ (z × z )) → Λ (S, T, d) (PS ×(PT ) ) d S T

σ − σ [(g1, g2), (l,V )] 7→ Ad(g1,g2)(V + (nS × (nT ) ) + l). is a (Gσ × Gσ)-equivariant isomorphism. 2) We have a disjoint union

[ [ Λ(S, T, d) = (Gσ × Gσ) · lσ , S1,d(S1),d1,V1(C) σ σ V ∈Λ (zS ×zT ) Π0⊂S1⊂S,ω(S1\Π0)=S1\Π0

h zσ where d1 = d|S1 , and V1 = V + {(x, γd(x)) : x ∈ S ∩ S1 }. 3) Λ(S, T, d) is an irreducible subvariety of Λ.

Proof. The proof of the theorem is based on the description of the complex variety L(S, T, d) in [9, Theorem 2.31]. The proof of 1) and 2) is similar to the proof of Theorem 5.4.4, and we omit it. Notice that Λ(S, T, d) is a real form of L(S, T, d), which is an irreducible component of L. By descent theory Λ(S, T, d) is irreducible too.

The variety of Lagrangian subalgebras can be represented as the ﬁnite union

[ [ Λ = Λ(S, T, d), ∈{0,1} (S,T,d) where (S, T, d) runs over the set of all σ-compatible generalized Belavin-Drinfeld

63 triples. The description of the irreducible components of Λ follows from the complex case result [9, Theorem 2.34]). The irreducible components of Λ are those Λ(S, T, d)’s not properly contained in some other such set.

Theorem 5.5.5. Λ(S, T, d) is an irreducible component of Λ unless |Π \ S| = 1,

γd1 T = d1(S) for some d1 ∈ I(Π, Π), d = d1|S and = (dimC h − dimC h ) mod 2.

Example 2. Let g0 = sl2(R). Then g = sl2(C), the corresponding adjoint group is G = PSL(2, C). The Satake diagram of g0 coincides with the Dynkin diagram of g.

Let σ be the complex conjugation of g with respect to g0. Then

  i 0 σ   G = PSL(2, R) ∪ Ad   · PSL(2, R) 0 −i has two connected components. The σ-compatible generalized Belavin-Drinfeld triples are (α, α, id), where α is a simple root of g, and (∅, ∅, id).

σ σ The unique orbit corresponding to the triple (α, α, id) is (G × G ) · (g0)id, where (g0)id is g0 embedded diagonally into g0 × g0. The orbit is isomorphic to Gσ, so the orbit has two connected components.

There are two orbits corresponding to the triple (∅, ∅, id). Let x 6= 0 ∈ h0 = zσ . Then Λ(h × h ) = {V ,V }, where V = (x, x) and V = (x, −x). We ∅ 0 0 + − + R − R have

σ σ 1 1 (G × G ) · l∅,∅,id,V ' RP × RP ( = −, +).

The orbits are closed and connected.

64 There are two irreducible components in Λ. The ﬁrst one is

σ σ σ σ σ σ (G × G ) · (g0)id = (G × G )(g0)id ∪ (G × G )l∅,∅,id,V+ .

It is the De Concini-Procesi compactiﬁcation of Gσ. The component is isomorphic to RP 3.

σ σ The second irreducible components is (G × G ) · l∅,∅,id,V− . The variety Λ has two connected components.

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