Lectures on Real Semisimple Lie Algebras and Their Representations (ESI Lectures in Mathematics & Physics)

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ESI Lectures in Mathematics and Physics Editors Joachim Schwermer (Institute for Mathematics, University of Vienna) Jakob Yngvason (Institute for Theoretical Physics, University of Vienna) The Erwin Schrödinger International Institute for Mathematical Physics Boltzmanngasse 9 A-1090 Wien Austria The Erwin Schrödinger International Institute for Mathematical Phyiscs is a meeting place for leading experts in mathematical physics and mathematics, nurturing the development and exchange of ideas in the international community, particularly stimulating intellectual exchange between scientists from Eastern Europe and the rest of the world. The purpose of the series ESI Lectures in Mathematics and Physics is to make selected texts arising from its research programme better known to a wider community and easily available to a worldwide audience. It publishes lecture notes on courses given by internationally renowned experts on highly active research topics. In order to make the series attractive to graduate students as well as researchers, special emphasis is given to concise and lively presentations with a level and focus appropriate to a student's background and at prices commensurate with a student's means.

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Arkady L. Onishchik

Lectures on Real Semisimple Lie Algebras and Their Representations M

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European Mathematical Society

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Author: Arkady L. Onishchik Faculty of Mathematics Yaroslavl State University Sovetskaya 14 150 000 Yaroslavl Russia

2000 Mathematics Subject Classification (primary; secondary): 17-01, 17B10, 17B20; 22E46

Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.ddb.de.

ISBN 3-03719-002-7 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2004 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C1 CH-8092 Zürich Switzerland Phone: +41 (0)1 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany 987654321

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii §1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

§2. Complexification and real forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 §3. Real forms and involutive automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 §4. Automorphisms of real semisimple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . 27 §5. Cartan decompositions and maximal compact subgroups . . . . . . . . . . . . . . . . 36 §6. Homomorphisms and involutions of complex semisimple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 §7. Inclusions between real forms under an irreducible representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 §8. Real representations of real semisimple Lie algebras . . . . . . . . . . . . . . . . . . . . . 64 §9. Appendix on Satake diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Introduction These notes reproduce the lectures which I gave in Masaryk University (Brno, Czech Republic) and in E. Schr¨ odinger Institute for Mathematical Physics (Vienna, Austria) during the autumn semesters 2001 and 2002, respectively. The main goal was an exposition of the theory of finite dimensional representations of real semisimple Lie algebras. ´ Cartan [3]. Iwahori [13] gave an The foundation of this theory was laid in E. ´ updated exposition of the E. Cartan’s work (see also [10], Ch. 7). This theory reduces the classification of irreducible real representations of a real Lie algebra g0 to description of the so-called self-conjugate irreducible complex representations of this algebra and to calculation of an invariant of such a representation (with values ±1) which is called the index. No general method for solving any of these two problems were given in [10, 13], but they were reduced to the case when g0 is simple and the highest weight of an irreducible complex representation is fundamental. A complete case-by-case classification for all simple real Lie algebras g0 was given (without proof) in the tables of Tits [24]. ´ Cartan in [2, 3] was to find all irreducible subalgebras The aim proclaimed by E. of the linear Lie algebras gln (C) and gln (R). As a continuation of this line, one should consider the works of Maltsev [17] and Dynkin [6] on semisimple subalgebras of simple complex Lie algebras, also based on the representation theory. A similar problem for real simple Lie algebras was studied in the paper of Karpelevich [15]. The results of this paper solve actually the problems on the self-conjugate complex representations and the index mentioned above. Note that the paper appeared before the publication of [13], but, unfortunately, still is not widely known. Our goal is to give a simplified (and somewhat extended and corrected) exposition of the main part of [15] and to relate it to the theory developped in [10, 13]. The first section contains some classical facts from the theory of Lie groups, Lie algebras and their representations, including the structure theory of complex semisimple Lie algebras, given without proofs. The necessary notation is also fixed there. In the main body of the lecture notes (§2 – §8) the exposition is given with detailed proofs. §2 deals with general facts on complexification and realification, real and complex structures and real forms. In particular, a general description of simple real Lie algebras in terms of simple complex ones is given and two main examples of real forms (normal and compact) in a complex semisimple Lie algebra are constructed. §3 is devoted to the correspondence between real forms of a complex semisimple ´ Cartan [4] Lie algebra g and involutive automorphisms of g discovered by E. which is the main tool in the subsequent study of real Lie algebras and their representations. Instead of Riemannian geometry exploited in [4], we use the elementary techniques of Hermitian vector spaces, as in [11], Ch. III or [19], Ch. 5. In §4, necessary facts about automorphisms of complex semisimple Lie algebras are given. As an example, we classify involutive automorphisms (and hence real forms) of the Lie algebra sln (C). At the same time, we do not give such a classi-

viii

Introduction

fication for other simple Lie algebras, referring to [11] or [19]. We also construct here the so-called principal three-dimensional subalgebra of a complex semisimple Lie algebra g and use it for studying the Weyl involution of g which corresponds to the normal real form of this algebra. In §5, we study the Cartan decompositions of a real semisimple Lie algebra g0 and of the corresponding adjoint linear group Int g0 . Then we prove the conjugacy of maximal compact subgroups of Int g0 (the elementary proof borrowed from [19] makes use of some properties of convex functions, instead of Riemannian geometry ´ Cartan). exploited in the original proof of E. §6 is devoted to the following problem. Suppose a homomorphism f of one complex semisimple Lie algebra g into another h be given. Choosing real forms g0 ⊂ g and h0 ⊂ h, one asks, when f (g0 ) ⊂ h0 . We prove the result of [15] which claims that this inclusion is equivalent to the relation f θ = θ
f if the classes of conjugate real forms and conjugate automorphisms are considered. Thus, the problem is reduced to that of extension of involutive automorphisms by homomorphisms. To get this reduction, we use, as in §3, the elementary techniques of Hermitian vector spaces. A more precise result is got in the case when f is a so-called S-homomorphism (in [15], it was proved for an irreducible representation into a classical linear Lie algebra h). In §7, we study the extension problem for involutive automorphisms in the main case, when f = ρ is an irreducible representation g → sln (C). First we consider the relation ρθ = θ
ρ in the case when θ
is an outer automorphism of sln (C), i.e., θ
(X) = −BX  B −1 , X ∈ sln (C), where B  = ±B. A condition of existence of such an extension in terms of the highest weight of ρ is found; it is expressed as invariance of the highest weight of ρ under an involutive automorphism s0 of the Dynkin diagram. An important problem is to calculate the sign j = ±1 in the above formula provided that this condition is satisfied. In [15], a simple explicit formula for this Karpelevich index j in the case when θ is an inner automorphism was proved. We generalize this formula to an arbitrary involutive θ. Then we consider the case of an inner θ
, getting, in particular, two formulas from [15] which express the signature of the invariant Hermitian form through the character of ρ. We do not reproduce the explicit formulas expressing this signature through the highest weight of ρ which were deduced in [15], using cumbersome calculations. We also omit the study of the extension problem for reducible representations g → sl(V ) and for representations g → so(V ) and g → sp(V ) which was made in [15], too. In §8, we give a classification of irreducible real representations of real semisimple Lie algebras following the method exposed in [10, 13]. It turns out that an irreducible complex representation ρ0 of a real semisimple Lie algebra g0 is self-conjugate if and only if the corresponding involution θ of g0 (C) extends to an outer involution of sln (C) by the complexified representation ρ0 (C). Moreover, the Cartan index of ρ0 coincides with the Karpelevich index of ρ0 (C). This allows to apply the results of §7 which leads to an explicit classification of irreducible real representations in terms of highest weights.

Introduction

ix

ˇ In §9, written by J. Silhan, a description of the symmetry s0 , introduced in §7, in terms of the Satake diagram of a real semisimple Lie algebra is given. This allows to determine self-conjugate complex representations by means of the Satake diagram. We conclude with some tables, giving in particular the indices of irreducible representations of simple complex Lie algebras. I would like to express my deep gratitude to Masaryk University (Brno) and E. Schr¨ odinger Institute for Mathematical Physics (Vienna) for hospitality and, in particular, to J. Slov´ ak and P. Michor for inviting me and for their interest in ˇ my lectures. I also thank J. Silhan who wrote the first version of these lecture notes and helped me very much in preparing the tables. My pleasant duty is to thank the EMS Publishing House, and in particular Dr. Thomas Hintermann, for including my work into one of lecture notes series. This work was partially supported by Russian Foundation for Basic Research, grant no. 01-01-00709, and by the grant no. 123.2003.01 for supporting scientific schools.

§1. Preliminaries In this section, some necessary facts from the theory of Lie groups and Lie algebras are formulated without proofs. We refer the reader for details to the corresponding text-books (see, e.g., [10, 11, 16, 19, 22]). I. Lie groups and Lie algebras We consider here finite dimensional Lie algebras over the field K = C or R. Let g be such a Lie algebra. Any x ∈ g determines the linear operator ad x in g given by (ad x)y = [x, y] , y ∈ g . The mapping ad : x → ad x is a homomorphism of Lie algebras g → gl(g) called the adjoint representation of g. Its image ad g is an ideal of the Lie algebra der g of all derivations of g. The Lie algebra ad g is called the adjoint algebra or the algebra of inner derivations of g. Consider the linear algebraic group Aut g of all automorphisms of g. For any x ∈ g we have exp(ad x) ∈ Aut g. The (normal) subgroup of Aut g generated by all exp(ad x), x ∈ g, is called the group of inner automorphisms of g and is denoted by Int g. The elements of Int g are called inner automorphisms of g, while the elements of Aut g \ Int g are called outer automorphisms. (I.1) The Lie subalgebras ad g ⊂ der g are tangent Lie algebras of the linear groups Int g ⊂ Aut g. Let G be a Lie group with tangent Lie algebra g. For any g ∈ G, let αg denote the inner automorphism h → ghg −1 of G generated by g. Then the automorphism Ad g = de αg ∈ Aut g is defined. The mapping Ad : g → Ad g is a representation of G in the vector space g called the adjoint representation of G. We have de Ad = ad. The image Ad G is a normal subgroup of Aut g called the adjoint linear group of G. (I.2) If G is connected, then Ad G = Int g and Ker Ad = Z(G) (the centre of the group G). A bilinear (or a sesquilinear) form b on a Lie algebra g is called invariant if all ad x, x ∈ g, are skew-symmetric relative to b, i.e., if b([x, u], v) = −b(u, [x, v]) ,

x, u, v ∈ g .

This is equivalent to the property b(αu, αv) = b(u, v) ,

α ∈ Int g , u, v ∈ g .

The most important example of an invariant bilinear form is the Killing form kg of g given by kg (u, v) = tr((ad u)(ad v)) , u, v ∈ g . (1)

§1. Preliminaries

2

(I.3) The Killing form kg is a symmetric invariant bilinear form on g satisfying kg (γu, γv) = kg (u, v) ,

γ ∈ Aut g , u, v ∈ g .

(I.4) For any ideal h of g, we have kg |h = kh . A Lie algebra g is called semisimple if g has no non-trivial solvable (or, which is equivalent, commutative) ideal. A Lie group is called semisimple if its tangent Lie algebra is semisimple. The following important Cartan criterion is valid: (I.5) A Lie algebra g over K = C or R is semisimple if and only if its Killing form kg is non-degenerate. (I.6) A Lie algebra g over K = C or R is semisimple if and only if g=

k


gi ,

i=1

where gi are non-commutative simple ideals. This implies that [g, g] = g. (I.7) If g is semisimple, then der g = ad g. This implies that Int g = (Aut g)◦ . A Lie algebra g is called reductive if g = g0 ⊕ z(g) , where g0 is semisimple and z(g) is the centre of g. This implies that [g, g] = g0 . Let G be a compact Lie group. Sometimes we will use the invariant integration of smooth (scalar or vector-valued) functions over G (see, e.g., [12], Ch. I, or [16], Ch. VIII). The  invariant integral of a function f is denoted by G f (g)dg; we suppose that G dg = 1. The following invariance conditions are satisfied: 

 f (gh)dg = G

 f (hg)dg =

G

f (g

−1

 )dg =

G

f (g)dg ,

h ∈ G.

G

(I.8) Let R : G → GL(V ) be a linear representation of a compact Lie group G in a real or complex vector space V . For any fixed v ∈ V , the vector  (R(g)v)dg ∈ V

v0 = G

is invariant under R, i.e., satisfies R(g)v0 = v0 , g ∈ G. Under a scalar product in a real or complex vector space V we mean a positive definite symmetric bilinear (or Hermitian) form on V . The following corollary of (I.8) is called Theorem of Weyl .

§1. Preliminaries

3

(I.9) Let R : G → GL(V ) be a linear representation of a compact Lie group G in a real or complex vector space V . For any fixed scalar product ( , ) on V , the function ( , )0 given by  (u, v)0 =

(R(g)u, R(g)v)dg ,

u, v ∈ V ,

G

is a scalar product on V , invariant under R, i.e., satisfying (R(g)u, R(g)v)0 = (u, v)0 , u, v ∈ V, g ∈ G. A real Lie algebra g is called compact if there exists an invariant scalar product on g. Clearly, any subalgebra of a compact Lie algebra is compact, too. (I.9) implies (I.10) If G is a compact Lie group, then its Lie algebra g is compact . We also formulate the following important properties of compact Lie algebras: (I.11) Any compact Lie algebra is reductive. (I.12). The Killing form kg of a compact Lie algebra g satisfies kg (x, x) ≤ 0, x ∈ g. A real Lie algebra g is compact semisimple if and only if kg is negative definite. (I.13) For any compact semisimple Lie algebra g, the Lie groups Aut g and Int g are compact. For any compact Lie algebra g, there exists a compact Lie group G such that g is the tangent Lie algebra of G. If g is compact semisimple, then each connected Lie group with the tangent Lie algebra g is compact . Let g be a real or complex Lie algebra and let θ ∈ Aut g be involutive, i.e., θ satisfies θ2 = e. Then θ is diagonalizable with eigenvalues ±1, and hence we have the eigenspace decomposition g = g+ ⊕ g− ,

where g± = {x ∈ g | θx = ±x} .

(2)

(I.14) The decomposition (2) is a Z2 -grading, i.e., [g+ , g+ ] ⊂ g+ , [g+ , g− ] ⊂ g− , [g− , g− ] ⊂ g+ . In particular, gθ = g+ is a subalgebra of g. Conversely, for any Z2 -grading g = k⊕p,

(3)

the linear transformation θ : x + y → x − y, x ∈ k, y ∈ p, is an involutive automorphism of g, and (3) is the corresponding eigenspace decomposition.

§1. Preliminaries

4

II. Structure of complex semisimple Lie algebras Let g be a complex semisimple Lie algebra. Choose a maximal toral subalgebra t of g, i.e., a maximal subalgebra consisting of semisimple elements. It is necessarily commutative, and g is decomposed into the direct sum of weight subspaces corresponding to the adjoint representation of t in the vector space g (see (III.4)). Then t coincides with the subspace corresponding to the weight 0 (i.e., with its centralizer in g). The non-zero weights are called the roots of g; they form a finite subset ∆ ⊂ t∗ (the system of roots). Thus, we have the so-called root space decomposition:
g=t⊕ gα , (4) α∈∆

where gα is the weight subspace (called the root space) corresponding to α ∈ ∆. (II.1) Any two maximal toral subalgebras of g are conjugate by an inner automorphism of g. Hence the root space decomposition is determined uniquely, up to inner automorphisms. The dimension dim t of any maximal toral subalgebra is called the rank of g and is denoted by rk g. Define the following real vector subspace of t: t(R) = {h ∈ t | α(h) ∈ R for all α ∈ ∆} .

(5)

Then the following properties hold: (II.2) t∗ = ∆C ; (II.3) t(R) is a real form of t; (II.4) t(R)∗ := ∆R is a real form of t∗ . As this notation suggests, t(R)∗ may be regarded as the real dual vector space for t(R). Remind (see (I.5)) that the Killing form k = kg is non-degenerate. (II.5) The Killing form k satisfies k(x, y) = 0 for two weight vectors x, y ∈ g corresponding to any weights λ, µ ∈ t∗ such that λ + µ = 0. It follows that t and the different vector subspaces gα ⊕ g−α , α ∈ ∆, are pairwise orthogonal, and the restrictions k|t and k|(gα ⊕ g−α ), are non-degenerate. The restriction k|t(R) is real and positive definite. Thus, we have a natural structure of Euclidean space on t(R). We often write (x, y) = k(x, y) for x, y ∈ t. It is also convenient to carry this bilinear form to the dual space t∗ . Consider the vector space isomorphism λ → uλ given by (uλ , h) = λ(h) , h ∈ t ; it maps t(R)∗ onto t(R). Now we define a non-degenerate form on t∗ by (λ, µ) := (uλ , uµ ) = λ(uµ ) = µ(uλ ) , λ, µ ∈ t∗ .

(6)

§1. Preliminaries

5

This form is real and positive definite on t(R)∗ . We also define the vector hλ =

2 uλ (λ, λ)

(7)

for any non-zero λ ∈ t(R)∗ . In particular, the vectors hα , α ∈ ∆, are called coroots. Clearly, hα ∈ t(R), α ∈ ∆. Also, any α ∈ ∆ is uniquely determined by its restriction to t(R) and, hence, may be identified with this restriction. (II.6) For any α ∈ ∆ we have −α ∈ ∆, but cα ∈ / ∆ for any c ∈ C, c = ±1. (II.7) α(hβ ) =

2(α,β) (β,β)

∈ Z for any α, β ∈ ∆.

(II.8) dim gα = 1 for any α ∈ ∆. It follows from (II.8) that we get a basis of g if we take a basis of t and a non-zero vector eα ∈ gα (a root vector ) for each root α ∈ ∆. By definition, we have (II.9) [h, eα ] = α(h)eα , h ∈ t. Other commutation relations are the following ones: (II.10) Let α, β ∈ ∆ be such that α + β = 0. Then  Nα,β eα+β , Nα,β = 0, if α + β ∈ ∆ , [eα , eβ ] = 0 otherwise. The simplest example of a non-commutative simple complex Lie algebra is g = sl2 (C). Denote       1 0 0 1 0 0 H= , E= , F = . 0 −1 0 0 1 0 Then t = HC is a maximal toral subalgebra of g. We have ∆ = {α, −α}, where α(H) = 2. Hence t(R) = HR . We have [H, E] = 2E ,

[H, F ] = −2F ,

[E, F ] = H .

Let again g be an arbitrary complex semisimple Lie algebra. A triple {e, h, f } of elements of g is called an sl2 -triple if [h, e] = 2e, [h, f ] = −2f, [e, f ] = h and h = 0. Such a triple spans the subalgebra e, h, f C of g which is isomorphic to sl2 (C). An isomorphism is given by E → e ,

H → h ,

F → f .

(II.11) Given a root vector eα ∈ gα , we can choose fα = e−α ∈ g−α in such a 2 way that k(eα , fα ) = (α,α) , which implies [eα , fα ] = hα . Then {eα , hα , fα } is an sl2 -triple.

6

§1. Preliminaries

(II.12) Given an automorphism θ ∈ Aut g such that θ(t) = t, the transposed transformation θ of t∗ satisfies θ (∆) = ∆ , θ(t(R)) = t(R) , θ(gα ) = g(θ
)−1 α . Any root α ∈ ∆ determines the hyperplane Pα = Ker α ⊂ t(R). Let rα denote the orthogonal reflection relative to Pα . It is given by rα (h) = h − α(h)hα ,

h ∈ t(R) .

(8)

Dually, we have the hyperplane Lα = {λ ∈ t(R)∗ | (λ, α) = 0}. The orthogonal reflection relative to Lα coincides with the transposed transformation rα and is given by 2(λ, α) α , λ ∈ t(R)∗ . rα (λ) = λ − (9) (α, α) The reflections rα are actually induced by inner automorphisms of g leaving t invariant. (II.13) Given a root α ∈ ∆, suppose that the root vectors eα and fα are chosen as in (II.11). Then rα = ϕα |t(R) , where ϕα = exp ad

π (eα − fα ) . 2

(II.14) The reflections rα (respectively, rα ) generate a finite group W of orthogonal transformations of t(R) (respectively, a finite group W ∨ of orthogonal transformations of t(R)∗ ). The mapping w → (w )−1 is an isomorphism of W onto W ∨ . The system of roots ∆ is invariant under W ∨ . Any of two groups defined in (II.14) is called the Weyl group of g. An element h ∈ t(R) is called regular if α(t) = 0 for all α ∈ ∆, and singular otherwise. The connected components of the set t(R)reg of regular elements of t(R) are called the Weyl chambers; they are open polyhedral cones. (II.14) implies that the Weyl group W permutes the Weyl chambers. (II.15) The natural action of W on the Weyl chambers is simply transitive. The property (II.13) suggests another definition of the Weyl group. Let us define the subgroups N = {ϕ ∈ Int g | ϕ(t) = t} and T = {exp(ad h) | h ∈ t} of Int g. (II.16) If ϕ ∈ N maps a fixed Weyl chamber onto itself, then ϕ ∈ T , and ϕ|t = id. The restriction homomorphism ϕ → ϕ|t(R) maps N onto the Weyl group W , and hence W N/T . Let us fix a Weyl chamber D. Since D is connected, for any root α ∈ ∆ we have either α(h) > 0 for all h ∈ D or α(h) < 0 for all h ∈ D. In the first case, α

§1. Preliminaries

7

is called positive (with respect to D), and in the second one it is called negative (with respect to D). The sets of positive and negative roots are denoted by ∆+ and ∆− = −∆+ respectively. A positive root α is called simple (with respect to D) if α = β + γ, where β, γ ∈ ∆+ . Let Π ⊂ ∆+ denote the subset of simple roots. We will write Π = {α1 , . . . , αl }. (II.17) Any α ∈ ∆+ has the form α = αi1 + . . .+ αir , where αi1 + . . .+ αik ∈ ∆+ for each k = 1, . . . , r. (II.18) The set Π is a basis of t(R)∗ . In particular, l = dimR t(R) = dimC t is  the rank of g. Any α ∈ ∆ can be uniquely written in the form α = li=1 ki αi , + where ki ∈ Z and either all ki
0 (for α ∈ ∆ ) or all ki  0 (for α ∈ ∆− ). The property (II.18) suggests to call the set of simple roots Π a base of the system of roots ∆. (II.19) The group W ∨ acts simply transitively on the subsets of positive roots and on the subsets of simple roots of ∆. Any α ∈ ∆ is simple with respect to an appropriate Weyl chamber. Given a fixed subset of simple roots Π ⊂ ∆, the Weyl group W is generated by rα , α ∈ Π. Let us fix a Weyl chamber D and consider the corresponding subsets ∆+ ⊃ Π of positive and simple roots. Define the linear form γ=

1  α. 2 +

(10)

α∈∆

(II.20) If β ∈ Π and α ∈ ∆+ , then rβ (α) ∈ ∆+ , except the case α = β, when = −β. We have

rβ (β)

rβ (γ) = γ − β , γ(hβ ) = 1 f or any β ∈ Π . Using the system Π = {α1 , . . . , αl }, one can construct a system of generators of the Lie algebra g in the following way. Denote hi = hαi ,

ei = eαi ,

fi = fαi ,

i = 1, . . . , l ,

(11)

where the root vectors eαi and fαi are chosen as in (II.11). Introducing the integers aij = αi (hj ) =

2(αi , αj ) (αj , αj )

(see (II.7)), we get a l × l matrix A = (aij ) called the Cartan matrix . (II.21) The elements hi , ei , fi , i = 1, . . . , l, form a system of generators of g and satisfy the relations [hi , hj ] = 0 , [hi , ej ] = aji ej , [hi , fj ] = −aji fj , [ei , fi ] = hi , [ei , fj ] = 0 f or i = j .

(12)

§1. Preliminaries

8

The set {hi , ei , fi | i = 1, . . . , l} is called the canonical system of generators of g associated with t and Π. The structure of g is completely determined by the Cartan matrix A. More precisely, we have (II.22) Suppose that two complex semisimple Lie algebras g and ˜g with canonical ˜ i , respectively, are given, and let A and A˜ be generator systems ei , fi , hi and e˜i , f˜i , h ˜ their Cartan matrices. If A = A, then there exists a unique isomorphism ϕ : g → ˜g such that ϕ(ei ) = e˜i , ϕ(fi ) = f˜i and ϕ(hi ) = ˜hi . For the entries aij , i = j, of the Cartan matrix only the values 0, −1, −2, −3 are possible. Moreover, the only possible values for mij = aij aji , i = j, are 0, 1, 2, 3, and the corresponding values for the angle between αi and αj are θij = π(1 − n1ij ), |α |

where nij = 2, 3, 4, 6, respectively. We also have |αji | = mij , whenever mij = 0 and |αj | ≥ |αi |. It is usual to depict the system of simple roots (or the Cartan matrix) by a graph, assigning to any αi , i = 1, . . . , l, a vertex and joining the i-th and j-th vertices by the edge of multiplicity mij (in the case mij = 0 the vertices are not linked). If |αj | > |αi |, then the edge is oriented by an arrow pointing from the j-th vertex towards the i-th one. This graph is called the Dynkin diagram of the Lie algebra g. As (II.22) shows, the Dynkin diagram determines g up to isomorphy. Let gi , i = 1, 2, be two complex semisimple Lie algebras. Consider g = g1 ⊕ g2 . Any maximal toral subalgebra t of g is of the form t = t1 ⊕ t2 , where ti is a maximal toral subalgebra of gi , i = 1, 2, the summands being orthogonal with respect to kg . Hence t∗ = t∗1 ⊕ t∗2 . Now, the system of roots ∆ of g decomposes as ∆ = ∆1 ∪ ∆2 , where ∆i is the system of roots of gi relative to ti , i = 1, 2. It follows that t∗ (R) = t∗1 (R)⊕t∗2 (R). Moreover, any Weyl chamber for g has the form D = D1 × D2 , where Di is a Weyl chamber for gi . Therefore we have Π = Π1 ∪ Π2 , where Πi is the system of simple roots of gi corresponding to Di and (α, β) = 0 for any α ∈ Π1 , β ∈ Π2 . (II.23) A semisimple complex Lie algebra g is simple if and only if its Dynkin diagram is connected. Generally, the decomposition of the Dynkin diagram into connected components corresponds to the decomposition of g into the direct sum of simple ideals (see (I.6)). The Dynkin diagrams of simple complex Lie algebras are listed in Table 1. III. Representations of Lie algebras Let g be a finite dimensional Lie algebra over the field K = C or R. We consider linear representations of g in finite dimensional vector spaces over K, i.e., homomorphisms of Lie algebras ρ : g → gl(V ). The number dim ρ = dim V is called the dimension of ρ. Two representations ρ : g → gl(V ) and ρ
: g → gl(V
) are called equivalent (or isomorphic) if there exists an isomorphism of vector spaces f : V → V
such that f ρ(x) = ρ
(x)f, x ∈ g . In this case we write ρ ∼ ρ
.

§1. Preliminaries

9

If ρ : g → gl(V ) is a representation, then we can choose a basis in V and define a homomorphism ρ˜ : g → gln (K), sending any x ∈ g to the matrix of ρ(x) relative to our basis. We will call ρ˜ a matrix form of ρ; it can be regarded as a representation of g in the vector space K n equivalent to ρ. Clearly, two representations are equivalent if and only if they have the same matrix form in appropriate bases. With any representation ρ : g → gl(V ) the dual (or contragredient ) representation ρ∨ : g → gl(V ∗ ) is associated. It is defined in the dual space V ∗ by (ρ∨ (x)λ)(v) = −λ(ρ(x)v) ,

x ∈ g,

λ ∈ V ∗, v ∈ V ,

or, equivalently, ρ∨ (x) = −ρ(x) ,

x ∈ g.

(13)

(III.1) If we choose dual bases in V and V ∗ , then the corresponding matrix ∨ of the representations ρ and ρ∨ are related by forms ρ˜ and ρ ∨ (x) = −ρ ρ ˜ (x) ,

x ∈ g.

A vector subspace W ⊂ V is called invariant under a representation ρ : g → gl(V ) if ρ(x)(W ) ⊂ W, x ∈ g. Restricting the operators ρ(x), x ∈ g, to an invariant subspace W , we get a representation of g in W , called the subrepresentation of ρ.

r Suppose that V = i=1 Vi , where Vi , i = 1, . . . , r, are vector subspaces, invariant under a representation ρ : g → gl(V ), and let ρi denote the subrepresentation in Vi . Then we say that ρ is the sum of ρi and write ρ = ri=1 ρi . A representation ρ : g → gl(V ) is called irreducible if V does not content any invariant subspace W, {0} = W = V . It is called completely reducible if for any invariant subspace W ⊂ V there exists an invariant W
⊂ V such that V = W ⊕ W
. r (III.2) A representation ρ is completely reducible if and only if ρ = i=1 ρi , where ρi are irreducible. In this case ρi are uniquely determined by ρ, up to equivalence. The representations ρi in (III.2) are called the irreducible components of ρ. (III.3) Any representation of a semisimple Lie algebra is completely reducible. Let us now consider representations of a complex semisimple Lie algebra g. Note that, by (I.6), we may always regard it as a homomomorphism of g into [gl(V ), gl(V )] = sl(V ). Let t be a maximal toral subalgebra of g. (III.4) For any representation ρ : g → gl(V ) we have the weight space decomposition
Vλ , V = λ∈Φρ

§1. Preliminaries

10

where Φρ ⊂ t∗ (R) is the system of weights of ρ and Vλ = {v ∈ V | ρ(h)v = λ(h)v ,

h ∈ t} = {0}

are the corresponding weight subspaces. A special case of the weight space decomposition (for ρ = ad) is the root space decomposition (4); we have Φad = ∆ ∪ {0}. (III.5) For any representation ρ we have Φρˆ = −Φρ . (III.6) λ(hα ) =

2(λ,α) (α,α)

∈ Z for any λ ∈ Φρ , α ∈ ∆.

(III.7) ρ(eα )Vλ ⊂ Vλ+α for any λ ∈ Φρ , α ∈ ∆. (III.8) For any θ ∈ Aut g such that θ(t) = t, we have Φρθ = (θ )−1 (Φρ ). The system Φρ is invariant under W ∨ . Let us fix a Weyl chamber D and consider the corresponding systems Π ⊂ ∆+ of simple and positive roots and the canonical generators (11) of g. A weight vector v ∈ V, v = 0, is called a highest vector for ρ if ρ(eα )v = 0 for any α ∈ ∆+ or, equivalently, if ρ(ei )v = 0, i = 1, . . . , l. The weight of v is called a highest weight of ρ. (III.9) If ρ is irreducible, then Φρ contains a unique highest weight Λ, and dim VΛ = 1. Two irreducible representations of g are equivalent if and only if they have the same highest weight. An arbitrary representation is determined up to equivalence by the system of highest weights of its irreducible components. A linear form λ ∈ t∗ (R) is called dominant if λ(hi ) ≥ 0, i = 1, . . . , l. (III.10) A highest weight of any representation is dominant. For any dominant Λ ∈ t∗ (R) such that Λi = Λ(hi ) ∈ Z there exists an irreducible representation of g with the highest weight Λ. Let { 1 , . . . , l } denote the base of t∗ (R) dual to the base {h1 , . . . , hl } of t(R). The linear forms i are called the fundamental weights. By (III.10), they are the highest weights of certain irreducible representations of g which are called the basic l representations. For any Λ ∈ t∗ (R) we have Λ = i=1 Λi i where Λi = Λ(hi ). The irreducible representation with the highest weight Λ can be described by writing coefficients Λi over the corresponding vertices of the Dynkin diagram of g. In this way we get the so-called Dynkin diagram of an irreducible representation. Let ρ : g → gl(V ) be an irreducible representation, and Λ be its highest weight. Then the highest vector vΛ ∈ VΛ generates V over g. More precisely, let us denote v∅ = vΛ , vi1 ,... ,ik = ρ(fi1 ) . . . ρ(fik )vΛ ,

1 ≤ i 1 , . . . , ik ≤ l .

§1. Preliminaries

11

(III.11) The vectors vi1 ,... ,ik span the vector space V . The canonical generators act on these vectors as follows: ρ(hi )v∅ = Λi v∅ , ρ(hi )vi1 ,... ,ik = (Λi − ai1 ,i − . . . − aik ,i )vi1 ,... ,ik , ρ(fi )vi1 ,... ,ik = vi,i1 ,... ,ik , ρ(ei )v∅ = 0 , ρ(ei )vi1 ,... ,ik = (δii1 ρ(hi ) + ρ(fi1 )ρ(ei ))vi2 ,... ,ik . Any λ ∈ Φρ may be written in the form λ = Λ − αi1 − . . . − αik , where k ≥ 0. Let gi , i = 1, 2, be two complex semisimple Lie algebras. Consider g = g1 ⊕ g2 . If ρi : gi → gl(Vi ), i = 1, 2, are linear representations, then one can define the representation ρ = ρ1 ⊗ ρ2 : g → gl(V1 ⊗ V2 ) by ρ(x1 , x2 )(v1 ⊗v2 ) = ρ1 (x1 )(v1 )⊗v2 +v1 ⊗ρ2 (x2 )(v2 ), xi ∈ gi ,

vi ∈ Vi , i = 1, 2 .

The representation ρ is called the tensor product of ρ1 and ρ2 . (III.12) If ρ1 and ρ2 are irreducible, then ρ = ρ1 ⊗ ρ2 is irreducible, too. Choose a maximal toral subalgebra ti ⊂ gi , i = 1, 2, and denote t = t1 ⊕ t2 . The highest weight of ρ relative to t is (Λ1 , Λ2 ), where Λi is the highest weight of ρi relative to ti , i = 1, 2. Any irreducible representation of g = g1 ⊕ g2 is equivalent to the tensor product ρ1 ⊗ ρ2 , where ρi is an irreducible representation of gi , i = 1, 2.

§2. Complexification and real forms Given a real vector space V0 , we can construct its complexification V = V0 (C) as the complex vector space V0 ⊗R C = V0 ⊕ iV0 . Clearly, any basis of V0 is a basis of V , and hence dimC V0 = dimR V . The mapping S : V → V , defined by S(u + iv) = u − iv ,

u, v ∈ V ,

is called the complex conjugation. On the other hand, any complex vector space V can be regarded as a real vector space, which is called its realification and is denoted by VR . Clearly, dimR VR = 2 dimC V . Consider again the complexification V = V0 (C) of a real vector space V0 . Clearly, the complex conjugation S given by (1) is an antiautomorphism (antilinear automorphism) of V , i.e., an automorphism of the real vector space VR satisfying S(cv) = c¯Sv, c ∈ C, v ∈ V . Also S is involutive, i.e., satisfies S 2 = e. For brevity, we will call such an antiautomorphism of any complex vector space an antiinvolution. Next, let us consider a complex vector space V . A real vector subspace V0 of VR is called a real form of V if VR = V0 ⊕ iV0 . One verifies easily that this is equivalent to the following property: the mapping V0 (C) → V sending u + iv ∈ V0 (C) to u + iv ∈ V, u, v ∈ V0 , is an isomorphism of complex vector spaces. This isomorphism identifies the complex conjugation defined in V0 (C) with an antiinvolution S of V , and we have V0 = V S = {v ∈ V | Sv = v} .

(1)

Generally, we call a real structure in a complex vector space V any antiinvolution S : V → V . Obviously, real structures in V are in bijective correspondence with real forms V0 ⊂ V (a real form V0 defines the corresponding complex conjugation S, while a real structure S defines the real form V S given by (1)). A complex structure in a real vector space V0 is, by definition, an automorphism J ∈ GL(V0 ) such that J 2 = −e. A complex vector space V may be regarded as the real vector space VR endowed with the complex structure J : v → iv, v ∈ V . Conversely, if a complex structure J in a real vector space V0 is given, then we may regard (V0 , J) as a complex vector space with the realification V0 , the multiplication by complex scalars being given by (a + bi)v = av + bJv, a, b ∈ R, v ∈ V0 . For any complex vector space V , let us denote by V¯ the complex vector space which coincides with V as an additive group, but is endowed with the following multiplication by complex scalars: c∗x = c¯x, x ∈ V¯ = V, c ∈ C. In other words, if J is the given complex structure in V , i.e., J : v → iv, v ∈ V , then V¯ = (VR , −J). The vector space V¯ is called the complex conjugate to V . Given a second complex vector space W , we easily see that any linear mapping α : V → W is a linear ¯ as well. If α is expressed by a matrix A relative to certain bases mapping V¯ → W ¯ relative the same bases is equal to A. ¯ of V and W , then the matrix of α : V¯ → W On the other hand, an antilinear mapping V → W can be interpreted as a linear ¯ or V¯ → W , and vice versa. mapping V → W

§2. Complexification and real forms

13

All these general notions can be introduced in the category of Lie algebras. Given a real Lie algebra g0 , we endow the complexification g = g0 (C) of the vector space g0 with the bracket extending the bracket of g0 , i.e., given by [x1 + iy1 , x2 + iy2 ] = [x1 , y1 ] − [y1 , y2 ] + i([x1 , y2 ] + [y1 , x2 ]),

x1 , y1 , x2 , y2 ∈ g0 .

Clearly, g is a complex Lie algebra. On the other hand, for any complex Lie algebra g, the realification gR of the vector space g is a real Lie algebra. Now, if g = g0 (C) is the complexification of a real Lie algebra g0 , then the complex conjugation σ, defined by σ(x + iy) = x − iy ,

x, y ∈ g0 ,

(2)

is an involutive antiautomorphism (antilinear automorphism) of the complex Lie algebra g (we call it an antiinvolution). Next, let g be a complex Lie algebra. A real subalgebra g0 of gR is called a real form of g if it is a real form of the complex vector space g, i.e., if gR = g0 ⊕ ig0 . In this case, g0 (C) is naturally identified with g. A real structure in a complex Lie algebra g is an antiinvolution σ : g → g. The complex conjugation (2) corresponding to a real form of g is a real structure in g. Conversely, any real structure σ in g defines the real form gσ . In this way, we get a bijection between real structures and real forms of g. Let us consider two real forms g0 and h0 of complex Lie algebras g and h, respectively. Clearly, any homomorphism h : g0 → h0 extends uniquely to a homomorphism h(C) : g → h of complex Lie algebras. The following proposition is the first (trivial!) step in the classification of real forms of a given complex Lie algebra up to isomorphy. Proposition 1. Consider two real forms g0 , g1 of a complex Lie algebra g, and let σ0 , σ1 be the corresponding real structures. Then g0 g1 if and only if σ1 = ασ0 α−1 for a certain α ∈ Aut g. Proof. By the above remark, any isomorphism h : g0 → g1 extends to an automorphism α = h(C) of g. Clearly, the antiautomorphisms ασ0 and σ1 α coincide with h on g0 . It follows that ασ0 = σ1 α, whence σ1 = ασ0 α−1 . Conversely, suppose that there exists α ∈ Aut g satisfying σ1 = ασ0 α−1 . One verifies easily that α(gσ0 ) = gσ1 , i.e., α(g0 ) = g1 .  A complex structure in a real Lie algebra g0 is a complex structure J in the vector space g0 satisfying J[x, y] = [x, Jy] ,

x, y ∈ g0 .

(3)

Clearly, any complex Lie algebra is endowed with the complex structure Jv = iv. Conversely, if a complex structure J in a real Lie algebra g0 is given, then we may regard (g0 , J) as a complex Lie algebra (with the same bracket); its realification is g0 . If g is a complex Lie algebra, then the complex conjugate vector space ¯g, endowed with the same bracket as g, is a complex Lie algebra as well. It is called

§2. Complexification and real forms

14

the complex conjugate Lie algebra. Clearly, the mapping id : g → g = ¯g is an antilinear isomorphism. Also, isomorphisms g → g¯ are antilinear automorphisms of g. In particular, g ¯ g, whenever g possesses a real form. Sometimes we will also deal with the following notion. A quaternion structure in a complex vector space V is an antiautomorphism J : V → V satisfying J 2 = −e. Thus, J is a complex structure on VR anticommuting with the given complex structure. Let us view the skew-field H of quaternions as a right vector space over C with the basis 1, j. Thus, a quaternion is an element of the form q = z + jw with ¯ If a quaternion z, w ∈ C, and the following relations hold: j 2 = −1, jw = wj. structure J in a complex vector space V is given, then V can be regarded as a right vector space over H, such that v(z + jw) = zv + w(Jv) ,

v ∈ V , z, w ∈ C .

Conversely, any quaternion vector space V is, in a natural way, a complex vector space endowed with a quaternion structure J, and linear endomorphisms V → V are the endomorphisms of the complex vector space commuting with J. Example 1. Here we describe certain real forms of the Lie algebras gln (C) and sln (C). We shall see later that our list contains all the real forms of sln (C) up to isomorphism. 1. An obvious real form of gln (C) is the subalgebra gln (R). The corresponding real structure is given by ¯ = (xij ). σ : X = (xij ) → X The restriction of σ to sln (C) gives the real form sln (R) of sln (C). 2. Let us consider a non-degenerate hermitian form h on Cn with signature (p, q), p + q = n. We may assume that h(z, w) = −z1 w ¯1 − . . . − zp w ¯p + zp+1 w ¯p+1 + . . . + zn w ¯n , z, w ∈ Cn . Consider the (real) Lie algebra up,q = {X ∈ gln (C) | h(Xz, w) + h(z, Xw) = 0

for all z, w ∈ Cn } .

¯ where Since h(z, w) = z  Ip,q w,  Ip,q =

−Ip 0

0 Iq

 ,

(4)

one sees easily that ¯  = 0} . up,q = {X ∈ gln (C) | XIp,q + Ip,q X In particular, u0,n = un is the algebra of skew-hermitian matrices. One deduces that up,q = gln (C)σ , where ¯  Ip,q , σ(X) = −Ip,q X

X ∈ gln (C) .

§2. Complexification and real forms

15

One proves easily that σ is a real structure, and hence up,q is a real form of gln (C). The elements of up,q are block matrices of the form   A B ¯ D , B where A ∈ up and D ∈ uq . Clearly, sup,q = up,q ∩ sln (C) is a real form of sln (C). 3. For n = 2m, one can construct a real form using quaternions. The mapping   z1 .  .       .  q1 z1 + jw1      zm    . .. Hm  q =  ..  =  →   ∈ C2m   .  w1   .  qm zm + jwm  ..  wm determines an isomorphism between Hm and C2m , both understood as right vector spaces over C. We will identify Hm and C2m by this isomorphism. Clearly, Hm is a right vector space over H, and the right multiplication q → qj in Hm is identified with the quaternion structure J in C2m given by     z1 −w ¯1  ..   ..   .   .      ¯m   z   −w J m=   w1   z¯1   .   .   ..   ..  wm z¯m or J(v) = Sm v¯ , v ∈ C2m , 

where Sm =

0 Im

−Im 0

 .

(5)

Consider the Lie algebra glm (H) of endomorphisms of the vector space Hm (or of quaternion n × n-matrices). Clearly, it is identified with the real subalgebra {X ∈ gl2m (C) | XJ = JX} of gl2m (C). Now, we define the mapping σ : gl2m (C) → ¯ m . This is an antiinvolution determining gl2m (C) by σ(X) = JXJ −1 = −Sm XS the real form gl2m (C)σ = glm (H) of the Lie algebra gl2m (C). The elements of this real form are block matrices of the form   A B ¯ A¯ , −B where A, B ∈ glm (C). A real form slm (H) of sl2m (C) is then formed by the above matrices with the condition Re tr A = 0. We want now to compare the Killing forms of a complex Lie algebra g, of a real form of g and of the complex conjugate Lie algebra ¯g.

16

§2. Complexification and real forms

Proposition 2. Let g be a complex Lie algebra. (i) If g0 is a real form of g, then kg |g0 = kg0 , and g is semisimple if and only if g0 is semisimple. (ii) kg¯ (x, y) = kg (x, y), x, y ∈ g. (iii) If σ is an antilinear automorphism of g, then kg (σx, σy) = kg (x, y), x, y ∈ g. Proof. (i) Any basis of the real vector space g0 is a basis of the complex vector space g. If x ∈ g0 , then ad x(g0 ) ⊂ g0 , and hence the operator ad x in g is expressed in such a basis by the same matrix as its restriction to g0 . This is also true for (ad x)(ad y), where x, y ∈ g0 . It follows that trg ((ad x)(ad y)) = trg0 ((ad x)(ad y)). Clearly, kg and kg0 have the same matrix in any basis of g0 . Hence, kg0 is nondegenerate if and only if kg is non-degenerate, and our second assertion follows from the Cartan criterion (I.5). (ii) Take x, y ∈ g. By a remark above, the matrix of the operator (ad x)(ad y) in ¯ g is complex conjugate to the matrix of this operator in g (with respect to the same basis). This implies our assertion. (iii) We may regard σ as an isomorphism of complex Lie algebras σ : ¯g → g.  By (I.3), we have kg¯ (x, y) = kg (σx, σy), and our assertion follows from (ii). We see from Proposition 2 (i) that the complexification operation gives a correspondence between real and complex semisimple Lie algebras. We will study now the corresponding question for simple Lie algebras. Let g be a real Lie algebra, and let z → z¯ denote the complex conjugation in its complexification g(C). For any m ⊂ g(C), let us denote m its image under the complex conjugation. The following properties follow immediately from the definitions. 1. For any subalgebra h ⊂ g its complexification h(C) is a subalgebra of g(C). 2. In the situation of assertion 1, h is an ideal of g if and only if h(C) is an ideal of g(C). 3. A vector subspace b ⊂ g(C) has the form b = a(C) for a subspace a ⊂ g if ¯ = b. In this case, a = b ∩ g. and only if b ¯ is a subalgebra (respecti4. For any subalgebra (or ideal) h ⊂ g(C), the subset h vely, an ideal) of g(C). 5. For any two vector subspaces a, b ⊂ g, we have [a, b](C) = [a(C), b(C)]. Proposition 3. If g is complex Lie algebra, then gR (C) g ⊕ ¯g. Proof. Let us denote gdbl = g ⊕ g¯. The mapping Σ : gdbl → gdbl given by Σ(x, y) = (y, x) is a real structure in gdbl , since Σ(c(x, y)) = (¯ cy, cx) = c¯(y, x) = c¯Σ(x, y), x, y ∈ g, c ∈ C. The corresponding real form is gd = {(x, x) | x ∈ g}. The mapping (x, x) → x is an isomorphism of real Lie algebras gd → gR , and hence gdbl gR (C).  Theorem 1. Let g be a complex simple Lie algebra. Then any real form of g and gR (whenever g is non-commutative) are real simple Lie algebras. Conversely, any real simple Lie algebra is either a real form of a complex simple Lie algebra or is isomorphic to gR , where g is a non-commutative complex simple Lie algebra.

§2. Complexification and real forms

17

Proof. Let g0 = 0 be a real Lie algebra, and let g = g0 (C) be simple. Then for any non-zero ideal a ⊂ g0 we get the non-zero ideal a(C) ⊂ g. Hence, a(C) = g, and a = g0 . Now, let g be a complex simple Lie algebra, and let a ⊂ gR be a real ideal such that 0 = a
gR . Then we get the real ideal ia of gR and two complex ideals a ∩ ia and a + ia of g. These ideals must satisfy a ∩ ia = 0 and a + ia = g. Thus, gR = a ⊕ ia, whence [a, ia] = 0. This implies that g is commutative. Now consider a real simple Lie algebra g0 = 0 and suppose that g = g0 (C) is not simple. We are going to prove that g0 admits a complex structure J, converting g0 into a complex Lie algebra. Fix a complex ideal 0 = a
g. Then we get the ideals ¯ a, b = a + ¯ a and c = a ∩ ¯ a of g. Since b = ¯b, ¯c = c, we have b = (b ∩ g0 )(C) and c = (c ∩ g0 )(C). As g0 = 0 is simple, this implies b ∩ g0 = g0 and c ∩ g0 = 0. It follows that g = b = a ⊕ ¯ a. Now we define the desired complex structure J by J(x + y¯) = ix − i¯ y, x, y ∈ a. Clearly, J 2 = −e, and we only have to verify the condition (3). For z1 = x1 + y¯1 and z2 = x2 + y¯2 , where xi , yi ∈ a, we get y1 , y¯2 ], whence [z1 , z2 ] = [x1 , x2 ] + [¯ [z1 , Jz2 ] = [x1 , ix2 ] − [¯ y1 , i¯ y2 ] = i[x1 , x2 ] − i[¯ y1 , y¯2 ] = J[z1 , z2 ] . It follows that g0 is the realification of the complex Lie algebra h = (g0 , J). Clearly, h is simple. If it is commutative, then h C. But in this case g0 R ⊕ R is not simple.  Let g be a complex semisimple Lie algebra. We are going now to construct two real forms of g using its canonical generators (see (1.11)). Consider the complex conjugate Lie algebra ¯ g. Clearly, ¯ g is semisimple, and any maximal toral subalgebra t of g is a maximal toral subalgebra of ¯g, too. Consider the root space decomposition (1.4) of g with respect to t. Clearly, any root subspace gα of g is the root subspace of ¯ g corresponding to the root α ¯ ∈ t∗ . Thus, the system of ¯ roots of ¯ g is ∆ = {α ¯ | α ∈ ∆}, where ∆ is the system of roots of g. It follows that the roots of ¯ g determine the same real form t(R) of t as the roots of g do, and α ¯ |t(R) = α|t(R), α ∈ ∆. Identifying α ∈ ∆ with its restriction to t(R), we get α ¯ = α. By Proposition 2 (ii) and (II.5), the Killing forms of g and of ¯g induce the same scalar product on t(R) and also on t(R)∗ . Now, choosing a Weyl chamber D of t(R) for g, we see that it is a Weyl chamber for ¯g, too. Denoting by Π = {α1 , . . . , αl } the corresponding system of simple roots for g (and for ¯g), it follows that the canonical system of generators {hi , ei , fi | i = 1, . . . , l} of g is at the same time a canonical system of generators of ¯g, with the same Cartan matrix A. By (II.22), we get an isomorphism σ : g → ¯g such that σ(hi ) = hi ,

σ(ei ) = ei ,

σ(fi ) = fi ,

i = 1, . . . , l .

2

(6) 2

Then σ is an automorphism of g leaving invariant our generators. Thus, σ = id, and so σ is a real structure in g. The corresponding real form gσ is the real subalgebra of g generated by {hi , ei , fi | i = 1, . . . , l}. It is called the normal (or split ) real form. Example 2. As in Example 1, consider the Lie algebra g = sln (C). As t, we may take the subalgebra of all diagonal matrices H = diag(x1 , . . . , xn ) , In particular, rk sln (C) = n − 1.

x1 + . . . + xn = 0 .

§2. Complexification and real forms

18

Let Eij denote the matrix with (i, j)-entry 1, all other entries being 0. Then [H, Eij ] = (xi − xj )Eij ,

H ∈ t.

It follows that ∆ = {xi − xj | i = j}, and exi −xj = Eij . Also hxi −xj = Eii − Ejj . Hence, t(R) is the set of all real diagonal matrices with trace 0. Let us fix the Weyl chamber D = {diag(x1 , . . . , xn ) | x1 < . . . < xn } ⊂ t(R) . Then ∆+ = {xi − xj | i < j} , Π = {α1 , . . . , αn−1 } ,

where

αi = xi − xi+1 , i = 1, . . . , n − 1 .

We may take the following canonical generators: hi = Eii − Ei+1,i+1 ,

ei = Ei,i+1 ,

fi = Ei+1,i ,

i = 1, . . . , n − 1 .

(7)

It follows that the normal real form is sln (R), and the corresponding real structure ¯ is given by σ(X) = X. To get another real form, we first construct an involutive automorphism of g which will be important in the sequel. With the Weyl chamber D fixed above we can associate the subset −D ⊂ t(R) which is, clearly, a Weyl chamber, too. The corresponding system of simple roots is −Π = {−α1 , . . . , −αl }, and its Cartan matrix coincides with that of Π. As a canonical system of generators of g, we may choose hi = h−αi , −fi , −ei , i = 1, . . . , l. By (II.22), we get a unique automorphism ω ∈ Aut g such that ω(hi ) = −hi ,

ω(ei ) = −fi ,

ω(fi ) = −ei ,

i = 1, . . . , l .

(8)

Clearly, ω 2 = id. The automorphism ω is called the Weyl involution. Now, ωσ = σω, since both sides are antiautomorphisms of g and coincide on the canonical generators. It follows that τ = ωσ = σω

(9)

is a real structure in g, giving a real form gτ . Example 3. Let us consider again the Lie algebra g = sln (C). In this case, the Weyl involution is given by ω(X) = −X  ,

X ∈ sln (C) .

In fact, the mapping X →  −X  is an automorphism of g, and one sees from (8) that it coincides with ω on the generators (7). Therefore, ¯ , τ (X) = −X

X ∈ sln (C) .

It follows that the corresponding real form gτ = sun .

§2. Complexification and real forms

19

Example 4. Due to Proposition 3, the existence of real structures on a complex semisimple Lie algebra g implies that gR (C) g ⊕ g or, equivalently, that gR is isomorphic to a real form of g ⊕ g. If we fix, e.g., the real structure τ on g defined by (9) and identify g with ¯ g via x → τ (x), then the real structure Σ given in the proof of this proposition is identified with Σ(x, y) = (τ (y), τ (x)) ,

x, y ∈ g .

(10)

The real form (g ⊕ g)Σ = {(x, τ (x)) | x ∈ g} is isomorphic to gR by the projection (x, τ (x)) → x. Example 5. Let g be the complex Lie algebra with basis e0 , e1 , e2 and commutation relations [e0 , e1 ] = e1 , [e0 , e2 ] = ie2 , [e1 , e2 ] = 0. Prove that g possesses no real forms.

§3. Real forms and involutive automorphisms By Proposition 2.1, the classification of real forms of a given complex Lie algebra g up to isomorphy is equivalent to the classification of antiinvolutions in g up to ´ Cartan proved (see [4]) that for a semisimple conjugacy by automorphisms of g. E. complex Lie algebra g, the word antiinvolutions in this trivial proposition may be replaced by involutions, i.e., involutive automorphisms of g. This description of real semisimple Lie algebras by involutive automorphisms of complex semisimple Lie algebras is a powerful tool in the study and classification of real Lie algebras, their subalgebras and homomorphisms. It is based on the existence of a compact real form of g, i.e., of a real form which is a compact Lie algebra. Let g be an arbitrary complex Lie algebra possessing real structures, and let us fix a real structure τ in g. Then there exists an obvious bijection between real structures in g and certain automorphisms of this Lie algebra. Namely, we assign to any real structure σ the automorphism θ = στ of g. We will say that the automorphism θ corresponds to the real structure σ. In particular, to σ = τ the identity automorphism e corresponds. Clearly, this correspondence depends on the choice of τ . The following properties of this correspondence are deduced directly. Proposition 1. (i) The correspondence σ → θ = στ is a bijection of the set of all antiinvolutions of g onto the set of all θ ∈ Aut g satisfying τ θτ = θ−1 . (ii) We have θ2 = e if and only if στ = τ σ. In this case, the real form gσ is invariant under τ and under θ. (iii) If θ corresponds to σ and α ∈ Aut g, then the automorphism θ
, corresponding to ασα−1 , has the form θ
= αθ(τ ατ )−1 .  Actually, the correspondence σ → θ is a special case of the description of the set of k-forms of an K-object X defined over a field k in terms of the Galois cohomology set H 1 (Γ, Aut X), where Γ is the Galois group of the field extension K ⊃ k (see [21], Ch. III). In our case, we deal with the cohomology set H 1 (Z2 , Aut g). The assertion (i) of Proposition 1 means that the function ¯0 → e, ¯1 → θ is a 1-cocycle 0, ¯ 1}, while (iii) expresses the cohomology relation between two cocycles, of Z2 = {¯ and (ii) asserts that invariant cocycles correspond to involutive automorphisms. Let now a complex semisimple Lie algebra g be given. In this case, a compact real form always exists (see Theorem 1 below), and we choose the corresponding real structure τ as the fixed real structure in the correspondence described above. ´ Cartan proved that any real structure σ in g is conjugate to a real strucE. ture commuting with τ , which gives an involution θ = στ by Proposition 1 (ii) (or, equivalently, that any Galois cohomology class contains an invariant cocycle). Following [11], Ch. III (see also [19], §5.1), we use for constructing such a structure a Hermitian scalar product in g associated with τ .

§3. Real forms and involutive automorphisms

21

More generally, with any real structure σ on g we associate the function hσ : g × g → C given by (1) hσ (x, y) = −k(x, σy) , x, y ∈ g , where k = kg is the Killing form of g. Clearly, it is linear in the first argument and antilinear in the second one. Proposition 2. (i) The function hσ is a Hermitian form coinciding with −kg on the real form gσ . (ii) hσ (x, y) = hσ (γx, γy), x, y ∈ g, for any γ ∈ Aut g satisfying γσ = σγ. (iii) hσ ([x, u], v) = −hσ (u, [x, v]) for x ∈ gσ , u, v ∈ g. (iv) The real form gσ is compact if and only if the Hermitian form hσ is positive definite. Proof. (i) Using Proposition 2.2 (ii), we get hσ (y, x) = −k(y, σx) = −k(x, σ(y)) = hσ (x, y) , and so hσ is hermitian. Clearly, hσ (x, y) = −k(x, y), whenever x, y ∈ gσ . (ii) follows trivially from the invariance of k under γ (see (I.3)), and (iii) from the invariance of k. (iv) If hσ is positive definite, i.e., hσ (x, x) > 0 for any non-zero x ∈ g, then, by (i), the restriction k|gσ is negative definite. But this restriction coincides with kgσ (Proposition 2.2 (i)), and hence gσ is compact by (I.12). Conversely, if gσ is compact, then, by (I.12), hσ is positive definite on the real form gσ . Since hσ is hermitian, it is positive definite on g.  Now we are going to show that each complex semisimple Lie algebra g has a compact real form. More precisely, we prove that the real form gτ constructed in §2 is compact. We will use the notation of §2. The real structure τ was defined by (2.9), in terms of the canonical generators ei , fi , hi corresponding to a simple root system Π = {α1 , . . . , αl }. We note some properties of τ . Proposition 3. We have τ |t(R) = − id, τ (gα ) = g−α , α ∈ ∆. Proof. The first relation follows from the definition of τ . To prove the second one, we apply τ to the equation [t, eα ] = α(h)eα ,

h ∈ t(R) .

Since α(h) ∈ R, we get [h, τ (eα )] = −α(h)τ (eα ) ,

h ∈ t(R) .

By linearity, this is also true for all t ∈ t, whence τ (eα ) ∈ g−α .



22

§3. Real forms and involutive automorphisms

Proposition 4. Any w ∈ W is of the form w = ϕ|t(R), where ϕ ∈ Int g has the following properties: ϕ(t) = t, ϕτ = τ ϕ. Proof. By (II.17), any w ∈ W is a product of reflections ri = rαi for some simple roots αi ∈ Π. Now, by (II.12), ri = ϕi |t(R), where ϕi = ϕαi = exp(ad π2 (ei − fi )). Thus, we may choose ϕ as a product of ϕi , and it is sufficient to verify that  ϕi τ = τ ϕi . Since τ ∈ Aut gR , we have τ ϕi τ −1 = exp(ad π2 τ (ei − fi )) = ϕi . Theorem 1. The real form gτ is compact. Proof. Due to Proposition 2 (iv), it suffices to show that the Hermitian form hτ is positive definite. Proposition 3 and (II.4) imply that the root space decomposition (1.4) is orthogonal with respect to hτ . Therefore we have to verify that hτ is positive definite on any gα , α ∈ ∆, and on t. By (II.11), we have hτ (ei , ei ) = −k(ei , −fi ) = k(ei , fi ) = (αi2,αi ) > 0, i = 1, . . . , l. Let α ∈ ∆ be an arbitrary root. By (II.19), there exists w ∈ W such w (α) = αi ∈ Π. Applying Proposition 4, we get ϕ ∈ Int g such that ϕ(t) = t, w = ϕ|t(R) and ϕτ = τ ϕ. Now, by (II.11), we have ϕ(gαi ) = gα . But ϕ leaves hτ invariant (Proposition 2 (iii)), and hence hτ is positive definite on gα . Finally, we see that hτ (h, h) = −k(h, −h) = k(h, h) > 0 for any non-zero h ∈ t(R) (see (II.4)). Since hτ is Hermitian, this is also true for any non-zero h ∈ t.  ¯ and ω(X) = −X  , and hence Example 1. For g = sln (C) we have σ(X) = X  ¯ τ (X) = −X (see Example 2.3). It follows that sun is a compact real form of sln (C). This is also evident from the fact that the corresponding Lie group SUn ⊂ SLn (C) is compact and from (I.10). Now we discuss some well-known facts from Hermitian geometry which will be useful in what follows. Let E be a finite dimensional vector space over the field K = C or R, endowed with a scalar product, i.e., a positive definite Hermitian (for K = C) or symmetric bilinear (for K = R) form ( , ). Denote n = dimK E. For any linear operator (endomorphism) α ∈ gl(E), the adjoint operator α∗ is defined by the condition (αx, y) = (x, α∗ y) , x, y ∈ E . A linear operator α is called self-adjoint if α∗ = α (one also says that α is Hermitian (in the complex case) or symmetric (in the real one). Let S(E) ⊂ gl(E) denote the vector subspace of all self-adjoint operators. Self-adjoint operators can be also described as linear operators α admitting a system of eigenvectors with real eigenvalues which forms an orthonormal basis of E. This is equivalent to the following

k property: E admits an orthogonal direct decomposition E = i=1 Eλi , where Eλi is the eigenspace of α corresponding to the eigenvalue λi ∈ R. A self-adjoint operator α is said to be positive definite if (αx, x) > 0 for all non-zero x ∈ E or, equivalently, if all its eigenvalues are positive. Let P(E) ⊂ GL(E) denote the subset of all positive definite self-adjoint operators. It is open in S(E).

k Take α ∈ S(E) and let E = i=1 Eλi be the corresponding eigenspace decomposition. One sees easily that the operator exp α admits the same eigenspace decomposition, with the eigenvalues exp λi > 0 instead of λi . Thus, exp α ∈ P(E), and we have the mapping exp : S(E) → P(E). This mapping is bijective. In

§3. Real forms and involutive automorphisms

23

fact, we can define the inverse mapping log sending a positive definite self-adjoint operator β with eigenvalues µi > 0 to the linear operator admitting the same eigenspace decomposition, but with the eigenvalues log µi ∈ R. Any α ∈ P(E) can be included into a unique real one-parameter subgroup β(t), t ∈ R, of the Lie group GL(E), lying in P(E) and satisfying β(1) = α. In fact, if β(t) is a one-parameter subgroup with these properties, then β
(0) ∈ S(E) and α = β(1) = exp β
(0). Thus, β
(0) = log α is uniquely determined by α. On the other hand, the formula β(t) = exp(t log α) gives, clearly, the desired subgroup. If t ∈ Z, then β(t) = αt . The same notation will be used for arbitrary t ∈ R. This is suggested by the following fact: if λi > 0 are the eigenvalues of α, then the operator β(t) can be given by the same eigenspace decomposition as α with the eigenvalues λti . If α,β ∈ P(E) satisfy β = γαγ −1 , where γ ∈ GL(E), then logβ = γ(logα)γ −1 and t β = γαt γ −1 for any t ∈ R. In the complex case the same is true for any invertible antilinear operator γ in E. This can be proved directly, using the eigenspace decompositions. Lemma 1. Let g be a complex or real Lie algebra of finite dimension endowed by a scalar product, and let θ ∈ Aut g ∩ P(g). Then θt ∈ Aut g for all t ∈ R. If g is semisimple, then θt ∈ Int g for all t ∈ R.

k Proof. Let g = i=1 gλi be the eigenspace decomposition for θ, where λi > 0 are the eigenvalues. If x ∈ gλi , y ∈ gλj , then θ[x, y] = [θx, θy] = [λi x, λj y] = λi λj [x, y] , whence [x, y] ∈ gλi λj . Therefore θt [x, y] = (λi λj )t [x, y] = [λti x, λtj y] = [θt x, θt y] for any t ∈ R. It follows that θt ∈ Aut g. The last assertion follows from (I.7).  A compact real structure in a complex Lie algebra is a real structure, whose corresponding real form is compact. In this case, any conjugate real structure is compact as well. From now on, we suppose that a compact real structure τ in a given complex semisimple Lie algebra g (existing by Theorem 1) is fixed. We will study the correspondence σ → θ = στ between antiinvolutions and automorphisms of g (see the beginning of the section). Let us regard g as the Hermitian vector space with the scalar product ( , ) = hτ (see (1)). Lemma 2. (i) For any antiinvolution σ, we have θ = στ ∈ S(g), and thus θ2 ∈ P(g)∩Aut g. (ii) For any involution θ ∈ Aut g, we have ψ = (θτ )2 ∈ P(g) ∩ Aut g. Proof. (i) To prove that θ∗ = θ, we get, using (I.3), (θx, y) = −k(θx, τ y) = −k(x, θ−1 τ y) = −k(x, τ στ y) = (x, θy) . for any x, y ∈ g. Clearly, θ2 is an positive definite automorphism.

24

§3. Real forms and involutive automorphisms

(ii) Again, using (I.3), we get (ψx, y) = −k(ψx, τ y) = −k(x, ψ −1 τ y) = −k(x, τ θτ θτ y) = −k(x, τ ψy) = (x, ψy) , for any x, y ∈ g. Thus, ψ ∗ = ψ. Now, hτ (ψx, x) = −k(ψx, τ x) = −k(θτ θτ x, τ x) = −k(τ θτ x, θτ x) = −k(θτ x, τ θτ x) = ((θτ )x, (θτ )x) > 0 for any x = 0 .



Given a complex Lie algebra g and two real structures σ, τ in g, we say that real forms gσ , gτ are compatible if στ = τ σ. Proposition 5. Let g be a complex semisimple Lie algebra. If two compact real forms of g are compatible, then they coincide. Proof. Let σ, τ be two compact real structures in g, such that στ = τ σ. We want to prove that σ = τ . To do this, consider the involution θ = στ . We have to show that θ = id (this is equivalent to σ = τ −1 = τ ). Since θ is an automorphism and gσ is a real form of g, it is sufficient to prove that θx = x for each x ∈ gσ . By Proposition 1 (ii), gσ is invariant under θ, and hence we have the eigenspace decomposition gσ = gσ+ ⊕ gσ− , where gσ± = {x ∈ g | θ(x) = ±x} (see (1.2)). To show that the second summand is 0, we use the positive definite Hermitian forms hσ and hτ given by (1). If there is a non-zero x ∈ gσ− , then hσ (x, x) > 0 and hτ (x, x) > 0. But hσ (x, x) = −k(x, x), while hτ (x, x) = −k(x, τ x) = k(x, x). This gives a contradiction.  Proposition 6. For any real structure σ in g, there exists α ∈ Int g such that  σ
= ασα−1 satisfies σ
τ = τ σ
, i.e., the real forms gσ = α(gσ ) and gτ are 1 compatible. We may choose α = ϕ− 4 , where ϕ = (στ )2 . Proof. Consider the automorphism θ = στ ∈ Aut g. By Lemma 2, ϕ = θ2 ∈ P(g) ∩ Aut g. By Lemma 1, we get the one-parameter subgroup ϕt ∈ Int g, t ∈ R. Using Proposition 1 (i), we see that τ ϕτ = ϕ−1 . This implies, by Lemma 2, that τ ϕt τ = ϕ−t , t ∈ R. Clearly, we also have θϕθ−1 = ϕ, whence θϕt θ−1 = ϕt , t ∈ R. We will search the desired inner automorphism α in the form α = ϕt for a certain t ∈ R. We can write (ασα−1 )τ = ϕt σϕ−t τ = ϕt στ ϕt = ϕt θϕt = ϕ2t θ , τ (ασα−1 ) = τ ϕt σϕ−t = ϕ−t τ σϕ−t = ϕ−t θ−1 ϕ−t = ϕ−2t θ−1 . These operators coincide if and only if ϕ4t = θ−2 = ϕ−1 , i.e., if t = − 14 . Thus, 1 α = ϕ− 4 is the desired automorphism. The last assertion follows from the property t of ϕ discussed before Lemma 1.  Now we can conclude that any two compact real structures are conjugate by an inner automorphism of g.

§3. Real forms and involutive automorphisms

25

Corollary. For any two compact real structures σ, τ in g there exists α ∈ Int g such that τ = ασα−1 . Thus, any complex semisimple Lie algebra admits a unique, up to conjugacy by inner automorphisms, compact real structure. The automorphism α can be chosen in the form described in Proposition 6. Proof. Apply Proposition 6 to the real structure σ and use Proposition 5.



Now we prove a similar proposition, where instead of the antiinvolution σ an involution is considered. Proposition 7. Given an involution θ ∈ Aut g, there exists α ∈ Int g such that 1 αθα−1 commutes with τ . We may choose α = ψ − 4 , where ψ = (θτ )2 . Proof. The proof is similar to that of Proposition 6. One considers the automorphism ψ = (θτ )2 ∈ P(g) ∩ Aut g (see Lemma 2 (ii)). Then one verifies that 1  α = ψ − 4 is the desired inner automorphism. Now we define the desired bijection between the conjugacy classes of antiinvolutions and involutions of a complex semisimple Lie algebra g. As above, we assume that a compact real structure τ is fixed. For any real structure σ, we can choose a real structure σ
= ασα−1 such that σ
τ = τ σ
, where α ∈ Int g is the inner automorphism constructed in Proposition 6. Then we assign to σ the automorphism θ = σ
τ , which is involutive due to Proposition 1 (ii). Theorem 2. Let g be a complex semisimple Lie algebra. The mapping σ → θ described above determines a bijection between the conjugacy classes of antiinvolutions and involutions by inner automorphisms (or by automorphisms) of g. This bijection does not depend on the choice of the compact real structure τ in g. Proof. First we shall prove that our mapping actually determines a mapping of conjugacy classes. Suppose that we have two antiinvolutions σ, σ1 which are conjugate by an inner automorphism. Then the corresponding antiinvolutions σ
, σ1
commuting with τ are also conjugate by an inner automorphism, i.e., σ1
= βσ
β −1 , where β ∈ Int g. Clearly, σ1
commutes with τ and βτ β −1 . By Corollary of Proposition 6, there exists γ ∈ Int g commuting with σ1
, such that γτ γ −1 = βτ β −1 . Then α = γ −1 β commutes with τ and satisfies σ1
= ασ
α−1 . Clearly, for the corresponding involutions θ = σ
τ and θ1 = σ1
τ we have θ1 = αθα−1 . Thus, we have a well defined mapping of conjugacy classes. Now we show that this mapping is bijective. First we prove the surjectivity. Let an involution θ ∈ Aut g be given. By Proposition 7, there exists α ∈ Int g such that θ
= αθα−1 satisfies θ
τ = τ θ
. Then σ = θ
τ is an antiinvolution. Since στ = τ σ, our mapping sends the conjugacy class of σ to that of θ
. Next we show the injectivity. Suppose that two antiinvolutions σ1 , σ2 are given and that θ1 = σ1
τ and θ2 = σ2
τ satisfy θ2 = βθ1 β −1 for a certain β ∈ Aut g. Here σi
are antiinvolutions conjugate to σi and commuting with τ . Clearly, we may assume that σi τ = τ σi and θi = σi τ, i = 1, 2. Then θ2 commutes with τ and βτ β −1 . By Corollary of Proposition 6, there exists γ ∈ Int g commuting with θ2 , such that γτ γ −1 = βτ β −1 . Then α = γ −1 β commutes with τ and satisfies θ2 = αθ1 α−1 . Clearly, σ2 = ασ1 α−1 .

26

§3. Real forms and involutive automorphisms

Finally, let us consider another compact real structure τ1 in g and the corresponding bijection between conjugacy classes of antiinvolutions and involutions. We claim that it coincides with the original bijection determined by τ . In fact, by Corollary of Proposition 6, we have τ1 = ατ α−1 , where α ∈ Int g. Let σ → σ
→ θ = στ , where σ
is conjugate to σ and commutes with τ , be the mapping giving the original bijection. Then, clearly, the new bijection is determined by the mapping σ → ασ
α−1 → θ1 = (ασ
α−1 )τ1 = αθα−1 . This implies our assertion. Clearly, the above argument also gives a bijection between the conjugacy classes of antiinvolutions and involutions by arbitrary automorphisms of g.  Example 2. By the bijection of Theorem 2, to the class of compact structures the identical involution θ = e = id corresponds. Now, if σ is the normal real structure given by (2.5), then the corresponding involutive automorphism is the Weyl involution ω = στ given by (2.7). Example 3. In Example 2.1 we described certain real structures in sln (C). All ¯  (see these structures commute with the compact real structure τ (X) = −X Example 1). We list the involutive automorphisms θ = στ corresponding to various real structures σ. ¯ we get θ(X) = ω(X) = −X . For the normal real structure σ(X) = X −1  −1 T ¯ For σ(X) = Sm XS m , n = 2m, we get θ(X) = −Sm X Sm = (AdSm )(−X ).  ¯ Ip,q we get θ(X) = Ip,q XIp,q . For σ(X) = −Ip,q X Example 4. Let g be a complex semisimple Lie algebra. Due to Example 2.4, we may regard gR as the real form of g ⊕ g corresponding to the antiinvolution Σ given by (2.9). Clearly, τ × τ is a compact real structure on g ⊕ g commuting with Σ. Thus, the involution, corresponding to the real form gR by Theorem 2, is θ = Σ(τ × τ ) : (x, y) → (y, x), x, y ∈ g.

§4. Automorphisms of complex semisimple Lie algebras We present here some main facts about the automorphism group Aut g of a complex semisimple Lie algebra g. Suppose that a maximal toral subalgebra t of g is chosen and let ∆ denote the corresponding system of roots. By (II.12), for any θ ∈ Aut g leaving t invariant, the transformation θ of t∗ (R) maps ∆ onto itself. Let us also choose a Weyl chamber in t(R), and let Π ⊂ ∆ denote the corresponding subset of simple roots. Consider the subgroups of Aut g defined by Aut(g, t) = {θ ∈ Aut g | θ(t) = t} , Aut(g, t, Π) = {θ ∈ Aut(g, t) | θ (Π) = Π} . Write Π = {α1 , . . . , αl } and let A = (aij ) denote the corresponding Cartan matrix. Any bijection s : Π → Π can be regarded as a permutation s ∈ Sl . The bijection is called an automorphism of Π if s leaves invariant the matrix A, i.e., if as(i)s(j) = aij for all (i, j). Denote by Aut Π the group of automorphisms of Π. If θ ∈ Aut(g, t, Π), then θ|t(R) is an orthogonal transformation, and hence (θ )−1 induces an automorphism of Π. Clearly, the mapping Φ : θ → (θ )−1 |Π is a homomorphism of the group Aut(g, t, Π) to Aut Π. The homomorphism Φ is surjective. Moreover, there exists a homomorphism Ψ : Aut Π → Aut(g, t, Π) such that ΦΨ = id. To see this, consider the canonical system of generators {hi , ei , fi | i = 1, . . . , l} given by (1.11). By (II.22), for any s ∈ Aut Π there exists a unique automorphism sˆ of g satisfying sˆ(ei ) = es(i) ,

sˆ(fi ) = fs(i) ,

sˆ(hi ) = hs(i) ,

i = 1, . . . , l .

(1)

Clearly, sˆ ∈ Aut(g, t, Π). Also Φ(ˆ s) = s. In fact, for any i, j we have s(hj )) = αi (hsˆ(j) ) = ai,s(j) = as−1 (i),j = s−1 (αi )(hj ) , (ˆ s (αi ))(hj ) = αi (ˆ  = sˆtˆ, and hence we get the homomorphism Ψ : whence sˆ |Π = s−1 . Now, st s → sˆ of Aut Π into Aut(g, t, Π) with the desired property ΦΨ = id. This implies that Φ is surjective, Ψ is injective, and we have the semidirect decomposition Aut(g, t, Π) = Ker Φ  Ψ(Aut Π). The automorphism Ψ(s) = sˆ defined by (1) is called the diagram automorphism of g corresponding to s ∈ Aut Π. In what follows, we denote by the same symbol s s )−1 = s the linear extension of s onto t(R)∗ . The above argument shows that (ˆ ∗ on t(R) . One also sees easily that any diagram automorphism commutes with the real structures σ, τ and the Weyl involution ω associated with Π (see §2). Proposition 1. The normal subgroup Ker Φ ⊂ Aut(g, t, Π) coincides with T = exp(ad t), and we have the semidirect decomposition Aut(g, t, Π) = T  Ψ(Aut Π) .

(2)

Proof. We only have to prove the assertion concerning Ker Φ. Clearly, exp(ad h) acts trivially on t for any h ∈ t. Conversely, suppose that θ ∈ Aut(g, t, Π) satisfies

28

§4. Automorphisms of complex semisimple Lie algebras

Φ(θ) = e. Then θ|t = id. By (II.12), θ(ei ) = ci ei , θ(fi ) = di fi , θ(hi ) = hi , where ci , di ∈ C \ {0}, i = 1, . . . , l. The relation [ei , fi ] = hi implies di = c−1 i , i = 1, . . . , l. Choosing h0 ∈ t in such a way that αi (h0 ) = log ci , i = 1, . . . , l, we see that θ and exp(ad h0 ) coincide on the generators ei , fi . Hence θ = exp(ad h0 ) ∈ T .  We are now going to extend this semidirect decomposition to the entire group Aut g. Theorem 1. Let g be a complex semisimple Lie algebra. Then Aut g = Int g  Ψ(Aut Π) ,

(3)

and the corresponding projection Aut g → Aut Π coincides with Φ on Aut(g, t, Π). Proof. First we prove that Aut g = (Int g) Aut(g, t, Π). If θ ∈ Aut g, then θ(t) is a maximal toral subalgebra of g, and, by (II.1), there exists ϕ ∈ Int g such that ϕθ(t) = t. By (II.12), ϕθ permutes the Weyl chambers in t(R). Let D be the Weyl chamber corresponding to Π. Then, by (II.15), we have w(ϕθ(D)) = D for an element w ∈ W . By (II.13), w = ψ|t(R) for an automorphism ψ ∈ Int g. Then ψϕθ(D) = D, and hence θ0 = ψϕθ ∈ Aut(g, t, Π). We got the desired decomposition θ = (ψϕ)−1 θ0 , where (ψϕ)−1 ∈ Int g. Since T ⊂ Int g, this decomposition together with (2) gives the decomposition Aut g = (Int g)Ψ(Aut Π) . Then (II.16) implies that Int g ∩ Ψ(Aut Π) = {e}. Hence, the latter decomposition is semidirect.  Corollary 1. We have Aut g/ Int g Aut Π .



In other words, the group Out g = Aut g/ Int g is isomorphic to the group of all symmetries of the Dynkin diagram of g. For the simple Lie algebras this group is indicated in Table 1. For the non-simple Lie algebras, there are also symmetries which permute isomorphic simple components of g. Corollary 2. There is the semidirect decomposition Aut(g, t) = N  Ψ(Aut Π) , where N = Aut(g, t) ∩ Int g.  Example 1. Consider the Lie algebra g = sln (C). We want to use Theorem 1 for the classification of involutive automorphisms of g. The result is that any such automorphism is conjugate in Aut g (by an inner automorphism) to one of the automorphisms listed in Example 3.3. If n ≥ 3, then we see from Table 1 that Aut Π Z2 . The automorphism ω : X → −X  is in this case an outer one. In fact, if ω is an inner automorphism, then ω = Ad C for a certain C ∈ SLn (C) (see (I.2)). Then −X  = CXC −1 , X ∈

§4. Automorphisms of complex semisimple Lie algebras

29

sln (C). But, e.g., X = diag(2, 0, . . . , 0, −1, −1) cannot satisfy to such an equation, since X and −X  have different eigenvalues. Let us classify first the outer involutive automorphisms. By Theorem 1, any outer automorphism has the form θ = ϕω, where ϕ ∈ Int g. Writing ϕ = Ad C, where C ∈ SLn (C), we have θ(X) = −CX  C −1 ,

X ∈ sln (C) .

The relation θ2 = id is equivalent to the condition X = (C(C  )−1 )X(C(C  )−1 )−1 ,

X ∈ sln (C) .

Thus, C(C  )−1 should commute with all X ∈ sln (C). This is true precisely in the case, when C(C  )−1 = λIn , where λ ∈ C× , which can be rewritten as C = λC  . Transposing this relation, we get C  = λC. It follows that C = λ2 C, whence λ = ±1. Thus, we get two different cases. I. If λ = 1, then we have C  = C. The reduction theory of symmetric bilinear forms implies that C = U U  , where the matrix U can be chosen from SLn (C). Then θ(X) = −U U  X  (U  )−1 U −1 = (Ad U )ω(Ad U )−1 (X) ,

X ∈ sln (C) .

Thus, θ = (Ad U )ω(Ad U )−1 is conjugate to ω by an inner automorphism of g. II. If λ = −1, then we have C  = −C and n = 2m. Using the reduction theory of skew-symmetric bilinear forms, we see that C = U Sm U  , where U ∈ SLn (C) and Sm is given by (2.5). It follows that θ = (Ad U )(Ad Sm )ω(Ad U )−1 is conjugate to (Ad Sm )ω by an inner automorphism of g. Note that these two cases give two different conjugacy classes of automorphisms. In fact, in the case I we have gθ son (C), while in the case II gθ spn (C), and these Lie algebras are non-isomorphic for any even n ≥ 2. Next, we consider the case III. θ is an inner involutive automorphism. Suppose that θ(X) = CXC −1 ,

X ∈ sln (C) ,

where C ∈ SLn (C). The condition θ2 = id is equivalent to C 2 X(C 2 )−1 = X, X ∈ sln (C), whence C 2 = λIn , where λ ∈ C× . Multiplying C by an appropriate scalar, we can assume that C 2 = In , but C ∈ / SLn (C). Then C is diagonalizable with eigenvalues ±1, i.e., C = U Ip,q U −1 , where U ∈ GLn (C) and Ip,q is given by (2.4). This implies that θ = (Ad U ) Ad Ip,q (Ad U )−1 is conjugate to Ad Ip,q in the group Aut g. By considering the subalgebras gθ , one sees easily that Ad Ip,q , p ≤ q, are pairwise non-conjugate. Now let us consider the case n = 2. Then all the automorphisms are inner ones, and so we have only the case III. One sees that θ is either trivial, or conjugate to Ad I1,1 . Thus, we have classified all involutive automorphisms up to conjugacy in Aut g.

30

§4. Automorphisms of complex semisimple Lie algebras

Using this classification, we can apply Theorem 3.2 to describe all the real forms of sln (C). We see that any real form is isomorphic to one of the forms described in Example 2.1. In the case n ≥ 3 all these forms are pairwise non-isomorphic. In the case n = 2 all the non-compact real forms are isomorphic: sl2 (R) su1,1 , while sl1 (H) = su2 . Now we illustrate Theorem 1 by determining the corresponding decomposition of the Weyl involution ω ∈ Aut g, defined by (2.8) in terms of the canonical generators {hi , ei , fi | i = 1, . . . , l}. Here we use the notation of the beginning of this section. Clearly, ω ∈ Aut(g, t), ω|t(R) = − id, and so ω(D) = −D is the Weyl chamber opposite to D. By (II.15), there exists a unique w0 ∈ W such that w0 (D) = −D. Since w02 (D) = D, we have w02 = e. Thus, (−w0 )(D) = D, and hence −w0 induces an involutive automorphism ν ∈ Aut Π. Then we get the diagram automorphism νˆ which is involutive as well. Following the proof of Theorem 1, we can write ω = ϕˆ ν, where ϕ ∈ N satisfies ϕ|t(R) = w0 . Our goal is to describe the automorphism ϕ explicitly. Consider the element  hα ∈ t(R) . (4) h= α∈∆+

Lemma 1. We have β(h) = 2 for any β ∈ Π. In particular, h ∈ D. Proof. The element h can be regarded as dual to the linear form 2γ (see (1.10)), and our assertion is similar to (II.20). To prove it, one first deduces from (1.7), (1.8) and (1.9) the relation rβ (hα ) = hrβ
(α) ,

α, β ∈ ∆

(we remind that rβ (α) ∈ ∆, due to (II.14)). Then we see that rβ (h) =



rβ (hα ) =

α∈∆+

 α∈∆+

hrβ
(α) .

If β ∈ Π, then, by (II.20), rβ (α) ∈ ∆+ , except of the case α = β, when rβ (β) = −β. This implies that rβ (h) = h − 2hβ , and our assertion follows from (1.8).  Clearly, we have h=

l 

ri hi ,

(5)

i=1

where ri ∈ N. The numbers ri are important invariants of the Lie algebra g. For simple complex Lie algebras, they are given in Table 4. We also define the elements e, f ∈ g by e=

l  √ ri ei , i=1

f=

l  √ ri fi . i=1

(6)

§4. Automorphisms of complex semisimple Lie algebras

31

Then {e, h, f } is an sl2 -triple. In fact, Lemma 1 implies easily the relations [h, ei ] = 2ei and [h, fi ] = −2fi , whence [h, e] = 2e and [h, f ] = −2f . Also [e, f ] = h, due to (1.12). It follows that the correspondence  E=

0 0

1 0



 → e ,

F =

0 1

0 0



 → f ,

H=

1 0

0 −1

 → h

(7)

determines a homomorphism q : sl2 (C) → g which maps sl2 (C) isomorphically onto s. The subalgebra s is called the principal three-dimensional subalgebra of g. We shall prove now that s is pointwise invariant under any diagram automorphism. Proposition 2. For any s ∈ Aut Π, we have sˆ(x) = x, x ∈ s. Proof. One sees easily that any s ∈ Aut Π permutes the positive roots and that sˆ(hα ) = h(s
)−1 (α) = hs(α) , α ∈ ∆+ . It follows that sˆ(h) = h. By (1) and (5), we have l l   sˆ(h) = ri hs(i) = ri hi = h . i=1

i=1

This implies the relations rs(i) = ri ,

i = 1, . . . , l , s ∈ Aut Π .

Using (1), (6) and (8), we get sˆ(e) = e, sˆ(f ) = f . Thus, sˆ|s = id.

(8) 

In the Lie group SL2 (C), take the element π g0 = exp (E − F ) = 2



0 1 −1 0

 .

One easily verifies that (Ad g0 )(X) = g0 Xg0−1 = −X  , X ∈ sl2 (C). Thus, ω0 = Ad g0 = exp(ad π2 (E−F )) is the Weyl involution of sl2 (C) (see Example 2.3). Using the principal three-dimensional subalgebra, we can define a similar automorphism ϕ ∈ Int g by π (9) ϕ = exp(ad (e − f )) . 2 To relate it to ω, we need the following simple lemma, which will be used later in various situations and therefore is formulated in a general form. Lemma 2. Suppose a homomorphism f : h → g of arbitrary Lie algebras be given. Then for any ψ ∈ Int h we have f ψ = ϕf , where ϕ ∈ Int g. If ψ = exp(ad x), x ∈ h, then we may take ϕ = exp(ad f (x)). If we have a homomorphism F : H → G of the corresponding connected Lie groups such that de F = f and if ψ = Ad g0 , g0 ∈ H, then we may take ϕ = Ad F (g0 ). Proof. If ψ = Ad g0 , where g0 ∈ H, then we may write F (αg0 (h)) = F (g0 hg0−1 ) = F (g0 )F (h)F (g0 )−1 = αF (g0 ) (F (h)), h ∈ G, i.e., F αg0 = αF (g0 ) F . Differentiating this relation, we get f ψ = (Ad F (g0 ))f , and so we may set ϕ = Ad F (g). In

32

§4. Automorphisms of complex semisimple Lie algebras

the case when ψ = exp(ad x) = Ad(exp x), x ∈ h, we have ϕ
= Ad F (exp x) = exp(ad f (x)). Note that the Lie group homomorphism always exists if we choose H to be simply connected.  Returning to our situation, we denote by G a connected Lie group with Lie algebra g. Since SL2 (C) is simply connected, we may consider a Lie group homomorphism Q : SL2 (C) → G such that de Q = q. Let us denote π g = Q(g0 ) = exp( (e − f )) ∈ G . 2

(10)

Due to the definition (9), ϕ = Ad g. Clearly, Lemma 2 implies the relation qω0 = ϕq .

(11)

Proposition 3. The automorphism ϕ given by (9) and the element g ∈ G defined by (10) possess the following properties: (i) ϕ(h) = −h, ϕ(e) = −f , ϕ(f ) = −e. (ii) ϕ(t(R)) = t(R), ϕ|t(R) = w0 . (iii) ϕ2 = e, g 2 = exp(πih) ∈ Z(G). (iv) ϕˆ s = sˆϕ for any s ∈ Aut Π. (v) ω = ϕˆ ν = νˆϕ. Proof. (i) follows immediately from (11). To prove (ii), we note that, by Lemma 1, h is a regular element of t(R), and hence t = zg (h). On sees from (i) that ϕ leaves t and hence t(R) invariant. By (II.16), ϕ|t(R) ∈ W . Since h ∈ D, we have ϕ(D) = −D, and therefore ϕ|t(R) = w0 . Clearly, g02 = −I2 = exp(πiH). Applying Q, we get g 2 = exp(πiq(H)) = exp(πih). It follows from Lemma 1 that Ad g 2 (ei ) = eπiαi (h) ei = e2πi ei = ei , i = 1, . . . , l. Similarly, Ad g 2 (fi ) = fi . It follows that ϕ2 = Ad g 2 = e. Therefore g 2 ∈ Z(G) (see (I.2)), and (iii) is proved. If s ∈ Aut Π, then, by Proposition 2, sˆϕˆ s−1 = exp(ad

π π sˆ(e − f )) = exp(ad (e − f )) = ϕ , 2 2

and (iv) is proved. Now we prove (v). It follows from (ii) that ϕ |Π = −ν, i.e., ϕ (αi ) = −αν(i) , i = 1, . . . , l. By (II.12), this implies that ϕ(gαi ) = g−αν(i) . Therefore ϕ(ei ) = ci fν(i) and ϕ(fi ) = di eν(i) , where ci , di ∈ C \ {0}. Also ϕ(hi ) = −hν(i) . Setting s = ν in (8), we get rν(i) = ri ,

i = 1, . . . , l .

Since ϕ(e) = −f , we have l l l    √ √ √ ri ei ) = ri ci fν(i) = − ri fi . ϕ( i=1

i=1

i=1

(12)

§4. Automorphisms of complex semisimple Lie algebras

33

It follows from (12) that ci = −1, i = 1, . . . , l. Similarly, ϕ(f ) = −e implies di = −1, i = 1, . . . , l. Thus, ϕ(hi ) = −hν(i) ,

ϕ(ei ) = −fν(i) ,

ϕ(fi ) = −eν(i) ,

i = 1, . . . , l .

Now it is easy to see that ω coincides with ϕˆ ν = νˆϕ on the canonical generators.  The involution ν ∈ Aut Π introduced above is important for the theory of representations of semisimple Lie algebras. We are going now to describe the explicit form of this involution. Proposition 4. (i) If g is a non-commutative simple complex Lie algebra, then ν is non-trivial precisely in the cases g = Al , l ≥ 2; D2m+1 , m ≥ 1; E6 . In these cases, ν is the only

s non-trivial symmetry of the Dynkin diagram. (ii) If g = i=1 gi , where gi are simple, then ν ∈ Aut Π induces the s corresponding involution νi on each component of the decomposition Π = i=1 Πi , where Πi is the system of simple roots of gi , i = 1, . . . , s (see (II.23)). (iii) ν lies in the centre Z(Aut Π). Proof. (i) We have only to investigate the cases g = Al , l ≥ 2, g = Dl , l ≥ 3, and g = E6 , since for other simple g the Dynkin diagram has no symmetries (see Table 1). 1) g = Al = sll+1 (C), l ≥ 2. We take t as in Example 2.2 and use the notation of this example. The Weyl group W is the group of all permutations of the diagonal elements xi of a matrix H = diag(x1 , . . . , xl+1 ) ∈ t. Clearly, w0 (diag(x1 , . . . , xl+1 )) = diag(xl+1 , . . . , x1 ) . This implies that ν(αi ) = αl−i+1 ,

i = 1, . . . , l .

2) g = Dl = so2l (C), l ≥ 3. This Lie algebra is the algebra of all block matrices of the form   X Y , where X ∈ gll (C) , Y  = −Y , Z  = −Z . Z −X  As a maximal toral subalgebra t, we may choose the subalgebra of diagonal matrices H = diag(x1 , . . . , xl , −x1 , . . . , −xl ) . The system of roots is ∆ = {±xi ± xj | i = j}. We choose the Weyl chamber D = {H | x1 > · · · > xl−1 > |xl |} . Then Π = {α1 , . . . , αl } ,

where αi = xi − xi+1 ,

i = 1, . . . , l − 1 ,

αl = xl−1 + xl .

34

§4. Automorphisms of complex semisimple Lie algebras

The Weyl group is generated by permutations of the diagonal elements xi and by the transformations of the form xi → −xi , xj → −xj , i = j; xk → xk , k = i, j. We shall consider separately the cases of even and odd l. If l = 2m, then w0 = −e ∈ W transforms D into −D. Thus, ν = id. If l = 2m + 1, then w0 ∈ W is given by w0 (diag(x1 , . . . , xl , −x1 , . . . , −xl )) = diag(−x1 , . . . , −xl−1 , xl , x1 , . . . , xl−1 , −xl ) . It follows that ν(αi ) = αi , i = 1, . . . , l − 2, ν(αl−1 ) = αl . 3) g = E6 . We refer to [1], Ch. VI, §4, for a proof of the relation −e ∈ / W which implies ν = id. (ii) Any Weyl chamber for g has the form D = D1 × . . . × Ds , where Di is a Weyl chamber for gi , and w0 = (w0 )1 × . . . × (w0 )s , where (w0 )i is the element of the Weyl group of gi mapping Di onto −Di . Thus, ν  = ν1 × . . . × νs , which implies our assertion. (iii) For any s ∈ Aut Π, we have ϕ˜ = sˆϕˆ s−1 ∈ Int g, where ϕ ∈ N satisfies ˜ = ϕ|t(R) ˜ = (s )−1 w0 s ∈ W . ϕ|t(R) = w0 (e.g., ϕ is given by (9)). By (II.16), w But clearly w(D) ˜ = −D. Therefore w ˜ = w0 , whence ν = −w0 = −s(w0 )s−1 = −1 sνs .  Proposition 1 and Theorem 1 are first steps in the study of automorphisms of complex semisimple Lie algebras which leads, in particular, to the classification of involutive automorphisms up to conjugacy in Aut g and hence to the classification of real semisimple Lie algebras (see, e.g., [11], [19]). We will not give the details of this classification and only prove an old result of Gantmacher [8] establishing the so called canonical presentation of an involutive automorphism. We formulate it in a modern form (it is actually a special case of Theorem 4.4.3 of [19]). The proof will use the following remark. Remark 1. Suppose a compact real form u of a complex semisimple Lie algebra g be given. Then we can construct a maximal toral subalgebra t of g in the following way: we choose an arbitrary maximal commutative subalgebra t0 of u and set t = t0 (C). By Proposition 3.1, (ii), the operators ad u, u ∈ u, in g are skewHermitian with respect to a scalar product, and hence diagonalizable with pure imaginary eigenvalues. It follows that t is a toral subalgebra. This is a maximal commutative subalgebra of g, since it coincides with the centralizer zg (t0 ). We also note that t(R) = it0 . In fact, all roots of g with respect to t are pure imaginary on t0 and real on it0 . Theorem 2. Let an involutive automorphism θ ∈ Aut g be given. Then (i) There exist a maximal toral subalgebra t ⊂ g and a subset Π of simple roots of the corresponding system of roots ∆ such that θ ∈ Aut(g, t, Π). (ii) θ is conjugate by an inner automorphism to an automorphism of the form s = sˆψ, where s ∈ Aut Π and ψ = exp(ad π2 it), the element t ∈ t θ1 = ψˆ satisfying sˆ(t) = t and α(t) ∈ 2Z, α ∈ ∆. (iii) Any involutive automorphism θ1 of the form described in (ii) commutes with the real structures σ, τ and the Weyl involution ω associated with Π (see §2).

§4. Automorphisms of complex semisimple Lie algebras

35

Proof. (i) First we show the existence of a maximal toral subalgebra t ⊂ g which is invariant under θ. By Proposition 3.7, there exists a θ-invariant compact form u of g. Since θ is involutive, we have the eigenspace decomposition u = u+ ⊕ u− , where u± corresponds to the eigenvalue ±1 (see (I.14)). Let a denote a maximal commutative subalgebra of u+ . Then the centralizer t = zg (a) = {x ∈ g | [x, a] = 0} is the desired maximal toral subalgebra. Let us denote t0 = t ∩ u = zu (a) . It suffices to show that t0 is commutative. In fact, it is easy to verify that t = t0 (C). If t0 is commutative, then, clearly, it is a maximal commutative subalgebra of u, and due to Remark 1, t is a maximal toral subalgebra of g. Obviously, θ(t) = t. By (I.11), t0 = z ⊕ s, where z and s are respectively the centre and the commutator subalgebra of t0 . Clearly, θ(t0 ) = t0 . Due to maximality of a, this implies that the corresponding eigenspace decomposition has the form t0 = a ⊕ b, where b ⊂ u− and a ⊂ z. Since this is a Z2 -grading (see I.14), we have s = [b, b] ⊂ a. But s is semisimple, and hence s = 0. Thus, t0 is commutative, and t is a θ-invariant maximal toral subalgebra. By Remark 1, t(R) = it0 . Next, we want to prove that θ leaves invariant a Weyl chamber in t(R). To do this, we note that ia contains a regular element. In fact, let ∆ denote the system of roots relative to t. If all the elements of ia are singular, then there exists α ∈ ∆ such that α(a) = 0. Hence, eα , e−α ∈ zg (a) = t, which contradicts to the fact that t is commutative (see (II.11)). Thus, there exists a Weyl chamber D ⊂ t(R) such that D ∩ ia = ∅. Further, any element of this intersection is invariant under θ. It follows that θ(D) = D, and hence the corresponding system of simple roots Π ⊂ ∆ satisfies θ (Π) = Π. Thus, θ ∈ Aut(g, t, Π). (ii) Applying Proposition 1, we see that θ = sˆϕ, where s ∈ Aut Π, s2 = id, and ϕ = exp(ad x), where x ∈ t. We also have θ|t = sˆ|t. Consider the corresponding eigenspace decomposition t = t+ ⊕ t− . Then x ∈ t decomposes as x = 12 (x + sˆ(x)) + 12 (x − sˆ(x)). Denoting u = 12 (x + sˆ(x)) ∈ t+ and v = − 12 x, we can write x = u + sˆ(v) − v. Then ad x = ad u + ad sˆ(v) + ad(−v) = ad u + sˆ(ad v)ˆ s + ad(−v), and so ϕ = exp(ad u)ˆ s exp(ad v)ˆ s(exp(ad v))−1 . Denoting ψ = exp(ad u), we have sˆψˆ s = exp(ad(ˆ s(u))) = ψ. Therefore, θ = exp(ad v)(ψˆ s)(exp(ad v))−1 is conjugate to θ1 = ψˆ s = sˆψ. Since ψ and sˆ commute, we have ψ 2 = e. It follows that ψ 2 (eα ) = exp(ad 2u)(eα ) = eα(2u) eα = eα for all α ∈ ∆, whence α(2u) ∈ 2πiZ, α ∈ ∆. Denoting u = π2 it, we get α(t) ∈ 2Z, α ∈ ∆, and θ1 is of the desired form. (iii) Since σ, τ, ω commute with sˆ, we only have to prove that they commute with ψ. Clearly, τ u = u, whence τ ψτ = ψ. Now, σu = −u, whence σψσ = ψ −1 = ψ. Finally, ωψ = ψω, since ω = στ . 

§5. Cartan decompositions and maximal compact subgroups Here we return to the correspondence between involutions (involutive automorphisms) and real forms of a complex semisimple Lie algebra established in §3. Let a complex semisimple Lie algebra g and a compact real structure τ in g be given, and let u = gτ denote the corresponding compact real form of g. Then any involution θ ∈ Aut g commuting with τ determines a real structure σ = τ θ. It is easy to understand, how to get the real form g0 = gσ directly from u and θ. The involution θ determines the eigenspace decomposition g = g+ ⊕ g− . Since θ commutes with τ and σ, both real forms u and g0 are stable under θ. Thus, we have the eigenspace decompositions g0 = (g0 )+ ⊕ (g0 )− , where (g0 )± = g± ∩ g0 , u = u+ ⊕ u− , where u± = g± ∩ u .

(1)

Let us denote k = (g0 )+ ,

p = (g0 )− .

Since θ = τ σ coincides with τ on g0 , we have k = u+ , p = iu− . Thus, the decompositions (1) have the form g0 = k ⊕ p ,

(2)

u = k ⊕ ip .

(3)

These decompositions are Z2 -gradings (see (I.14)), i.e., [k, k] ⊂ k ,

[k, p] ⊂ p ,

[p, p] ⊂ k .

(4)

Denote by k the Killing form kg and its restriction to g0 which coincides with kg0 by Proposition 2.2, and consider the real bilinear form bθ (x, y) = −k(x, θy) ,

x, y ∈ g0 .

(5)

Clearly, bθ is the restriction of the positive definite Hermitian form hτ and hence is a scalar product (a positive definite symmetric bilinear form). It follows that k(x, x) < 0

for x ∈ k , x = 0 ;

k(y, y) > 0

for y ∈ p , y = 0 .

(6)

Now let us start with a real semisimple Lie algebra g0 . A direct sum decomposition (7) g0 = k ⊕ p is called a Cartan decomposition if it is a Z2 -grading and if the Killing form k = kg0 satisfies (6). As we have seen above, the decomposition (2) of g0 constructed with the help of a compact real form u of the complexification g = g0 (C), compatible with g0 , is a Cartan decomposition. We will prove that any Cartan decomposition of g0 can by obtained in this way.

§5. Cartan decompositions and maximal compact subgroups

37

Suppose that a Cartan decomposition (7) is given. By (I.14), the involutive transformation θ of g0 defined by θ(x + y) = x − y ,

x ∈ k, y ∈ p.

(8)

is an automorphism of g0 . Extend θ to an (involutive) automorphism of g = g0 (C) denoted by the same symbol. Clearly, θσ = σθ, where σ is the complex conjugation in g relative g0 . Therefore τ = θσ is an antiinvolution commuting with σ. We claim that it is compact. In fact, for any x ∈ k, y ∈ p we have hτ (x + y, x + y) = −k(x + y, τ (x + y)) = −k(x + y, θ(x + y)) = −k(x + y, x − y) = −k(x, x) + k(y, y) . Due to (6), this value is positive whenever x+ y = 0. Since hτ is a Hermitian form, it follows that it is positive definite on g = g0 (C). Further, if τ (x + y) = x + y for x ∈ k(C), y ∈ p(C), then σ(x − y) = x + y, whence x ∈ k, y ∈ ip. Thus, u = gτ is described by (3), and therefore k = u+ , p = iu− . Theorem 1. Each Cartan decomposition of a real semisimple Lie algebra g0 has the form (2), where k = u+ , p = iu− , for a compact real form u of g0 (C), compatible with g0 . Any two Cartan decompositions of g0 are conjugate by an inner automorphism of g0 . Proof. The first assertion has been already proved. Suppose that two Cartan decompositions of g0 are given, and let τ and τ1 be the compact real structures on g = g0 (C), commuting with the complex conjugation σ relative to g0 and determining our decompositions. By Corollary of Proposition 3.6, there exists ψ ∈ Int g such that τ1 = ψτ ψ −1 and ψσ = σψ. Then ψ leaves g0 invariant and maps gτ onto gτ1 . It follows that ψ transforms the first Cartan decomposition 1 into the second one. Actually, this automorphism is expressed as ψ = ϕ 4 , where t {ϕ | t ∈ R} is a 1-parameter subgroup of Int g consisting of positive definite symmetric operators with respect to hτ . We have ϕt = exp(t ad z), where z ∈ g, and the condition ψσ = σψ implies σ(ad z)σ = ad(σ(z)) = ad z. Thus, z ∈ g0 , and ψ|g0 is the inner automorphism exp( 14 ad z) of g0 .  Example 1. Example 4.1 allows to write down the Cartan decompositions g0 = k⊕p for all real forms g0 of sln (C). (i) For g0 = sln (R): k = {X ∈ sln (R) | X  = −X} = son ,

p = {X ∈ sln (R) | X  = X} .

(ii) For g0 = slm (H), n = 2m: ¯  = −X} = spm , k = {X ∈ slm (H) | X

¯  = X} . p = {X ∈ slm (H) | X

(iii) For g0 = sup,q :    X 0 k= | X ∈ up , Y ∈ uq , tr X + tr Y = 0 , 0 Y    0 Z p= | Z is a (p × q)-matrix . Z¯  0

38

§5. Cartan decompositions and maximal compact subgroups

Example 2. Let g be a complex semisimple Lie algebra. Let us find a Cartan decomposition of the real Lie algebra gR . Consider gR as a real form of g ⊕ g (see Examples 2.4 and 3.4). Then one should take the compact real structure τ × τ on g ⊕ g, where τ is a compact real structure on g, and the involution θ : (x, y) → (y, x), x, y ∈ g. Clearly, we get k = {(x, x) | x ∈ u} ,

p = {(ix, −ix) | x ∈ u} .

The projection (x, τ (x)) → x maps k onto u and p onto iu, and thus we get the Cartan decomposition in the form gR = u ⊕ iu .

(9)

Let us prove some properties of the Cartan decompositions which will be useful in what follows. We remind that a Cartan decomposition of a real semisimple Lie algebra g0 gives rise to the scalar product (5). We regard g0 as the euclidean vector space with this scalar product. Proposition 1. (i) θαθ = (α∗ )−1 for any α ∈ Aut g0 . In particular, θ∗ = θ. (ii) ad θ(x) = −(ad x)∗ for any x ∈ g0 . Proof. (α−1 x, y) = −k(α−1 x, θy)

(i)

= −k(x, αθy) = (x, θαθy) for any x, y ∈ g0 . Applying this to α = θ, we see that θ∗ = θ. (ii)

((ad x)y, z) = −k((ad x)y, θz) = k(y, (ad x)θz) = −(y, θ(ad x)θz) = −(y, (ad θ(x))z)

for any x, y, z ∈ g0 .



Now we go over to the multiplicative Cartan decompositions of real semisimple Lie groups. For simplicity, we will consider only the automorphism groups of real semisimple Lie algebras, referring to [11, 19] for the general case. The basic fact here is the well-known polar decomposition theorem for linear operators in an euclidean vector space E. As in §3, we denote by S(E) and P(E) respectively the vector space of symmetric linear operators and its subset of positive definite operators. We remind that we have the bijective mapping exp : S(E) → P(E) and denote by log its inverse. Let also O(E) denote the orthogonal group of E. Lemma 1. The mapping exp : S(E) → P(E) is real bianalytic, and thus log is an analytic mapping. Proof. Since exp is analytic, it suffices to prove that dα0 exp : S(E) → S(E) is an isomorphism for any α0 ∈ S(E). Take α ∈ S(E) such that (dα0 exp)α = 0.

§5. Cartan decompositions and maximal compact subgroups

39

Denoting β(t) = α0 + tα and γ(t) = exp β(t), we have β(t)γ(t) = γ(t)β(t) for all t ∈ R. Differentiating this relation at t = 0, we get α exp α0 + α0 γ
(0) = γ
(0)α0 + (exp α0 )α. Since γ
(0) = (dα0 exp)α = 0, this implies that α commutes with exp α0 , and hence with α0 . Therefore γ(t) = (exp α0 )(exp tα), whence 0 = γ
(0) = (exp α0 )α, and so α = 0.  Proposition 2. Let E be an euclidean vector space. Then the mapping µ : O(E) × P(E) → GL(E) given by µ(k, p) = kp and the mapping µ ◦ (id × exp) : O(E) × S(E) → GL(E) are bianalytic. Proof. By Lemma 1, exp : S(E) → P(E) is bianalytic. We repeat the well-known argument from linear algebra. Take u ∈ O(E) and p = exp s ∈ P(E), where s ∈ S(E), and denote a = up = u exp s. Then a∗ = pu−1 , whence p2 = exp 2s = a∗ a. It follows that 1 1 1 (10) s = log(a∗ a), p = exp( log(a∗ a)) , u = a exp(− log(a∗ a)) . 2 2 2 Thus, p, s and u are expressed uniquely as analytic functions in a.  Let now g0 be a real semisimple Lie algebra, endowed with a Cartan decomposition (7). Consider the involutive automorphism θ of g0 given by (8) and the bilinear form (5), where k is the Killing form of g0 . We regard g0 as the euclidean vector space with the scalar product (x, y). Let us denote K = Aut g0 ∩ O(g0 ) ,

P = Aut g0 ∩ P(g0 ) .

Clearly, K is a compact linear Lie group. By Proposition 1, K = {α ∈ Aut g0 | θαθ = α}, and ad k ⊂ ad g0 is the Lie subalgebra corresponding to K. The multiplication mapping µ from Proposition 2 sends K × P to Aut g0 . Now, Proposition 1 (ii), implies that ad z ∈ S(g0 ), and hence exp(ad z) ∈ P for any z ∈ p. Theorem 2. The mappings µ : K × P → Aut g0 and µ(id × exp ad) : K × p → Aut g0 are bianalytic and induce bianalytic maps K ◦ × P → Int g0 and K ◦ × p → Int g0 . We have K ◦ = K ∩ Int g0 = {α ∈ Int g0 | θαθ = α} . Proof. To prove our claim concerning Aut g0 , it suffices to verify that for any a ∈ Aut g0 the operators s ∈ S(g0 ) and p ∈ P(g0 ) expressed by (10) belong to ad p and Aut g0 , respectively. By Proposition 1 (i), we see that a∗ ∈ Aut g0 , and therefore p2 ∈ Aut g0 . By Lemma 3.1, pt ∈ Int g0 . Hence s = log p = ad z for a certain z ∈ g0 . Since (ad z)∗ = ad z, we see from Proposition 1 (ii) that θ(z) = z, i.e., z ∈ p. Since the mapping µ(id × exp ad) : K × p → Aut g0 is a homeomorphism, it should map the connected components of K × p onto those of Aut g0 . It follows that K ◦ × p is mapped onto Int g0 . Hence µ(K ◦ × P ) = Int g0 . The decomposition Int g0 = K ◦ P implies the last assertion of the theorem. 

40

§5. Cartan decompositions and maximal compact subgroups

Corollary. The subgroups K and K ◦ are maximal compact subgroups of the Lie groups Aut g0 and Int g0 , respectively. Proof. Suppose that we have a linear group L such that K  L ⊂ Aut g0 . By Theorem 2, L = K(L ∩ P ), where L ∩ P = {e}. Take p ∈ L ∩ P, p = e. Then the sequence (p, p2 , . . . ) in L has no limit points, due to Lemma 1, since the sequence (log p, 2 log p, . . . ) is unbounded. Therefore L is non-compact. The same argument  applies to the subgroup K ◦ of Int g0 . Our next aim is to prove that any two maximal compact subgroups of the group Aut g0 or Int g0 are conjugate. Note that Theorem 1 implies the following weaker assertion: any two maximal compact subgroups of Int g0 corresponding to Cartan decompositions are conjugate in Int g0 . But it is not clear, whether each maximal compact subgroup of Int g0 corresponds to a Cartan decomposition. Theorem 3. For any compact subgroup L of Aut g0 or Int g0 , there exists q ∈ P such that qLq −1 ⊂ K. Any two maximal compact subgroups in Aut g0 or Int g0 are conjugate by an element of Int g0 . The idea of the proof is as follows. We define an analytic action α → Tα of the linear Lie group G = Aut g0 on the manifold P given by Tα (p) = αpα∗ ,

α ∈ G,

p∈P.

(11)

Note that αpα∗ ∈ P = G ∩ P(g0 ). In fact, α∗ ∈ G by Proposition 1 (i), (αpα∗ )∗ = αpα∗ , and (αpα∗ (x), x) = (p(α∗ (x)), α∗ (x)) > 0 ,

x ∈ g0 \ {0} ,

since p is positive definite. Also Tαβ = Tα Tβ , Te = id. This action is transitive. In 1 1 fact, any p ∈ P may be written as p = p 2 ep 2 = T 12 (e). The stabilizer of the point p e ∈ P is K, and hence the homogeneous space P of the group G can be identified with G/K (or with G◦ /K ◦ , where the identity component G◦ = Int g0 ). The crucial fact is that any compact subgroup L ⊂ G leaves invariant a point q ∈ P . If −1 it is proved, then one sees easily that T −1 1 LT 1 (e) = e, and so T 1 LT 1 ⊂ K. In 2 2 q2

q

q2

q

the original Cartan’s proof of the conjugacy theorem (see, e.g., [11]), the existence of a fixed point was shown by a nice geometrical argument, using the G-invariant Riemannian metric on the symmetric space P = G/K. The proof which follows (see [19]) is quite elementary. As suggested above, we have to find a point in P which is fixed under the subgroup L by the action (11). This point will be constructed as a minimum point of an L-invariant function on P . This function possesses a certain convexity property which implies the uniqueness of its minimum point. Thus, it should be fixed under L. Let us consider the more general situation, when an arbitrary euclidean vector space E of dimension n is given. We define the following analytic function on P(E) × P(E): (12) r(β, γ) = tr(βγ −1 ) , β, γ ∈ P(E) .

§5. Cartan decompositions and maximal compact subgroups

41

Further, for any compact subset Ω ⊂ P(E), we define the function ρΩ (β) = max r(β, γ) , γ∈Ω

β ∈ P(E) .

(13)

One proves easily that it is continuous. Let us give an explicit expression of r. For a fixed β ∈ P(E), choose an orthonormal basis v1 , . . . , vn of E such that β(vi ) = λi vi , i = 1, . . . , n. For any γ ∈ P(E), we have γ −1 ∈ P(E); let γ −1 be expressed by the matrix (gij ) relative to our basis. Then λi > 0, gii > 0, i = 1, . . . , n, and we have r(β, γ) =

n 

λi gii ,

γ ∈ P(E) .

(14)

i=1

We use this formula for establishing some properties of functions r and ρΩ . Let β denote the usual norm of a linear operator β. For β ∈ P(E), we have β = maxi λi , where λi , i = 1, . . . , n, are the eigenvalues of β. Lemma 2. Let us fix a compact subset Ω ⊂ P(E). Then there exists b > 0 such that (15) ρΩ (β) ≥ bβ , β ∈ P(E) . Proof. We fix β ∈ P(E) and use the expression (14). Since the set of all orthonormal bases (i.e., O(E)) and Ω are compact, there exists a constant b > 0 such that gii ≥ b, i = 1, . . . , n, for all γ ∈ Ω and all orthonormal bases of E. Then r(β, γ) ≥ b

n 

λi ≥ bβ ,

β ∈ P(E) ,

γ ∈ Ω.

i=1



Clearly, this implies (15).

Lemma 3. For any compact subset Ω ⊂ P(E), the function ρΩ attains its minimum on each subset F ⊂ P(E) which is closed in S(E). Proof. Take any β0 ∈ F . Then the set B = {β ∈ F | ρΩ (β) ≤ ρΩ (β0 )} is compact. In fact, B is closed in S(E), since ρΩ is continuous, and is contained in the ball β ≤ 1b ρΩ (β0 ), due to Lemma 2. Let β1 ∈ B be a minimum point of ρΩ on B. Since β0 ∈ B, we have ρΩ (β1 ) ≤ ρΩ (β0 ) < ρΩ (β) for all β ∈ F \ B. Thus, β1 is the desired minimum point.  Now we recall the following classic definition. A function f (t), t ∈ R, is called strictly convex if for any a, b ∈ R, a < b, the graph of f between a and b lies under the line segment linking the point (a, f (a)) with (b, f (b)), i.e., if f (t) < f (a) +

b−t t−a f (b) − f (a) (t − a) = f (a) + f (b) , b−a b−a b−a

a < t < b.

(16)

It is well known that a smooth function f satisfying f

(t) > 0 for all t ∈ R is strictly convex.

42

§5. Cartan decompositions and maximal compact subgroups

Lemma 4. Let F (t, s) be a continuous function on R × Ω, where Ω is a compact space, and define f (t) = maxs∈Ω F (t, s), t ∈ R. If F (t, s) is a strictly convex function of t for any fixed s ∈ Ω, then f is strictly convex. Proof. For any t ∈ R, choose a point s(t) ∈ Ω such that F (t, s(t)) ≥ F (t, s), s ∈ Ω. Then for a < t < b we get, using (16), f (t) = F (t, s(t)) < F (a, s(t))

t−a b−t t−a b−t + F (b, s(t)) ≤ f (a) + f (b) . b−a b−a b−a b−a 

We need this lemma to establish the following important fact. Lemma 5. Choose an β ∈ P(E) and a compact Ω ⊂ P(E). Then the function f (t) = ρΩ (β t ), t ∈ R, is strictly convex. Proof. Using an appropriate orthonormal basis of E, we can write, due to (14), r(β t , γ) =

n 

gii λti =

i=1

n 

gii et log λi ,

i=1

where λi > 0 are the eigenvalues of β. Since gii > 0, the analytic function t → r(β t , γ) is strictly convex for any fixed γ ∈ P(E). By Lemma 4, f (t) =  maxγ∈Ω r(β t , γ) is strictly convex, too. Proof of Theorem 3. We apply the above considerations to the euclidean space g0 with the scalar product (5). The set P = exp(ad p) is closed in Int g0 due to Theorem 2. Note that tr ad x = 0 for any x ∈ g0 , since ad g0 g0 coincides with its commutator subalgebra. Thus, ad g0 ⊂ sl(g0 ). Using this inclusion, one deduces that Int g0 ⊂ SL(g0 ) is closed in gl(g0 ). It follows that P is closed in S(g0 ). Now we consider the function r on P × P which is the restriction of (12) and the function ρΩ on P given by (13), where Ω is a compact subset of P . Applying Lemma 3 to the subset F = P , we see that there exists β0 ∈ P such that ρΩ (β0 ) ≤ ρΩ (β) for all β ∈ P . We want to prove that β0 is the only minimum point of ρΩ . One sees immediately from (11) that r is invariant under the action α → Tα of G = Aut g0 on P , i.e., r(Tα (β), Tα (γ)) = r(β, γ) ,

α ∈ G.

ρΩ (Tα (β)) = ρTα−1 (Ω) (β) ,

α ∈ G.

It follows that (17)

Since the action is transitive, we may assume that β0 = e. Suppose that β ∈ P is another minimum point of ρΩ . Consider the function f (t) = ρΩ (β t ) ,

t ∈ R.

§5. Cartan decompositions and maximal compact subgroups

43

Clearly, 0 and 1 are two minimum points of f . But f is strictly convex, due to Lemma 5, and hence f (t) < f (0) = f (1) for 0 < t < 1. This gives a contradiction. Now, take as Ω the orbit L(e) = {αα∗ | α ∈ L} of L under the action (11). This is an L-invariant compact subset of P . By (17), the corresponding function ρΩ is L-invariant. Thus, L(β0 ) = β0 , and the existence of the point fixed under L is proved. As we saw above, this implies that qLq −1 ⊂ K for a certain q ∈ P ⊂ G◦ = Int g0 . Now, if L is a maximal compact subgroup of G, then qLq −1 is maximal compact as well, and hence qLq −1 = K. If L is a maximal compact subgroup of  G◦ , then qLq −1 ⊂ K ∩ G◦ = K ◦ , whence qLq −1 = K ◦ . Example 3. Let g be a complex semisimple Lie algebra. By Example 2, any Cartan decomposition of the real Lie algebra gR has the form (9), where u is a compact real form of g. Consider the Lie group G = Int g = Int gR . Theorem 2 implies that the connected Lie subgroup U ⊂ G corresponding to the subalgebra ad u ⊂ ad gR is a maximal compact subgroup of G. Applying Theorem 3, we see that any maximal compact subgroup of G corresponds to a compact real form of g. To finish, we note that the following generalization of Theorem 3 is true: any two maximal compact subgroups of a Lie group G with a finite number of connected components are conjugate by an element of G◦ (see, e.g., [9]).

§6. Homomorphisms and involutions of complex semisimple Lie algebras Here we start to study the behavior of real forms of complex semisimple Lie algebras under homomorphisms of these algebras. More precisely, suppose a homomorphism f : g → h of complex semisimple Lie algebras and a real form g0 of g be fixed. We would like to know, for which real forms h0 of h the inclusion f (g0 ) ⊂ h0 holds. We will prove a theorem due to Karpelevich [15], giving an answer in terms of the involutive automorphisms corresponding to the given real forms. First we will make some preliminary remarks. Let a homomorphism f : g → h of arbitrary complex Lie algebras be given.  Let σ and σ
be real structures in g and h, respectively. Then f (gσ ) ⊂ hσ if and
only if f σ = σ f . In fact, if the latter relation holds, then we get immediately  f (x) = σ
(f (x)) for any x ∈ gσ . Conversely, suppose that f (gσ ) ⊂ hσ . Then f σ
σ and σ f coincide on the real form g . Since both mappings are antilinear, they should coincide on the entire vector space g. Generally, we say that a mapping a : g → g extends by f to a mapping a
: h → h, whenever f a = a
f . In this case, we write a ↑f a
. The subscript f may be omitted if it is clear which homomorphism is considered. We have proved above the following Proposition 1. Let f : g → h be a homomorphism of complex semisimple Lie algebras and let σ and σ
be real structures in g and h, respectively. Then f (gσ ) ⊂  hσ if and only if σ ↑f σ
.  Now we note certain simple properties of the extension relation, assuming that f : g → h is fixed. Proposition 2. (i) If a ↑ a
and b ↑ b
, then aa
↑ bb
. −1 (ii) If a ↑ a
and a and a
are invertible, then a−1 ↑ a
.

(iii) An operator a in h extends idg , i.e., idg ↑ a if and only if a
|f (g) = id. (iv) For any ϕ ∈ Int g, there exists ϕ
∈ Int h such that ϕ ↑ ϕ
. If ϕ = exp(ad x), where x ∈ g, then we may take ϕ
= exp(ad f (x)). (v) If idg ↑ ψ, where ψ ∈ P(h) relative to a compact real structure in h, then idg ↑ ψ t , t ∈ R. Proof. The assertions (i)–(iii) are trivial. Let us prove (iv). Let G and H be connected Lie groups with tangent Lie algebras g and h such that there exists a homomorphism F : G → H satisfying de F = f (it always exist, if G is simply connected). By (I.2), Int g = Ad G and Int h = Ad H. If ϕ = Ad g, where g ∈ G, then we may write F (αg (h)) = F (ghg −1 ) = F (g)F (h)F (g)−1 = αF (g) (F (h)), h ∈ G, i.e., F αg = αF (g) F . Differentiating this relation, we get f ϕ = (Ad F (g))f , and so we may set ϕ
= Ad F (g). In the case when ϕ = exp(ad x) = Ad(exp x), x ∈ g, we have ϕ
= Ad F (exp x) = exp(ad f (x)). To prove (v), we note that, by (iii), idg ↑ ψ means ψ|f (g) = id. Thus, f (g) lies in the eigenspace of ψ corresponding to the eigenvalue 1. Clearly, all ψ t , t ∈ R, have the only eigenvalue 1 in this space. Hence idg ↑ ψ t , t ∈ R. 

§6. Homomorphisms and involutions

45

Now we will consider a homomorphism f : g → h of complex semisimple Lie algebras. Let a real structure σ on g extend to a real structure σ
on h, i.e., assume  that f (gσ ) ⊂ hσ . Then, by Proposition 2 (iv), for any ϕ ∈ Int g there exists ϕ
∈  −1 Int h such that ϕ
σ
ϕ
↑f ϕσϕ−1 or, equivalently, f (ϕ(gσ )) ⊂ ϕ
(hσ ). So we may consider the extension relation between conjugacy classes of real structures, where the conjugacy by inner automorphisms is meant, and we see that it corresponds to the inclusion relation between conjugacy classes of real forms in g and h. Our next aim is to replace in this correspondence the conjugacy classes of real structures by the conjugacy classes of involutions, using Theorem 3.2. The crucial step is the following proposition giving a compact extension of any compact real structure. Proposition 3. Let f : g → h be a homomorphism of complex semisimple Lie algebras. (i) If a compact real structure τ on g is given, then there exists a compact real structure τ
on h such that τ ↑f τ
. (ii) If τ

is another compact real structure on h extending τ , then there exists β ∈ Int h such that τ

= βτ
β −1 and e ↑f β. Proof. (i) Let us denote by u the compact real form gτ of g. Consider the simply connected Lie group G with tangent Lie algebra g. There exists a homomorphism F : G → H = Int h such that de F = ad ◦f . The connected Lie subgroup U ⊂ G corresponding to the subalgebra u is compact (see (I.13)), and hence the subgroup F (U ) ⊂ H is compact as well. Therefore F (U ) is contained in a maximal compact subgroup V of H. By Example 5.3, any maximal compact subgroup of H corresponds to a compact real form v of h. Since F (U ) ⊂ V , we have ad f (u) ⊂ ad v, whence f (u) ⊂ v. By Proposition 1, τ extends to the compact structure τ
on h corresponding to v. (ii) Let us apply Corollary of Proposition 3.6 to the compact real structures τ
1 and τ

. It claims that τ

= βτ
β −1 , and β = ϕ 4 , where ϕ = (τ

τ
)2 . Clearly, 2  e = (τ τ ) ↑f ϕ. By Proposition 2 (v), this yields e ↑f β. Remark 1. If f = ρ is a linear representation, i.e., h = sl(W ), then for proving (i) you can use, instead of Theorem 5.3, Theorem of Weyl (I.9) claiming that there exists a Hermitian scalar product in W , invariant under a given compact linear group. The real form v ⊃ ρ(u) will consist of all skew-Hermitian operators, relative to a scalar product in W invariant under R(U ), with zero trace. Here R is the representation of the Lie group G such that de R = ρ. Let us fix a compact real structure τ in g and a compact real structure τ
in h such that τ ↑f τ
. Consider the correspondence between antiinvolutions and involutions in g defined in §3. It assigns to any real structure σ in g an involutive automorphism θ = σ1 τ ∈ Aut g, where σ1 is a real structure commuting with τ and having the form σ1 = ασα−1 , where α ∈ Int g. We will say that θ is an involution corresponding to σ (or to the real form gσ ), though θ is determined only up to conjugacy by inner automorphisms. As was shown in Theorem 3.2, this correspondence gives rise to a bijection between antiinvolutions and involutions. Using the compact real structure τ
, we can define a similar correspondence for h. We will now prove the following theorem (see [15]).

46

§6. Homomorphisms and involutions

Theorem 1. Let f : g → h be a homomorphism of complex semisimple Lie algebras, and let two real forms g0 ⊂ g and h0 ⊂ h be given. We also suppose that two compact real structures τ in g and τ
in h, such that τ ↑f τ
, are fixed. Let θ ∈ Aut g denote an involution corresponding to g0 . If f (g0 ) ⊂ h0 , then θ extends by f to an involution θ
∈ Aut h corresponding to h0 . Conversely, if θ extends to an involution θ
∈ Aut h corresponding to h0 , then f (g0 ) is contained in a real form of h which is conjugate to h0 by an inner automorphism. Proof. Let us denote by σ and σ
the real structures determining g0 and h0 , respectively. By Proposition 6.3, there exists such an α ∈ Int g that σ1 = ασα−1 commutes with τ . Then θ = σ1 τ is an involution corresponding to g0 . First suppose that f (g0 ) ⊂ h0 . Then σ ↑ σ
. By Proposition 2 (iv), α ↑ α
, where α ∈ Int h, and hence σ1 ↑ α
σ
α
−1 . Therefore we may assume that σ1 = σ, and θ = στ . Consider now ϕ = (σ
τ
)2 ∈ Aut h ∩ P(h) (see Lemma 3.2 (i)). 1 By Proposition 3.6, σ1
= βσ
β −1 , where β = ϕ− 4 , commutes with τ
, and then θ
= σ1
τ
is an involution corresponding to h0 . But by Proposition 2 (i) e = θ2 ↑ ϕ, and, by Proposition 2 (v), this yields e ↑ β. Using Proposition 2 again, we see that σ ↑ σ1
, and hence θ ↑ θ
. Conversely, suppose that θ extends to an involution θ
corresponding to h0 . Consider ψ = (θ
τ
)2 ∈ Aut h ∩ P(h) (see Lemma 3.2 (ii)). By Proposition 3.7, 1 θ1
= γθ
γ −1 , where γ = ψ − 4 , commutes with θ
. By Proposition 2 (i), e = σ 2 ↑ ψ, which implies that e ↑ γ. It follows that θ ↑ θ1
, and hence σ1 ↑ σ1
= θ1
τ
. By Proposition 2 (iv), α ↑ α
, where α ∈ Int h. Therefore σ ↑ α
−1 σ1
α
,  −1 and hence f (g0 ) ⊂ α
(hσ1 ). On the other hand, θ
corresponds to both real  structures σ
and σ1
, and Theorem 3.2 implies that hσ1 is conjugate to h0 by an inner automorphism.  Corollary 1. Let f : g0 → h0 be a homomorphism of real semisimple Lie algebras and let a Cartan decomposition g0 = k ⊕ p be given. Then there exists a Cartan decomposition h0 = k
⊕ p
such that f (k) ⊂ k
and f (p) ⊂ p
. Proof. Consider the complexification of our homomorphism f (C) : g → h, where g = g0 (C), h = h0 (C), and denote by σ, σ
the corresponding complex conjugations in g and h, respectively. By Theorem 5.1, the given decomposition of g0 is the eigenspace decomposition for an involution θ of g, commuting with σ and restricted to g0 , and τ = σθ is a compact real structure. By Proposition 3 (i), we have τ ↑f (C) τ
for a certain compact real structure τ
in h. Applying the first assertion of Theorem 1, we get an involution θ
and an automorphism β ∈ Int h such that θ
= (βσ
β −1 )τ
, e ↑f (C) β, θ ↑f (C) θ
and τ ↑f (C) τ
. Then we have τ ↑f (C) τ1
= β −1 τ
β and θ ↑f (C) θ1
= β −1 θ
β. The desired Cartan decomposition of h0 may be constructed as the eigenspace decomposition for θ1
|h0 .  Corollary 1 (often referred to as the canonical embedding property) was first proved in [14] using methods of Riemannian geometry and is equivalent to the following fact. Let G0 be a connected semisimple Lie subgroup of a connected semisimple real Lie group H0 and let K
denote the connected Lie subgroup of H0 corresponding to the compact part k
of a Cartan decomposition of its Lie algebra

§6. Homomorphisms and involutions

47

h0 = k
⊕ p
. Then G0 has a totally geodesic orbit in the Riemannian symmetric space H0 /K
. It was also proved independently by Mostow [18]. The above results were specified in [15] in an important case, when f is given by an irreducible linear representation of g. This means that h ⊂ gl(V ) is a linear complex Lie algebra and f : g → h ⊂ gl(V ) is an irreducible linear representation of g in the complex vector space V . A natural generalization of this situation gives the following definition. We say that a homomorphism f : g → h of complex Lie algebras is an S-homomorphism if for any ϕ ∈ Int h which satisfies e ↑f ϕ (or, which is the same, is identical on f (g)), we have ϕ = e. Lemma 1. Suppose that f : g → h ⊂ gl(V ) is an irreducible linear representation of g in the complex vector space V . Then f is an S-homomorphism. Conversely, if a linear representation f : g → gl(V ) of a semisimple complex Lie algebra g is an S-homomorphism into gl(V ) or sl(V ), then f is irreducible. Proof. Suppose that an inner automorphism ϕ ∈ Int h is identical on f (g). We have ϕ(X) = (Ad C)X = CXC −1 , X ∈ h, for an element C of the connected Lie subgroup H ⊂ GL(V ) corresponding to h. By our assumption, CXC −1 = X for any X ∈ g. Since f is irreducible, C = λe, where λ ∈ C× , by the Schur Lemma. Hence ϕ = e. If g is semisimple and a linear representation f : g → gl(V ) is reducible, then f leaves invariant a non-trivial direct sum decomposition V = V1 ⊕ V2 . Constructing the linear operator a in V such that a(v) = v, v ∈ V1 and a(v) = −v, v ∈ V2 , we get a non-trivial ϕ = Ad a ∈ Int gl(V ) such that ϕ|f (g) = id. Clearly, there exists c ∈ C such that ca ∈ SL(V ), and ϕ = Ad(ca).  Proposition 4. Let f : g → h be an S-homomorphism of complex semisimple Lie algebras, and let τ be a compact real structure on g. Then there exists a unique compact real structure τ
on h such that τ ↑ τ
. If a real structure σ
on h extends a real structure σ on g satisfying στ = τ σ, then σ
τ
= τ
σ
, and the involution θ
= σ
τ
extends the involution θ = στ . Proof. By Proposition 3, there exists a compact real structure τ
on h which extends τ . By the same proposition, any compact real structure τ

with the same property has the form τ

= ϕτ
ϕ−1 , where ϕ ∈ Int h extends e. In our case ϕ = e, and hence τ

= τ
. Suppose now that a real structure σ
extends a real structure σ on g such that στ = τ σ. An argument from the proof of Theorem 1 shows that σ
commutes with a compact real structure τ

which extends τ . But τ
is the only compact real structure with this property, and hence τ

= τ
.  This proposition implies, in particular, that if an S-homomorphism is given, then for any compact real form u of g there exists a unique compact real form v of h such that f (u) ⊂ v. We give now a strong generalization of this fact. Let g be a complex semisimple Lie algebra and θ1 , θ2 ∈ Aut g. If θ1 , θ2 are conjugate by an inner automorphism ϕ ∈ Int g, then they belong to the same coset of the group Aut g modulo Int g. In fact, we have θ2 = ϕθ1 ϕ−1 = θ1 (θ1−1 ϕθ1 )ϕ−1 ∈ θ1 Int g. We will say that two real forms (or real structures) of g are of the same kind if the corresponding involutions belong to the same coset modulo Int g

48

§6. Homomorphisms and involutions

or, equivalently, determine the same automorphism of the Dynkin diagram of g (see §4). The above remark shows that this definition is correct, and any two real forms conjugate by an inner automorphism are of the same kind. The real forms corresponding to inner involutions will be said to be of the first kind . If Aut g/ Int g Z2 , then the real forms corresponding to outer involutions will be said to be of the second kind . Thus, any real form of a simple complex Lie algebra g, except of g so8 , is of the first or of the second kind. The following theorem was proved in [15] in the case, when g is simple, non-isomorphic to so8 , and f is given by an irreducible representation. Theorem 2. Suppose that f : g → h is an S-homomorphism. Let g0 be a real form of g. Assume that we have two real forms h0 , h1 of h such that f (g0 ) ⊂ h0 and f (g0 ) ⊂ h1 . If h0 and h1 are of the same kind, then h0 = h1 . This is true, in particular, if h0 and h1 are conjugate by an inner automorphism of h. Proof. Let σ, σ1 and σ2 denote the real structures determining g0 , h1 and h2 , respectively. Then σ ↑ σ1 and σ ↑ σ2 . The involution θ corresponding to g0 is defined by θ = στ , where τ is a compact real structure on g such that στ = τ σ. By the proof of Theorem 1, the involutions θi which extend θ and correspond to hi , i = 1, 2, are given by θ1 = σ1 τ
and θ2 = σ2 τ
, where τ
is the only compact structure on h which satisfies τ ↑ τ
(see Proposition 4). By Proposition 2 (i), e = θ2 ↑ θ1 θ2 . But θ1 θ2 ∈ Int h, since θ1 and θ2 belong to the same coset modulo Int h. Therefore θ1 θ2 = e, and θ1 = θ2 , whence σ1 = σ2 .  The name “S-homomorphism” is suggested by the notion of S-subalgebra introduced by Dynkin in [6]. A subalgebra f of a complex semisimple Lie algebra h is called regular if f is normalized by a maximal toral subalgebra of h. In particular, any subalgebra of maximal rank (i.e., containing a maximal toral subalgebra of h) is regular. A subalgebra f ⊂ h is called an R-subalgebra if it is contained in a proper regular subalgebra, and an S-subalgebra otherwise. Proposition 5. A homomorphism f : g → h of complex semisimple Lie algebras is an S-homomorphism if and only if f (S) is an S-subalgebra in h, i.e., is not contained in any proper regular subalgebra. Proof. First we prove that for any semisimple R-subalgebra f there exists a nontrivial ϕ ∈ Int h such that ϕ|f = id. If f is an R-subalgebra, then f ⊂ ˜f, where ˜f is a maximal regular subalgebra. It follows (see [6]) that ˜f is either a maximal parabolic or a maximal semisimple subalgebra of maximal rank. In the first case, f lies in a Levi subalgebra of ˜f, and hence its centralizer in h is non-trivial. In the second case, one proves that ˜f = hϕ for a non-trivial ϕ ∈ Int h of a finite order (see [9], Ch. 6). Conversely, suppose that for a semisimple subalgebra f ⊂ h there exists a nontrivial ϕ ∈ Int h such that ϕ|f = id. Then f ⊂ hϕ  h. If the automorphism ϕ is semisimple, then ϕ = exp(ad h), where h = 0 lies in a maximal toral subalgebra t of h. Then t ⊂ hϕ , and hence hϕ is regular, and f is an R-subalgebra. In the general case, the Jordan decomposition in the algebraic group Aut h gives ϕ = ϕs exp ad y, where ϕs is a semisimple inner automorphism, and y ∈ hϕ is a nilpotent element of h. If ϕs = e, then hϕ ⊂ hϕs  h, and by the above argument f is an R-subalgebra.

§6. Homomorphisms and involutions

49

If ϕs = e, then hϕ = zh (y) is the centralizer of a nilpotent element y = 0. By Morozov’s theorem, we can include y into a sl2 -triple {e0 = y, h0 , f0 } spanning a simple three-dimensional subalgebra s ⊂ h. It is known (see [9], §6.2) that the centralizer zh (s) is a maximal reductive subalgebra of zh (y). By Maltsev’s theorem, we may assume that f ⊂ zh (s). Then f ⊂ zh (h0 ). But h0 is semisimple, and, by the above argument, zh (h0 ) is regular. Thus, f is an R-subalgebra.  Example 1. Suppose that h ⊂ gl(V ) is a semisimple complex linear Lie algebra. If a subalgebra f ⊂ h is irreducible, then it is an S-subalgebra of h, see Lemma 1. By the same lemma, the converse is true if h = sl(V ). One also can prove that any semisimple S-subalgebra of so(V ), dim V odd, or of sp(V ), is irreducible. But this is false for h = so(V ), dim V even. For example, the subalgebra  so2r+1  (C) ⊕ X 0 so2s+1 (C) ⊂ h = so2(r+s+1) (C), r, s ≥ 1, consisting of matrices , is an 0 Y S-subalgebra. This gives an example of an S-homomorphism into h = so2m (C) which is a reducible linear representation.

§7. Inclusions between real forms under an irreducible representation In this section, the following problem will be studied. Let ρ : g → sl(V ) be a linear representation of a complex semisimple Lie algebra g in a complex vector space V of dimension n, and let an involution θ ∈ Aut g be given. We want to know, to which involutions θ
∈ Aut sl(V ) does θ extend by ρ. As Theorem 6.1 shows, this problem is equivalent to that of finding inclusions between real forms of complex Lie algebras g and sl(V ) sln (C) determined by ρ. We mainly will consider the case, when ρ is irreducible. Then, by Theorem 6.2, the extending involution θ
is unique in its coset modulo inner automorphisms. Let us choose a maximal toral subalgebra t of a given complex semisimple Lie algebra g and a Weyl chamber D ∈ t∗ (R). As usually, we denote by Π ⊂ ∆+ the corresponding subsets of simple and positive roots and by hi , ei , fi , i = 1, . . . , l, the corresponding canonical generators of g. Let also σ, ω and τ = σω be defined by (2.6), (2.8) and (2.9). For g = sln (C), we denote these (anti)involutions by σ0 , ω0 , τ0 , respectively, and define them as in Examples 2.2 and 2.3: ¯, σ0 (X) = X

ω0 (X) = −X  ,

¯ . τ0 (X) = −X

(1)

Irreducible representations of g will be described by their highest weights relative to t and D (see (III.9)). The following simple lemma will be useful. Lemma 1. Let ρ be an irreducible representation of a complex semisimple Lie algebra g. Then, for any s ∈ Aut Π, we have Φρˆs = s(Φρ ). The highest weight of the representation ρˆ s is s(Λ), where Λ is the highest weight of ρ. Proof. The first assertion follows immediately from (III.8), since sˆ = s−1 (see s corresponding §4). Moreover, any weight vector vλ of ρ is a weight vector of ρˆ to the weight s(λ) ∈ Φρˆs . In particular, consider the highest vector vΛ of ρ. Then s. Thus, s(Λ) is the highest weight of (4.1) implies that vΛ is a highest vector of ρˆ ρˆ s.  In our study of the extension problem, we will also use the representation R : G → SLn (C) of the simply connected Lie group G with Lie algebra g, such that de R = ρ. Two inclusions between real forms are easy to establish. Due to Proposition 6.3, for any compact real form u of g there exists a compact real form u
of sl(V ) such that ρ(u) ⊂ u
. By Remark 6.1, u
is the subalgebra of all skew-Hermitian operators with zero trace with respect to a scalar product in V invariant under R(U ), where U is the connected Lie subgroup of G corresponding to u. Also, by (III.11), V admits a basis consisting of weight vectors such that the images of canonical generators ρ(hi ), ρ(ei ), ρ(fi ), i = 1, . . . , l, are expressed by real matrices (to get such a basis, you may choose any maximal linearly independent subset of the set of vectors vi1 ,... ,ik ). Then the corresponding matrix form ρ˜ of ρ satisfies ρ˜(g0 ) ⊂ sln (R), where g0 is the normal real form of g. Now we are going to present these facts in a more precise form. Clearly, we may replace a given representation ρ : g → sl(V ) by an equivalent one. Usually we will replace ρ by an equivalent representation in Cn , i.e., by its

§7. Inclusions between real forms

51

matrix form ρ˜ : g → sln (C) relative to a basis of V . If we change the basis, then ρ(x), x ∈ g, where the the new matrix form will be x → C ρ˜(x)C −1 = (Ad C)˜ matrix C ∈ GLn (C) is the transition matrix from the new basis to the old one. After choosing an appropriate basis in V , we will usually denote the matrix form of a representation by the same symbol ρ. Proposition 1. Let ρ : g → sl(V ) be a linear representation of a complex semisimple Lie algebra g. Then there exists a basis of V such that for the corresponding matrix form ρ : g → sln (C) the following conditions are satisfied:

ρ(x), x ∈ t ,

σ ↑ρ σ0 ;

(2)

τ ↑ρ τ0 ;

(3)

are diagonal matrices.

(4)

Under the conditions (2) and (3) we have ω ↑ ρ ω0 .

(5)

The conditions (2), (3) are equivalent, respectively, to the inclusions ρ(gσ ) ⊂ sln (R) ,

(6)

ρ(g ) ⊂ sun .

(7)

τ

Proof. First note that the equivalence of (2) and (6) and of (3) and (7) follows from Proposition 6.1. Clearly, (6) means that ρ(gσ ) leaves invariant the real span of the chosen basis, while (7) means that the basis is orthonormal with respect to a Hermitian scalar product invariant under R(U ). Finally, the condition (4) means that all the vectors of the basis are weight vectors with respect to t. Due to (III.2) and (III.3), we may assume that ρ is irreducible. As was mentioned above, it follows from (III.11) that V admits a basis in which all ρ(x), x ∈ gσ , are expressed by real matrices. Then the corresponding matrix form of ρ satisfies (6), and hence (2). Now, it follows from Proposition 6.4 that there exists a unique compact real structure τ
on sln (C) which extends τ . Since στ = τ σ, we see from the same proposition that σ0 τ
= τ
σ0 . But the standard compact real structure τ0 also commutes with σ0 . By Proposition 3.6, τ
= ϕτ0 ϕ−1 , where ϕ ∈ Int sln (C) and ϕσ0 = ϕσ0 . Then the matrix representation ρ1 = ϕ−1 ρ satisfies both (2) and (3). In fact, we have ρ1 σ = ϕ−1 ρσ = ϕ−1 σ0 ρ = σ0 ϕ−1 ρ = σ0 ρ1 , ρ1 τ = ϕ−1 ρτ = ϕ−1 τ
ρ = τ0 ϕ−1 ρ = τ0 ρ1 . Now, ϕ = Ad C, where C ∈ GLn (C), and hence ρ1 (x) = C −1 ρC is the matrix form of ρ corresponding a new basis of V . We also have to satisfy the condition (4) (it was satisfied by our first basis, but may be lost by changing it). We see from (6) and (7) that in the constructed basis we have iρ(x) ∈ sun , ρ(x) ∈ sln (R) for any x ∈ t(R). This means that all

52

§7. Inclusions between real forms

these ρ(x) are symmetric real matrices. Since they form a commutative family, there exists a matrix D ∈ On such that all the matrices from Dρ(t(R))D−1 are real diagonal ones. Going over to the new basis with transition matrix D−1 , we complete the proof.  It is clear (and was used at the end of the proof) that if we change a basis of V satisfying the conditions (2) and (3) of Proposition 1, using a real orthogonal transition matrix, then the new basis satisfies the same conditions. The following proposition shows that this is the only way to get a new basis satisfying the conditions (2) and (3), provided that ρ is irreducible. Proposition 2. Let ρ : g → sl(V ) be an irreducible linear representation of a complex semisimple Lie algebra g. Suppose that we fix a basis of V such that (2) and (3) are satisfied. If we have another basis of V with the same property, then the transition matrix is of the form C = cB, where c ∈ C× and B ∈ On . Proof. Let C denote the transition matrix to the new basis. Consider the representation ρ
= ψ −1 ρ : g → sln (C), where ψ = Ad C. Clearly, the relations τ ↑ρ τ0 and σ ↑ρ σ0 imply, respectively, τ ↑ρ ψτ0 ψ −1 and σ ↑ρ ψσ0 ψ −1 . It follows from Proposition 6.4 that ψτ0 ψ −1 = τ0 . Thus, σ
= ψσ0 ψ −1 commutes with τ0 , and we get an involution ω
= σ
τ0 . Clearly, ω ↑ρ ω
. Since ω
= ψω0 ψ −1 , (5) and Theorem 6.2 imply that ω
= ω0 . Hence, σ
= σ0 . Thus, ψ commutes with τ0 , σ0 and ω0 . This allows to deduce our assertion, using simple calculations with matrices. For any X ∈ sln (C), we have ¯ −1 = CXC −1 = C¯ X ¯ C¯ −1 , C XC whence C = aC, a ∈ C. Applying the √ complex conjugation, we get C = aC, whence aa = 1, i.e., |a| = 1. Then C1 = a C ∈ GLn (R). Thus, we may assume that C ∈ GLn (R). Exploiting ψω0 = ω0 ψ, we get C(−X  )C −1 = −(CXC −1 ) = −C 

−1

X C  ,

for any X ∈ sln (C). It follows that C  C = bIn , where b ∈ R. Clearly, b > 0, and hence B = √1b C ∈ On .  Before studying our main problem, we consider an application of Proposition 1 to dual representations. Proposition 3. For any linear representation ρ of a complex semisimple Lie algebra g we have ρ∨ ∼ ρω ∼ ρˆ ν, where ν ∈ Aut Π is the automorphism of the Dynkin diagram of g described in Proposition 4.4. If ρ is irreducible with highest weight Λ, then the highest weights of ρ∨ is ν(Λ). Proof. The condition (5) means that ρ(ω(x)) = −ρ(x) . Thus, in the basis, chosen in Proposition 1, ρω has the same matrix form as ρ∨ (see (III.1)). It follows that

§7. Inclusions between real forms

53

ρ∨ ∼ ρω. Now, by Proposition 4.3 (v), ω = ϕˆ ν , where ϕ = exp(ad π2 (e − f )) (see (4.9)). Using Proposition 6.2 (iv), we get ρω = ρϕˆ ν = Ad R(g)ρˆ ν ∼ ρˆ ν, where g ∈ G is given by (4.10). By (5), this implies our first assertion. The claim about highest weights follows from Lemma 1.  A representation ρ is called self-dual if ρ∨ ∼ ρ. Corollary 1. An irreducible linear representation ρ with highest weight Λ is selfdual if and only if ν(Λ) = Λ.  In the sequel we will consider linear representations in their matrix form, i.e., as homomorphisms ρ : g → sln (C). Let θ be an involution of g, and suppose that there exists an involution θ
of sln (C) such that θ ↑ρ θ
. If ρ is irreducible, then only one outer θ
and only one inner θ
extending θ may exist, due to Theorem 6.2. Let g0 denote a real form of g corresponding to θ (see §3). Suppose that θ
is an outer automorphism. As we saw in Example 4.1, θ
= (Ad B)ω0 , where ω0 is given by (1), and B ∈ GLn (C) satisfies B  = ±B. The number  +1 if B  = B , j(g, θ, ρ) = −1 if B  = −B is called the Karpelevich index of the pair (θ, ρ). If j(g, θ, ρ) = 1, then ρ(g0 ) is conjugate to a subalgebra of sln (R). If j(g, θ, ρ) = −1, then n = 2m is even, and ρ(g0 ) is conjugate to a subalgebra of slm (H). If θ
is an inner automorphism, then, by Example 4.1, θ
is conjugate (by an inner automorphism) to Ad Ip,q for p ≤ q, the matrix Ip,q being given by (1.4). In this case, ρ(g0 ) is conjugate to a subalgebra of sup,q . The number q − p ≥ 0 is called the signature of the pair (θ, ρ). As usually, we consider the real forms of g and the corresponding involutions up to conjugacy by inner automorphisms of g (see Theorem 3.2). Therefore we may assume that the given involution θ is of the form described in Theorem 4.2 (ii). More precisely, θ = ψˆ s , s ∈ Aut Π , ψ = exp(ad u) , π where u = it ∈ t , sˆ(t) = t , α(t) ∈ 2Z for all α ∈ ∆ . 2

(8)

In particular, ψ is an involutive inner automorphism commuting with σ, τ and ω (see Theorem 4.2 (iii)). Let us denote z = exp u ∈ G. Proposition 4. (i) The element z 2 = exp(πit) lies in the centre Z(G) of G. (ii) Let ρ : g → sln (C) be an irreducible representation with highest weight Λ. Then R(z)2 = eπiΛ(t) In . (iii) If the condition (4) is satisfied, then 1

R(z) = e 2 πiΛ(t) A , where A is a diagonal matrix from On , A2 = In .

(9)

54

§7. Inclusions between real forms

Proof. (i) We have Ad z 2 = (Ad z)2 = ψ 2 = e, and hence z 2 ∈ Z(G), by (I.2). (ii) By the Schur Lemma, (i) implies R(z)2 = R(z 2 ) = aIn , where a ∈ C. Applying R(z)2 to the highest vector vΛ of ρ, we get R(z)2 vΛ = e2Λ(u) vΛ = eπiΛ(t) vΛ . Thus, a = eπiΛ(t) . 1 (iii) It follows from (4) that the matrix A = e− 2 πiΛ(t) R(z) is diagonal. By (ii), A2 = In . Therefore the diagonal elements of A are ±1, and hence A ∈ On .  We want now to find the outer automorphisms of sln (C) to which θ extends by an irreducible representation ρ and to deduce a formula expressing the index j(g, θ, ρ) in terms of the highest weight of ρ. Assuming that θ is of the form (8), we denote (10) s0 = sν = νs (see Proposition 4.4 (iii)). Clearly, s20 = e. Theorem 1. Let ρ : g → sln (C) be an irreducible representation of a complex semisimple Lie algebra g with highest weight Λ, and let θ denote an involutive automorphism of g of the form (8). An outer involutive automorphism θ
of sln (C) satisfying θ ↑ρ θ
exists if and only if s0 (Λ) = Λ .

(11)

j(g, θ, ρ) = (−1)Λ(t+h) ,

(12)

Under this condition, we have

where h ∈ t(R) is given by (4.4). Proof. By Proposition 1, we may assume that ρ satisfies the conditions (2) – (4). We will also use the notation of Proposition 4 and Proposition 4.3. Let us suppose that ρθ = θ
ρ for an outer involutive θ
∈ Aut sln (C). By Example 4.1, θ
= (Ad B)ω0 , where B ∈ SLn (C) satisfies B  = jB, j = j(g, θ, ρ). Using Proposition 6.2 (iv), we get ρθ = ρ exp(ad u)ˆ s = (Ad R(z))ρˆ s, whence (Ad R(z))ρˆ s = (Ad B)ω0 ρ .

(13)

Thus, ρˆ s ∼ ω0 ρ ∼ ρ∨ ∼ ρˆ ν (see Proposition 3), whence s(Λ) = ν(Λ), due to Lemma 1. Thus, (11) holds. Conversely, suppose that s(Λ) = ν(Λ), whence ρˆ s ∼ ρ∨ . Then ρθ = (Ad R(z))ρˆ s ∼ ρ∨ = ω 0 ρ . Hence there exists B ∈ GLn (C) satisfying ρθ = (Ad B)ω0 ρ. We show that the 2 outer automorphism θ
= (Ad B)ω0 is involutive. Clearly, e = θ2 ↑ρ θ
. But
2
2 θ ∈ Int sln (C). Since ρ is irreducible, θ = e (see Lemma 6.1).

§7. Inclusions between real forms

55

To calculate the index, consider again the relation (13). Using Proposition 4 (iii), we may replace R(z) in this formula by a diagonal matrix A ∈ On such that A2 = In . Clearly, sˆ commutes with σ and τ , and hence the left side of (12) satisfies the conditions (2) and (3). Clearly, the same is true for ω0 ρ. Therefore, by Proposition 2, we may assume that B ∈ On . Under this assumption, we clearly have B 2 = jIn . (14) Now, by (5), ω0 ρ = ρω. Using Proposition 4 (iii) and Proposition 4.3 (v), we get from (13) the following relation: (Ad A)ρˆ s = (Ad B)ρ exp(ad

π (e − f ))ˆ ν. 2

By Proposition 6.2 (iv), we can rewrite it in the following form: ρˆ s = Ad(ABR(g))ρˆ ν,

(15)

where g ∈ G is given by (4.10). We also note that R(g) = exp ρ( π2 (e − f )) ∈ SOn . In fact, σ(e − f ) = e − f , τ (e − f ) = −(e − f ), and by (6) and (7), the matrix ρ( π2 (e − f )) is real skew-symmetric. Moreover, Proposition 4.3 (iii) implies the following relation: R(g)2 = exp(πiρ(h)) = (−1)Λ(h) In .

(16)

In fact, since g 2 ∈ Z(G), the Schur Lemma implies that R(g)2 = R(g 2 ) is a scalar matrix. Applying this operator to the highest vector vΛ of ρ, we get R(g)2 vΛ = exp(πiρ(h))vΛ = eπiΛ(h) vΛ = (−1)Λ(h) vΛ , since Λ(h) =

l 

ri Λi ∈ Z ,

(17)

i=1

by (4.5). One more relation we can obtain if we apply the first and the last terms of (12) to u = π2 it ∈ g. We get ρ(u) = (Ad B)(−ρ(u)) = −Bρ(u)B −1 . By exponentiating, one gets R(z) = BR(z)−1 B −1 . Using (9), we get BAB −1 = eπiΛ(t) A. It follows that eπiΛ(t) = ±1. Hence Λ(t) ∈ Z and BAB −1 = (−1)Λ(t) A .

(18)

Now we will restrict both sides of (15) to the principal 3-dimensional subalgebra s ⊂ g constructed in §4. By Proposition 4.2, any canonical automorphism sˆ, s ∈ Aut Π, acts on s trivially. It follows from (14) that the matrix C = ABR(g)

56

§7. Inclusions between real forms

commutes with any ρ(x), x ∈ s. Hence C leaves invariant any weight subspace of ρ|s. The weights of ρ|s are restrictions of the weights of ρ to hR and may be identified with the corresponding eigenvalues of the operator ρ(h). Thus, for l any λ ∈ Φρ we identify λ|hR with the integer λ(h) = i=1 ri λ(hi ), where ri are defined by (4.5). In particular, to the highest weight Λ of ρ there corresponds the non-negative integer m = Λ(h) given by (17). By (III.11), any other weight of ρ has the form λ = Λ − αi1 − . . . − αik , k > 0, and, due to Lemma 4.1, the corresponding weight of ρ|s is identified with λ(h) = m − 2k < m. Thus, the highest vector vΛ ∈ Cn is the only, up to a scalar factor, weight vector of ρ|s corresponding to the weight m. It follows that CvΛ = cvΛ for a certain c ∈ C. We claim that c2 = 1. In fact, by the condition (4), the standard basis of Cn consists of weight vectors of ρ. But vΛ , up to a complex factor, should enter into this basis, since dim VΛ = 1. Thus, we may assume that vΛ is one of the basic vectors and, in particular, that vΛ ∈ Rn . Thus, vΛ is a real eigenvector of the matrix ABR(g) ∈ On , and hence c = ±1. Now we calculate the matrix C 2 . Note that C commutes with R(g), since g belongs to the connected Lie subgroup of G corresponding to s. Using (18), (14) and (16), we may write C 2 = (AB)2 R(g)2 = (−1)Λ(t) B 2 R(g)2 = j(−1)Λ(t+h) In . Comparing this with the above result, we see that j(−1)Λ(t+h) = 1, but this is equivalent to (12).  Remark 1. One sees from the proof of Theorem 1 that the index formula (12) can also be written in another form. As we know, g 2 = exp(πih) and z 2 = exp(πit) belong to the centre Z(G) of G, and hence R(g 2 ) and R(z 2 ) are scalar matrices. We proved, actually, that their possible values are ±In and that R(g 2 ) = (−1)Λ(h) In ,

R(z 2 ) = (−1)Λ(t) In .

(19)

Note that the integer Λ(h) can be calculated by the formula (17). Then (12) can be expressed in the following form: R(g 2 z 2 ) = j(g, θ, ρ)In . In the following examples, some special cases will be considered. Example 1. Let us consider the Lie algebra g = sll+1 (C), l > 1. First we suppose that the involution θ is an outer one. Since ν is the only non-trivial automorphism of Π (Proposition 4.4 (i)), we have s = ν, s0 = e, and the condition (11) is satisfied for any highest weight Λ. By Example 4.1, there are two conjugacy classes of real forms of the second kind represented by sll+1 (R) and slm (H), 2m = l + 1, the corresponding involutions θ being, respectively, ω0 and (Ad S)ω0 , where S is the matrix Sm given by (2.5) or any other non-singular skew-symmetric matrix. In the case θ = ω0 , we need not use Theorem 1 to calculate the index. In fact, sll+1 (R) is the normal real form of g, and it was proved in Proposition 1 that any

§7. Inclusions between real forms

57

representation ρ, expressed in an appropriate basis, maps the normal real form into sln (R). Thus, j(g, θ, ρ) = 1. In the second case, we may set θ = νˆ. In fact, by Proposition 4.3, we have νˆ = (Ad g −1 )ω0 , where g = exp( π2 (e − f )). Clearly, g ∈ SO2m . One sees from (17) and Table 4 that 1 (h) = 2m − 1 for g = sl2m (C), and by (19) g 2 = −I2m . It follows that g and g −1 are skew-symmetric, and hence we may set θ = (Ad g −1 )ω0 = νˆ. By (12), we get j(g, θ, ρ) = (−1)Λ(h) . Thus, ρ maps slm (H) into sln (R) if Λ(h) is even, and into sl n2 (H) if Λ(h) is odd. By Remark 1, equivalent conditions are R(−I2m ) = In and R(−I2m ) = −In , respectively. From (17) and Table 4 one gets the explicit formula j(g, θ, ρ) = (−1)Λ1 +Λ3 +...+Λ2m−1 . If g0 is a real form of the first kind, then, by Example 4.1, g0 = sup,l+1−p , θ = Ad Ip,l+1−p , where p = 0, . . . , l. Since s = e, the condition (11) is satisfied, whenever Λi = Λl+1−i , i = 1, . . . , l. In the notation of Example 2.2 we have θ = exp(ad π2 it), where t ∈ t(R) is given by αp (t) = 2, αi (t) = 0 for i = p. Using Table 4, one gets from (12) the following result:  1 if l = 2m, j(g, θ, ρ) = (−1)(m+p)Λm if l = 2m − 1. Example 2. Suppose that g = so2l (C), l ≥ 4. We will use the notation of the proof of Proposition 4.4. As is well known (see, e.g., [11] or [19]), there exist, up to isomorphy, the following real forms of the first kind: 1. g0 = so2k,2(l−k) , θ = Ad diag(−Ik , Il−k , −Ik , Il−k ), where k = 0, . . . , l − 2. 2. g0 = u∗l (H), θ = Ad Sl . Since s = e, we see from Proposition 4.4 that for l = 2m the condition (11) is always satisfied, while for l = 2m + 1 it is satisfied, whenever Λ2m = Λ2m+1 . Now, in the case 1 θ = exp(ad π2 it), where t ∈ t(R) is given by αk (t) = 2, αi (t) = 0 for i = k. One gets from (12) and Table 4 the formula  j(g, θ, ρ) =

(−1)(m+k)(Λ2m−1 +Λ2m ) if l = 2m, 1 if l = 2m + 1.

In the case 2, θ is conjugate to exp(ad π2 it), where t ∈ t(R) is given by αl (t) = 2, αi (t) = 0 for i < l. Using again (12) and Table 4, we get j(g, θ, ρ) = (−1)Λ1 +Λ3 +...+Λ2m−1 . One also has the following real forms of the second kind: 3. g0 = so2k+1,2(l−k)−1 , θ = Ad diag(−Ik , Il−k , −Ik+1 , Il−k−1 ), where k = 0, . . . , l − 2. If l = 2m, then s0 = s = e, and (11) is satisfied, whenever Λ2m−1 = Λ2m . If l = 2m + 1, then s0 = e, and (11) is always satisfied.

58

§7. Inclusions between real forms

To calculate the index, we should present θ in the form (8). It is easy to verify that the non-trivial diagram automorphism sˆ is given by sˆ = Ad T , where the matrix T ∈ O2l is determined by the transposition of two vectors el , e2l of the standard basis of C2l . One also proves that θ is conjugate to exp(ad π2 it)ˆ s, where t ∈ t(R) is given by αk (t) = 2, αi (t) = 0 for i = k. Then (12) and Table 4 imply the following result:  1 if l = 2m, j(g, θ, ρ) = (−1)(m+k)(Λ2m +Λ2m+1 ) if l = 2m + 1. Example 3. Suppose that g = so2l+1 (C), l ≥ 2. This is the algebra of all block matrices of the form   0 U V  −V  X Y  , X ∈ gll (C), Y  = −Y, Z  = −Z, U, V ∈ Cl . −U  Z −X  As a maximal toral subalgebra t, one may choose the subalgebra of diagonal matrices H = diag(0, x1 , . . . , xl , −x1 , . . . , −xl ). The system of roots is ∆ = {±xi ± xj (i = j), ±xi }. We will use the system of simple roots Π = {α1 , . . . , αl }, where αi = xi − xi+1 , i = 1, . . . , l − 1, αl = xl . It is well known (see, e.g., [11] or [19]) that there exist, up to isomorphy, only the following real forms (all are of the first kind): g0 = so2k,2(l−k)+1 , θ = Ad diag(−Ik , Il−k , −Ik , Il−k+1 ), where k = 0, . . . , l. The condition (11) is always satisfied. We can write θ = exp(ad π2 it), where t ∈ t(R) is given by αk (t) = 2, αi (t) = 0 for i = k. Using (12) and Table 4, we get 1

j(g, θ, ρ) = (−1)(k+ 2 l(l+1))Λl . Example 4. Consider the case g = sp2l (C), l ≥ 2. This is the algebra of all block matrices of the form   X Y , where X ∈ gll (C), Y  = Y, Z  = Z. Z −X  As a maximal toral subalgebra t one may choose the subalgebra of diagonal matrices H = diag(x1 , . . . , xl , −x1 , . . . , −xl ). The system of roots is ∆ = {±xi ± xj (i = j), ±2xi }. We will use the system of simple roots Π = {α1 , . . . , αl }, where αi = xi − xi+1 , i = 1, . . . , l − 1, αl = 2xl .

§7. Inclusions between real forms

59

It is well known (see, e.g., [11] or [19]) that there exist, up to isomorphy, only the following real forms (all are of the first kind): 1. g0 = spp,l−p , θ = Ad diag(−Ip , Il−p , −Ip , Il−p ), p = 0, . . . , l − 1. 2. g0 = sp(R), θ = ω0 |g. The condition (11) is always satisfied. In the case 1, we have θ = exp(ad π2 it), where t ∈ t(R) is given by αp (t) = 2, αi (t) = 0 for i = p. Using (12) and Table 4, we get Λ1 +Λ3 +...+Λ2[ 1 (l+1)]−1

j(g, θ, ρ) = (−1)

2

.

In the case 2, θ is conjugate to exp(ad π2 it), where t ∈ t(R) is given by αl (t) = 2, αi (t) = 0 for i < l. It follows that j(g, θ, ρ) = 1. Example 5. Let G be a simply connected complex semisimple Lie group and g its Lie algebra. Let ρ be an irreducible representation of g satisfying the condition (11). Due to Remark 1, the Karpelevich index of ρ may be found by calculating the value R(w) of the corresponding representation R of G on the element w = (gz)2 ∈ Z(G). This leads to the following simple observation: if the order of Z(G) is odd, then j(g, θ, ρ) = 1. In fact, in this case w should have an odd order, and R(w)2 = In implies R(w) = In . The order of Z(G) is odd for the simple Lie algebras sl2m+1 (C) (Z(G) Z2m+1 ), E6 (Z(G) Z3 ), G2 , F4 , E8 (Z(G) = {e}) and their direct sums. Thus, under the condition (11), ρ always maps any real form of each of these algebras into sln (R). Example 6. Let g be a complex semisimple Lie algebra. Consider an irreducible representation ρ : g⊕g → sl(V ). Consider the involution θ : (x, y) → (y, x) of g⊕g which corresponds to the real form of this algebra isomorphic to gR (see Example 3.4). Using Theorem 1, one can find the condition, under which θ extends to an outer involutive automorphism of sl(V ), and calculate the index. Any irreducible representation ρ of g ⊕ g has the form ρ = ρ1 ⊗ ρ2 , where ρi , i = 1, 2, are two irreducible representations of g (see (III.12)). If t is a maximal toral subalgebra of g, then we choose t ⊕ t as a maximal toral subalgebra of g ⊕ g. Then the highest weight of ρ is Λ = (Λ1 , Λ2 ), where Λi is the highest weight of ρi , i = 1, 2. Clearly, θ = sˆ is a canonical automorphism determined by the transposition s of two copies of the system Π of simple roots of g, and s(Λ) = (Λ2 , Λ1 ). By (10) and Proposition 4.4 (ii), s0 (Λ) = (ν(Λ2 ), ν(Λ1 )). It follows that the condition (11) has the form Λ2 = ν(Λ1 ) or, equivalently, ρ2 ∼ ρ∨ 1. Λ1 (h)+ν(Λ1 )(h) Using (12), we see that j(g ⊕ g, sˆ, ρ1 ⊗ ρ∨ = (−1)2Λ1 (h) = 1. 1 ) = (−1) Thus, under the above condition, ρ always maps our real form into sln (R). Due to Theorem 2.1, a real semisimple Lie algebra g0 is simple precisely in the following two cases: 1) g0 (C) is simple; 2) g0 = gR , where g is a simple complex Lie algebra. Let ρ be an irreducible representation of g0 (C). The second case is studied in Example 6. In the first case, the involution s0 and the index j(g, θ, ρ) can be calculated using Examples 1 – 5. The results are presented in Table 5. Note that

60

§7. Inclusions between real forms

so2k+1,2(l−k) so2(l−k),2k+1 , and therefore the index formula for g0 = sop,2l+1−p , established in Example 7.3 in the case of even p, can be applied to the case of odd p. This gives the corresponding formula in Table 5. The same problem for an arbitrary real semisimple Lie algebra is reduced to the case of a simple g0 with the help of the following proposition. Proposition 5. Let gk , k = 1, 2, be two complex semisimple Lie algebras, θk an involutive automorphism of gk and s0k ∈ Aut Πk the corresponding involution of the system of simple roots Πk of gk (see (10)). Then the involution s0 induced by s01 and s02 on the system of simple roots Π = Π1 ∪ Π2 of g = g1 ⊕ g2 corresponds to the automorphism θ = θ1 × θ2 of g. If ρk is an irreducible linear representation of gk , k = 1, 2, then j(g, θ, ρ1 ⊗ ρ2 ) = j(g1 , θ1 , ρ1 )j(g2 , θ2 , ρ2 ). Proof. Let tk be a maximal toral subalgebra of gk , k = 1, 2. We may suppose that θk has the form (8), i.e., θk = exp(ad π2 itk )ˆ sk , where sk ∈ Aut Πk and tk ∈ tk , k = 1, 2, satisfy the conditions of (8). Then t = t1 ⊕ t2 is a maximal toral subalgebra and Π = Π1 ∪ Π2 is a system of simple roots of g. One verifies easily that θ = θ1 × θ2 = exp(ad π2 it)ˆ s, where t = (t1 , t2 ) ∈ t and s = s1 s2 ∈ Aut Π also satisfy the conditions of (8). By Proposition 4.4 (ii) and (iii), ν = ν1 ν2 , where νk is the corresponding involution of Πk , k = 1, 2, and s0 = sν = s01 s02 . Clearly, (4.4) implies that h = (h1 , h2 ), where hk ∈ tk (R) is the sum of the positive coroots of gk , k = 1, 2. Now, if Λk is the highest weight of ρk , then, by (III.12), Λ = (Λ1 , Λ2 ) is the highest weight of ρ = ρ1 ⊗ ρ2 . Using (12), we get j(g, θ, ρ) = (−1)Λ(t+h) = (−1)(Λ1 ,Λ2 )((t1 ,t2 )+(h1 ,h2 )) = (−1)Λ1 (t1 +h1 )+Λ2 (t2 +h2 ) = j(g1 , θ1 , ρ1 )j(g2 , θ2 , ρ2 ).  Next, we apply Theorem 1 to study of bilinear invariants of an irreducible representation. Let ρ : g → gl(V ) be a linear representation of a complex Lie algebra g and b a bilinear form in V . We say that b is invariant under ρ if b(ρ(x)u, v) + b(u, ρ(x)v) = 0 ,

x ∈ g,

u, v ∈ V .

(20)

An example is the Killing form kg invariant under the adjoint representation ρ = ad (see (1.1)). Bilinear forms on V are in a natural bijection with linear mappings β : V → V ∗ . Namely, if β : V → V ∗ is a linear mapping, then the function b(u, v) = β(u)(v) ,

u, v ∈ V,

is a bilinear form on V , and any bilinear form is obtained in this way. Clearly, nondegenerate bilinear forms correspond to isomorphisms V → V ∗ . Further, bilinear forms invariant under ρ correspond to the mappings β satisfying βρ(x) = ρ∨ (x)β ,

x ∈ g.

§7. Inclusions between real forms

61

In particular, a non-degenerate invariant bilinear form exists if and only if ρ is self-dual. The Schur Lemma implies that if ρ is irreducible, then the invariant bilinear form (if it exists) is unique, up to a complex factor. Moreover, any nonzero invariant bilinear form is non-degenerate. Choose a basis in V and denote by B the matrix of b in this basis. Replacing the representation ρ by its matrix form, we see that (19) is expressed by Bρ(x) + ρ(x) B = 0 ,

x ∈ g.

(21)

In the case when det B = 0, the relation (21) can also be written in the form ρ = (Ad B −1 )ω0 ρ , which means that the automorphism θ
= (Ad B)ω0 extends e by ρ. If g is semisimple, then we may apply Theorem 1. From its proof we know that we can assume B ∈ On and B  = jB, where j = j(g, e, ρ) = ±1. Thus, the invariant form b is symmetric or skew symmetric, depending on the value of j. Summarizing and applying Theorem 1, we get the following assertion. Theorem 2. If an irreducible representation ρ of a complex Lie algebra g admits a non-zero invariant bilinear form b, then b is non-degenerate and unique up to a complex factor. Such a form exists if and only if ρ is self-dual. If g is semisimple, then this condition is equivalent to ν(Λ) = Λ, where Λ is the highest weight of ρ. Moreover, b(v, u) = (−1)Λ(h) b(u, v) ,

u, v ∈ V ,

(22) 

where h is given by (4.4).

A representation ρ of a Lie algebra g is called orthogonal if it admits a symmetric non-degenerate invariant bilinear form, and symplectic if it admits a skewsymmetric non-degenerate invariant bilinear form. In the first case, ρ is equivalent to a homomorphism g → son (C), and in the second one to a homomorphism g → spn (C), n even. Theorem 2 allows to describe all orthogonal and symplectic linear representations of complex semisimple Lie algebras. Note that this description is due to E.B. Dynkin [5, 7]. A similar result for Chevalley groups see in Steinberg [23], §12. To conclude, we consider the case when the extending involution is inner. In particular, we prove two formulas (see [15]) for the signature of (θ, ρ) which express this signature through the character of the representation R. We remind that the character of R is the function χR : G → C defined by χR (g) = tr R(g) ,

g ∈ G.

One of these formulas also exploits the invariant integration over a maximal compact subgroup of G (see §1 I).

§7. Inclusions between real forms

62

Theorem 3. Let ρ : g → sln (C) be an irreducible representation with highest weight Λ, and let an involution θ = ψˆ s ∈ Aut g of the form (8) be given.
(i) We have θ ↑ρ θ for an involution θ
∈ Int sln (C) if and only if s(Λ) = Λ. In particular, any involution θ ∈ Int g extends to an involution θ
∈ Int sln (C). (ii) Under the condition of (i), the signature q − p of the pair (θ, ρ) is determined by  χR (gΘ(g)−1 )dg ,

(q − p)2 = n

(23)

U

where Θ is the automorphism of G satisfying de Θ = θ and U is a maximal compact subgroup of G invariant under Θ. (iii) If θ = ψ = exp(ad u) ∈ Int g, then the signature can also be expressed by the formula (24) q − p = |χR (exp u)| . Proof. (i) Denoting ψ = Ad z, where z = exp u ∈ G, we can write the extension condition in the form ρθ = (Ad R(z))ρˆ s = (Ad B)ρ ,

(25)

where B ∈ GLn (C). If such an extension exists, then ρˆ s ∼ ρ, whence s(Λ) = Λ, by Lemma 1. Conversely, if s(Λ) = Λ, then ρˆ s ∼ ρ, and there exists B ∈ GLn (C) 2 2 satisfying (25). Then θ
= Ad B extends θ. Now, e = θ2 ↑ρ θ
, whence θ
= e by
Lemma 6.1. In the case s = e, we may take θ = Ad R(z). (ii) Suppose that (25) is satisfied. Then R(Θ(g)) = BR(g)B −1 ,

g ∈ G.

(26)

By Example 4.1, we may assume that B = CIp,q C −1 for a certain C ∈ GLn (C) and p ≤ q. In particular, B 2 = In , and tr B = q − p. Clearly, u = gτ is a compact real form of g invariant under θ, and the corresponding maximal compact subgroup U ⊂ G is invariant under Θ. Applying (I.8) to the representation g → Ad R(g) of U in the vector space gln (C), we get the matrix  B0 =

(R(g)BR(g)−1 )dg ∈ gln (C)

(27)

U

satisfying R(g)B0 R(g)−1 = B0 . Clearly, R(g)|U is irreducible, and the Schur Lemma implies B0 = λIn , where λ ∈ C. From (27) we see that  (tr B)dg = tr B = q − p.

tr B0 = U

Therefore, B0 =

q−p n In .

Since B = B −1 , we get from (26)

B(R(g)BR(g)−1 ) = R(Θ(g))R(g)−1 = R(Θ(g)g −1 ) ,

g ∈ G.

§7. Inclusions between real forms

63

The integration over U gives q−p B= BB0 = n



R(Θ(g)g −1 )dg.

U

Calculating the traces, we get (23). (iii) Using the Schur Lemma, we get R(z) = cB = cCIp,q C −1 , where c ∈ C× (see the proof of (ii)). Comparing the determinants of both sides, we see that |c| = 1, by Proposition 4 (iii). It follows that q − p = tr Ip,q = | tr R(z)| = |χR (z)| = |χR (exp u)| .



Using tedious calculations with the Weyl character formula (see [22]) and integration, in [15] explicit formulas for the signature of irreducible representations of classical Lie algebras were deduced from (23) and (24). We will not reproduce them here.

§8. Real representations of real semisimple Lie algebras The main goal of this section is the classification of irreducible real representations of real semisimple Lie algebras. We start with general notions concerning complexification, realification and complex conjugation of representations, which are quite similar to those discussed in §2 in connection with Lie algebras. Certain simple facts similar to Proposition 2.3 and Theorem 2.1 will be established. Let g0 be an arbitrary real Lie algebra. In this section, representations ρ : g0 → gl(V0 ) of g0 in a real vector space V0 will be studied. We call them generally real representations of g0 . Real homomorphisms ρ : g0 → gl(V ), where V is a complex vector space, will also be considered; they are called complex representations of the real Lie algebra g0 . If g is a complex Lie algebra, then its representations in a complex vector space studied in §7 will also be called complex representations. Respectively, we have two complexification operations applied to a real representation ρ : g0 → gl(V0 ). Let us denote V = V0 (C) and g = g0 (C). First, we can extend any ρ(x), x ∈ g0 , to a complex linear operator in V , obtaining a complex representation ρC : g0 → gl(V ). Second, we can extend ρC to a homomorphism of complex Lie algebras ρ(C) : g → gl(V ), i.e., to a complex representation of g. Now, if we begin with a complex representation ρ : g0 → gl(V ) of a real Lie algebra g0 , then we also can extend it to a complex representation ρ(C) : g → gl(V ). On the other hand, we may regard ρ as a representation of g0 in the real vector space VR , and thus we get a real representation ρR : g0 → gl(VR ). This is the realification operation. Also, any complex representation ρ : g → gl(V ) of a complex Lie algebra g gives rise to a real representation ρR : gR → gl(VR ). A real (complex) representation is called irreducible if the representation space does not contain any non-zero proper real (respectively, complex) invariant vector subspaces. The equivalence of two real (complex) representations is defined, using real (respectively, complex) isomorphisms of representation spaces. It is easy to establish certain implications between the irreducibility of ρ, ρC , ρ(C) and ρR and between the equivalence of two representations ρ1 and ρ2 , of their complexifications and realifications (see Tables 2 and 3). A complex structure J in a real vector space V0 is said to be invariant under a real representation ρ of g0 in V0 if ρ(x)J = Jρ(x), x ∈ g0 . In this case, we may regard ρ as a complex representation of g0 in (V0 , J), and the original real representation ρ will be its realification. Similarly, a real (or quaternion) structure S in a complex vector space V is said to be invariant under a complex representation ρ : g0 → gl(V ) if ρ(x)S = Sρ(x), x ∈ g0 . If a real structure S in V is invariant under ρ, then the real form V0 = V S of V is invariant, and the real subrepresentation ρ0 : g0 → gl(V0 ) of ρ satisfies ρC 0 = ρ. If a quaternion structure in V is invariant under ρ, then V admits a structure of the vector space over H, invariant under ρ, which means that ρ determines a homomorphism g0 → glm (H), where 2m = dimC V . Let ρ : g0 → gl(V ) be a complex representation of a real Lie algebra g0 . Then ρ may be regarded as a complex representation of g0 in the complex conjugate vector space V¯ (see §2). We denote this representation by ρ¯ and call it the complex conjugate to the representation ρ. We distinguish between these two representations,

§8. Real representations of real semisimple Lie algebras

65

since they act in distinct complex vector spaces and, as we shall see later, are not necessarily equivalent. At the same time, (¯ ρ)R = ρR . Let us find the matrix form of the complex conjugate representation. Choose a basis v1 , . . . , vn of V and denote by C(x) = (cij (x)) the corresponding matrix of ρ(x), x ∈ g0 . Then ρ¯(x) is expressed in the same basis, regarded as a basis of V¯ , by the complex conjugate matrix C(x). Thus, ρ¯(x) has the following matrix form: ρ¯ = σ0 ρ ,

(1)

¯ in the Lie algebra gln (C) is denoted. where by σ0 the real structure X → X Now we will give a description of irreducible real representations of a real Lie algebra g0 in terms of its irreducible complex representations, using the notions introduced above. The results and their proofs are similar to the description of simple real Lie algebras in terms of simple complex ones given by Theorem 2.1. Proposition 1. For any complex representation ρ : g0 → gl(V ), we have (ρR )C ∼ ρ + ρ¯ . Proof. Consider the complex vector space W = V ⊕ V¯ and define the mapping S : W → W by S(u, v) = (v, u), u, v ∈ V . One sees easily that S is a real structure in W invariant under ρ + ρ¯. The corresponding real form W S coincides with Wd = {(u, u) | u ∈ V }, and the projection (u, u) → u, u ∈ V , gives an isomorphism of ¯. the real subrepresentation ρ0 induced on Wd onto ρR . Thus, (ρR )C ∼ ρC 0 = ρ+ρ  Theorem 1. Any irreducible real representation ρ : g0 → gl(V0 ) of a real Lie algebra g0 satisfies precisely one of the following two conditions: (i) ρC is an irreducible complex representation; (ii) ρ = ρ
R , where ρ
is an irreducible complex representation admitting no invariant real structures. Conversely, any real representation ρ satisfying (i) or (ii) is irreducible. Proof. Suppose that ρ is irreducible, but ρC is reducible. We want to prove that V0 admits a complex structure invariant under ρ. Fix a non-zero complex vector subspace W  V = V0 (C) invariant under ρC . Using the complex conjugation with ¯, Y = W +W ¯ respect to V0 , we can construct the complex vector subspaces W C ¯ ¯ ¯ and Z = W ∩ W of V invariant under ρ . We have Y = Y, Z = Z, whence Y = (Y ∩ V0 )(C) and Z = (Z ∩ V0 )(C). Since ρ is irreducible, this implies ¯ , and that Y ∩ V0 = V0 and Z ∩ V0 = {0}. It follows that V = Y = W ⊕ W ¯ | u ∈ W }. Then we can define a complex structure J on V0 by V0 = {u + u J(u + u¯) = i(u − u ¯), u ∈ W . Clearly, we have ρ(x)J(u + u¯) = ρ(x)i(u − u ¯) = i(ρC (x)u − ρC (x)¯ u) = i(ρC (x)u − ρC (x)u) = J(ρC (x)u + ρC (x)u) = Jρ(x)(u + u ¯) , This means that J is invariant under ρ.

x ∈ g0 , u ∈ W .

66

§8. Real representations of real semisimple Lie algebras

It follows that ρ may be regarded as a complex representation ρ
in (V0 , J) and is the realification of ρ
. Clearly, ρ
is irreducible (Table 3). If ρ
admits an invariant real structure S, then V0S is invariant under ρ, which gives a contradiction. Thus, the condition (ii) is satisfied. By Proposition 1, (ii) implies that ρC is reducible, and hence (i) is not satisfied. Conversely, if (i) is satisfied, then ρ is irreducible, due to Table 2. Suppose that (ii) is satisfied, i.e., ρ = ρ
R , where ρ
: g0 → gl(V ) is an irreducible complex representation. If ρ is reducible, then there exists a non-zero real vector subspace W  V which is invariant under ρ. Then we can construct the ρ-invariant real vector subspace iW and the ρ-invariant complex vector subspaces W ∩ (iW ) and W + iW . Since ρ
is irreducible, we can deduce that V = W ⊕ (iW ). Thus, W is an ρ-invariant real form of V . Clearly, the corresponding real structure is invariant under ρ, which is a contradiction.  Thus, all irreducible real representations ρ of a given real Lie algebra g0 split into two disjoint classes I and II which are characterized by the conditions (i) and (ii) of Theorem 1, respectively. The following corollary describes these classes from another point of view. Corollary 1. The class I consists of all irreducible real representations ρ which admit no invariant complex structure. In this case, ρC is an irreducible complex representation, admitting an invariant real structure. The class II consists of all irreducible real representations ρ which admit an invariant complex structure, i.e., have the form ρ = ρ
R , where ρ
is an irreducible  complex representation. In this case, ρ
admits no invariant real structures. As Theorem 1 shows, the following question is important for our study: which irreducible complex representations ρ : g0 → gl(V ) admit invariant real structures? By definition, an invariant real structure S : V → V is an isomorphism V → V¯ , satisfying ρ(x)S = S ρ¯(x), x ∈ g0 , i.e., an isomorphism of ρ onto the complex conjugate representation ρ¯. A complex representation of g0 is called self-conjugate whenever ρ ∼ ρ¯. We see that any complex representation admitting an invariant real structure is self-conjugate. The converse is not true, in general. Proposition 2. Let ρ : g0 → gl(V ) be an irreducible complex representation of a real Lie algebra g0 . (i) If S is an antiautomorphism V → V such that Sρ(x) = ρ(x)S, x ∈ g0 , then S 2 = ce, where c ∈ R× . (ii) If S1 is another antiautomorphism V → V satisfying S1 ρ(x) = ρ(x)S1 , x ∈ g0 , then S1 = dS, where d ∈ C× , and S12 = c1 e, where ε = sgn c = sgn c1 does not depend on the choice of S. (iii) If S1 and S2 are two real structures invariant under ρ, then S1 = dS2 , where √ d ∈ C, |d| = 1, and V S1 = d V S2 . Proof. (i) Clearly, S 2 ∈ GL(V ) satisfies ρ(x)S 2 = S 2 ρ(x), x ∈ g0 . By the Schur Lemma, we have S 2 = ce, where c ∈ C× . For any v ∈ V , we have c(Sv) = S 2 (Sv) = S(S 2 v) = S(cv) = c¯(Sv), and thus c = c¯. Hence c ∈ R. (ii) If we have another antiautomorphism S1 : V → V such that S1 ρ(x) = ρ(x)S1 , x ∈ g0 , then S1 S −1 is an automorphism commuting with all ρ(x), and

§8. Real representations of real semisimple Lie algebras

67

the Schur Lemma implies that S1 S −1 = de, where d ∈ C× . We get S1 = dS, and ¯ 2 = (|d|2 c)e. Clearly, sgn(|d|2 c) = sgn c. hence S12 = (dS)(dS) = (dd)S (iii) By = dS2 , and S12 √ = S22 = e implies |d| = 1. If x ∈ V S2 , √ (ii), we 1have S1 √ √  then S1 ( d x) = d S1 x = d x, and thus d x ∈ V S1 . Let ρ : g0 → gl(V ) be a self-conjugate irreducible complex representation. We define the Cartan index of ρ as ε(ρ) = sgn c = ±1, where c is defined by the following condition: we have S 2 = ce, where S is an antiautomorphism of V commuting with ρ. This sign is uniquely determined, due to Proposition 2. Setting S0 = √1 S, we get an antiautomorphism of V commuting with ρ and satisfying |c|

S02 = ε(ρ)e. If ε(ρ) = 1, then S0 is a real structure in V invariant under ρ. If ε(ρ) = −1, then S0 is a quaternion structure in VR invariant under ρ, and V admits a structure of the vector space over H, invariant under ρ. Thus, ρ determines a homomorphism g0 → glm (H), where 2m = dimC V . Summarizing, we see that an irreducible complex representation ρ : g0 → gl(V ) admits an invariant real structure if and only if ρ is self-conjugate and its Cartan index is equal to 1. In this case, the invariant real structure is determined uniquely, up to a complex factor, and hence the real form V0 of V , invariant under ρ, is determined uniquely, up to multiplication by a complex number. We can now classify the irreducible real representations described in Theorem 1 up to equivalence. Theorem 2. (i) Two irreducible real representations ρ1 and ρ2 of the class I are equivalent if C and only if ρC 1 and ρ2 are equivalent. (ii) Two irreducible representations ρ1 = (ρ
1 )R and ρ2 = (ρ
2 )R of the class II are equivalent if and only if ρ
1 ∼ ρ
2 or ρ
1 ∼ ρ
2 . (iii) A representation of the class I cannot be equivalent to one of the class II. C Proof. (i) The implication ρ1 ∼ ρ2 ⇒ ρC 1 ∼ ρ2 is obvious (see Table 2). Conversely, C suppose that ρ1 and ρ2 are of the class I and that ρC 1 ∼ ρ2 . Let α : V1 → V2 be an isomorphism of the representation spaces of ρC and ρC 1 2 respectively, C (x) = ρ (x)α, x ∈ g . Denote by S a real structure in Vi invariant satisfying αρC 0 i 1 2
−1 under ρC , i = 1, 2. Then S = α S α is an invariant real structure in V1 , and 2 1 i √ S1 S1
Proposition 2 implies that S1 = dS1 , √ where |d| = 1, and V1 = d V1 . It follows √ that d α(V1S1 ) = V2S2 . Therefore d α determines an equivalence between ρ1 and ρ2 . (ii) If ρ1 ∼ ρ2 , then, by Proposition 1, ρ
1 + ρ
1 ∼ ρ
2 + ρ
2 . Therefore ρ
1 ∼ ρ
2 or ρ
1 ∼ ρ
2 . Conversely, any of two equivalences ρ
1 ∼ ρ
2 and ρ
1 ∼ ρ
2 implies ρ1 = (ρ
1 )R ∼ (ρ
2 )R = ρ2 . (iii) is evident. 

From now on, we assume that g0 is a real semisimple Lie algebras. Theorems 1 and 2 reduce the problem of classification of irreducible real representations of g0 to the following ones: to describe the complex conjugate representation ρ¯ in terms of an irreducible complex representation ρ and, in particular, to determine the self-conjugate irreducible complex representations of g0 ; to calculate the Cartan

68

§8. Real representations of real semisimple Lie algebras

index of a self-conjugate irreducible complex representation. We will see now that these two problems can be solved if we use the results of §7. We will regard g0 as a real form of the complex semisimple Lie algebra g = g0 (C). The Cartan correspondence described in §3 assigns to this real form an involution θ of g. For any irreducible complex representation ρ0 of g0 , we denote by ρ = ρ0 (C) the corresponding complex representation of g which is irreducible, too, and thus is determined, up to equivalence, by its highest weight Λ. We will say that Λ is the highest weight of ρ0 . Clearly, Λ determines ρ0 up to equivalence, as well. In the following theorem, we use the notation of §7. In particular, by s one denotes the involutive automorphism of Π which corresponds to the involution θ by Theorem 4.1. We also put s0 = sν ∈ Aut Π. By Proposition 4.4 (iii), s0 = νs is involutive. The Cartan index of ρ0 will be denoted by ε(g0 , ρ0 ). Theorem 3. Let ρ0 : g0 → sl(V ) be an irreducible complex representation of a ¯ denote the highest weights of the real semisimple Lie algebra g0 , and let Λ, Λ ¯ representations ρ0 , ρ0 , respectively. Then Λ = s0 (Λ). In particular, ρ0 is selfconjugate if and only if s0 (Λ) = Λ. In this case, the Cartan index of ρ0 is expressed by ε(g0 , ρ0 ) = j(g, θ, ρ) , where ρ = ρ0 (C). Proof. Let σ denote the complex conjugation in g = g0 (C) with respect to g0 . By definition, θ = στ , where τ is a compact real structure in g commuting with σ. Due to Proposition 7.1, we may choose a basis of V such that the conditions (7.3) and (7.5) are satisfied (note that the notation σ does not coincide with that from Proposition 7.1!). We may replace ρ by this matrix form in this basis, regarding it as a homomorphism g → sln (C). By (1), we have ρ0 = σ0 ρ0 . Denoting ρ¯ = ρ0 (C), we have (2) ρ¯ = σ0 ρσ . In fact, both sides of this equality are complex homomorphisms g → sln (C) which coincide on the real form g0 . Multiplying both sides of (2) by θ from the right, we deduce ρ¯θ = σ0 ρτ = σ0 τ0 ρ = ω0 ρ = ρ∨ , where ω0 is given by (7.1). By Theorem 4.1, θ = ψˆ s, where ψ ∈ Int g and ¯ = ν(Λ), due to Proposition 7.3. s ∈ Aut Π. It follows that ρˆ ¯s ∼ ρ∨ , whence s(Λ) ¯ = s0 (Λ). Hence, Λ Now suppose that ρ0 is self-conjugate, i.e., that s(Λ) = ν(Λ). Then we can apply Theorem 7.1. In particular, θ ↑ρ θ
, where θ
is an outer automorphism of sln (C), i.e., there exists a matrix B ∈ GLn (C) such that ρθ = (Ad B)ω0 ρ .

(3)

As was shown in the proof of this theorem, we may assume that B ∈ On satisfies (7.13). One deduces from (3) that ρστ = (Ad B)σ0 τ0 ρ = (Ad B)σ0 ρτ ,

§8. Real representations of real semisimple Lie algebras

69

whence ρσ = (Ad B)σ0 ρ. Restricting both sides to g0 , we get ρ0 = (Ad B)σ0 ρ0 .

(4)

Let S0 : (z1 , . . . , zn ) → (¯ z1 , . . . , z¯n ) denote the standard real structure in Cn . Clearly, the real structure σ0 in the Lie algebra gln (C) may be written as σ0 (X) = S0 XS0 , where X ∈ gln (C) is viewed as a linear operator in Cn . Define the antilinear operator S = BS0 in Cn . Then (4) may be written as ρ0 (x) = Sρ0 (x)S −1 ,

x ∈ g0 .

Since B is real, BS0 = S0 B, whence S 2 = B 2 . Therefore we get from (7.13) S 2 = j(g, θ, ρ)In . Using Proposition 2, we see that ε(g0 , ρ0 ) = j(g, θ, ρ).



Summarizing, we see that Theorems 1, 2, 3 and Theorem 7.1 give the following classification of irreducible real representations of real semisimple Lie algebras. Let g0 be a real form of a complex semisimple Lie algebra g, and let θ ∈ Aut g be the corresponding involution written in the form (7.8). Suppose an irreducible representation ρ in a complex vector space V be given, and let Λ be its highest weight. Denote ρ0 = ρ|g0 . Then ρ0 determines an irreducible real representation of g0 , and here three different cases are to be distinguished. 1. The real case. If s0 (Λ) = Λ ,

ε(g0 , ρ0 ) = (−1)Λ(t+h) = 1 ,

(5)

then ρ0 leaves invariant a real form V0 of V and induces there an irreducible representation ρ0 |V0 of g0 . 2. The quaternion case. If s0 (Λ) = Λ ,

ε(g0 , ρ0 ) = (−1)Λ(t+h) = −1 ,

(6)

then the realification (ρ0 )R of ρ0 acting in VR is irreducible, and VR admits a structure of quaternion vector space invariant under (ρ0 )R . 3. The complex case. If (7) s0 (Λ) = Λ , then the realification (ρ0 )R of ρ0 acting in VR is irreducible, and ρ0 admits neither real nor quaternion invariant structures. All irreducible real representations of real semisimple Lie algebras can be obtained in one of these ways. In the real case, we get a bijection between the dominant weights Λ satisfying (5) and the irreducible representations ρ0 |V0 of g0 regarded up to equivalence. In the quaternion case, a similar assertion is true. In the complex case, two dominant weights Λ1 , Λ2 satisfying (7) determine equivalent real representations (ρ0 )R if and only if either Λ2 = Λ1 or Λ2 = s0 (Λ1 ).

70

§8. Real representations of real semisimple Lie algebras

Example 1. Let g0 be a compact semisimple Lie algebra. We may regard it as a compact real form of a complex semisimple Lie algebra g, corresponding to the trivial involution θ = e. Consider an irreducible representation ρ : g → sl(V ) with highest weight Λ. The condition (5) is ν(Λ) = Λ, (−1)Λ(h) = 1, it means that ρ is orthogonal (see Theorem 7.2). In this case, we get an irreducible representation g0 → son ⊂ sln (R), where n = dim V . The condition (6) means that ρ is symplectic, and we get an irreducible representation g0 → spm ⊂ sl2n (R), n = 2m. The condition (7) means that ρ is not self-dual. Then we get an irreducible representation g0 → sun ⊂ sl2n (R). Example 2. Let g0 = gR be a realification of a complex semisimple Lie algebra g. As in Examples 2.4 and 3.4, we will identify g0 with the real form of g ⊕ g corresponding to the involution s : (x, y) → (y, x). The embedding g0 → g ⊕ g has the form x → (x, τ x), where τ is a fixed compact real structure in g. To apply our classification, consider an irreducible complex representation ρ : g ⊕ g → sl(V ). Using the notation of Example 7.6, we may write ρ = ρ1 ⊗ ρ2 , where ρ1 , ρ2 are two irreducible complex representations of g, and Λ = (Λ1 , Λ2 ), where Λi is the highest weight of ρi , i = 1, 2. If Λ2 = ν(Λ1 ) (or ρ2 ∼ ρ∨ 1 ), then the Cartan index is 1, by Example 7.6 and Theorem 3. Therefore, we get an irreducible representation of g0 in a real form of V . These real irreducible representations are in 1-1 correspondence with the dominant weights Λ1 (or Λ2 ). The quaternion case is impossible. If Λ2 = ν(Λ1 ) (or ρ2  ρ∨ 1 ), then we get a real irreducible representation of g0 in VR . These real irreducible representations are determined by ordered pairs (Λ1 , Λ2 ) of dominant weights of g, and the pairs (Λ1 , Λ2 ) and (ν(Λ2 ), ν(Λ1 )) give equivalent representations. We also note that in the case ρ2 ∼ ρ∨ 1 the vector space V can be identified with End V1 , where V1 is the space of the representation ρ1 . Let us replace ρ1 by its matrix form such that (3.7) is satisfied. Then End V1 is identified with glm (C), where m = dim ρ1 , and ρ acts by the formula ρ(x, y)Z = ρ1 (x)Z − Zρ1 (y) , Therefore



ρ(x, τ x)Z = ρ1 (x)Z + Zρ1 (x) ,

x, y ∈ g . x, y ∈ g .

It follows that the corresponding irreducible representation of g0 acts in the real form V0 = {Z ∈ glm (C) | Z¯  = Z} consisting of Hermitian matrices. Example 3. Let ρ : g → gl(V ) be an irreducible complex representation of a complex semisimple Lie algebra g and dim V > 1. Then the real representation ρR of gR is irreducible. In fact, in the opposite case one proves, as in Theorem 1, the existence of a real structure S in V , invariant under ρ. For any x ∈ g, we have ρ(ix)S = iρ(x)S = iSρ(x). But, on the other hand, ρ(ix)S = Sρ(ix) = S(iρ(x)) = −iSρ(x). Thus, ρ is trivial, and dim V = 1. This construction is actually a special case of Example 2, since ρ = ρ ⊗ ρ2 , where ρ2 is the trivial representation of dimension 1. In the case, when g0 is a real form of a simple complex Lie algebra, the involution s0 giving the self-conjugacy condition of an irreducible complex representation ρ0

§8. Real representations of real semisimple Lie algebras

71

of g0 is shown in Table 5. It can be also read from the Satake diagrams of g0 which are given in this Table, using Theorem 9.1 (see Appendix). Table 5 also contains the values of the corresponding Cartan index. The calculation is based on Examples 7.1 – 7.5. The index is expressed through the coordinates Λi of the highest weight Λ of ρ0 in the basis of fundamental weights, following the numeration of simple roots admitted in [19]. The real forms g0 are denoted as in [11] or [19]. Together with Example 2, this solves the classification problem of irreducible real representations of simple real Lie algebras. The general case is reduced to this one by the following proposition which is an immediate corollary of Proposition 7.5 and Theorem 3.

s Proposition 3. Consider the Lie algebra g0 = i=1 gi , where gi are non-commutative simple real Lie algebras, and its representation ρ0 = ρ1 ⊗ . . . ⊗ ρs , where ρi is an irreducible complex representation of gi . Then ρ0 ∼ ρ¯0 if and only if ρi ∼ ρ¯i for each i = 1, . . . , s. Under this condition, we have ε(g0 , ρ0 ) =

s 

ε(gi , ρi ).

i=1

 In conclusion, we will deduce a formula reducing the calculation of the Cartan index to the case when the given highest weight is a fundamental one. Let us fix g0 and regard ε(ρ0 ) as a function ε(Λ) of the highest weight Λ of ρ = ρ0 (C). By Theorem 3, this function is defined on the subsemigroup Γ(g0 ) of the semigroup of all dominant weights Λ of g = g0 (C) given by the equation s0 (Λ) = Λ. By the same theorem, ε(ρ0 ) coincides with the Karpelevich index j(g, θ, ρ) which is expressed by (7.12). It follows that ε(Λ + Λ
) = ε(Λ)ε(Λ
), i.e., ε is a homomorphism of Γ(g0 ) into the group {1, −1}. We also note the following property of the index. Lemma 1. For any dominant weight Λ of g, we have ε(Λ + s0 (Λ)) = 1. Proof. We should prove that (Λ + s0 (Λ))(t + h) is even. Using sˆ(t) = t, we get (Λ + s0 (Λ))(t + h) = Λ(t + ν  t) + Λ(h) + Λ(s 0 h) . By Proposition 4.2, s 0 h = h. Since Λ(h) ∈ Z, we have only to prove that Λ(t + ν  t) = (Λ+ν(Λ))(t) ∈ 2Z. Let ρ be an irreducible representation of g with highest weight Λ. Then −ν(Λ) = w0 (Λ) ∈ Φρ (this is the so-called lowest weight of ρ). It follows that Λ + ν(Λ) = Λ − w0 (Λ) is a sum of simple roots of g. By (7.8), its value on t is even.  Clearly, the function ε is completely determined by its values on any generators of the semigroup Γ(g0 ). To construct such generators, one can use the fundamental weights. Let 1 , . . . , l denote the fundamental weights corresponding to the system of simple roots Π = {α1 , . . . , αl }. Then any dominant weight Λ can be written as l  Λ= Λ i i , i=1

72

§8. Real representations of real semisimple Lie algebras

where Λi = Λ(hi ) are non-negative integers. The involution s0 permutes the weights i . Therefore we can write them as 1 , . . . , r , r+1 , . . . , 2r , 2r+1 , . . . , l , where s0 (i) = r + i ,

i = 1, . . . , r ;

s0 (k) = k ,

k = 2r + 1, . . . , l .

Then the weights i + r+i ,

i = 1, . . . , r ;

k ,

k = 2r + 1, . . . , l ,

generate the semigroup Γ(g0 ). By Lemma 1, ε( i + r+i ) = 1, i = 1, . . . , r. Therefore, ε is completely determined by the values ε( k ), k = 2r + 1, . . . , l. More precisely, l   ε( k )Λk = ε( i )Λi . (8) ε(Λ) = k=2r+1

s0 (i)=i

´ Cartan [3] and was proved by Iwahori [13] (see This formula goes back to E. also [10], Ch. 7). It was used in the tables of Tits [24], were the indices of representations were given without proof. More precisely, these tables contain a description of self-conjugate irreducible representations of real forms of simple complex Lie algebras and of the “real” and “quaternion” fundamental weights. The same information you can obtain from Table 5 below.

§9. Appendix on Satake diagrams Another way to describe the classification of real semisimple Lie algebras is provided by the Satake diagrams, originally introduced in [20]. The Satake diagram of a real semisimple Lie algebra g0 has the same sets of vertices and edges as the Dynkin diagram of its complexification g (although coloured in black or white) and possibly some arrows, relating white vertices. It is natural to ask about the relation between these data and the symmetries of the Dynkin diagram discussed above. In this Appendix, we will show how to read the involution s0 = νs (see (7.10)) of the Dynkin diagram of g from the Satake diagram of its real form g0 which corresponds to an involutive automorphism θ of g inducing the symmetry s ∈ Aut Π by Theorem 4.1. This involution appears in Theorems 7.1 and 8.3 and, in particular, allows to determine all self-conjugate complex irreducible representation of the real Lie algebra g0 . The answer is formulated in Theorem 1 below. First we recall the construction of Satake diagrams (see, e.g., [20] or [19], §5.4, for more details). Let us suppose that g0 is a real semisimple Lie algebra with the complexification g. The Cartan correspondence described in §3 implies that there are the following data: a compact structure τ , a real structure σ and an involution θ of g, such that σ = τ θ, g0 = gσ and τ , σ and θ commute pairwise. Thus, θ(g0 ) = g0 , and there is the Cartan decomposition g0 = k ⊕ p (see §5). Choose a subalgebra a of g0 which lies in p and is maximal among all such subalgebras. Then a is commutative and any maximal commutative subalgebra t0 ⊂ g0 containing + a satisfies θ(t0 ) = t0 and has the form t0 = t+ 0 ⊕ a, where t0 ⊂ k. Moreover, t = t0 (C) ⊂ g is a maximal toral subalgebra of g which is invariant under τ, σ and θ, and we have t(R) = it+ 0 ⊕ a. Let ∆ ⊂ t∗ be the system of roots of g relative to t. Since θ ∈ Aut(g, t), we have an involution θ : ∆ → ∆, due to (II.12). The roots from the subset ∆c = {α ∈ ∆ | θ (α) = α} = {α ∈ ∆ | α | a = 0}

(1)

are called compact roots, while the roots from ∆nc = ∆ \ ∆c are called noncompact ones. The system of positive roots ∆+ can be chosen in such a way that θ (α) ∈ −∆+ for each non-compact α ∈ ∆+ . To obtain such a system ∆+ , it suffices to take the set of roots which are positive with respect to the lexicographical ordering in t(R)∗ determined by a base of t0 such that its first elements constitute a base of a. Let Π ⊂ ∆+ be the corresponding set of simple roots, and let us denote ± ± ± ∆± c = ∆ ∩ ∆c , ∆nc = ∆ ∩ ∆nc , Πc = Π ∩ ∆c , Πnc = Π ∩ ∆nc .

Then, by the choice of ∆+ , we have − θ (∆+ nc ) = ∆nc .

The following important property was proved in [20]:

(2)

74

§9. Appendix on Satake diagrams

Lemma 1. There exists an involution κ : Πnc → Πnc such that for any α ∈ Πnc we have  θ (α) = −κ(α) − cαγ γ, γ∈Πc



where cαγ are non-negative integers.

The Satake diagram of g0 is defined as the Dynkin diagram corresponding to Π, where compact roots are denoted by black circles •, non-compact roots by white circles ◦ and any pair α, κ(α) ∈ Πnc , such that α = κ(α), is linked by an arrow. It is known that the Satake diagram determines g0 uniquely, up to isomorphy, and that g0 is simple if and only if its Satake diagram is connected. Satake diagrams for all real simple Lie algebras can be found, e.g., in [19], see also Table 5 below. By Corollary 2 of Theorem 4.1, we have θ = ϕˆ s, where ϕ ∈ N = Aut(g, t)∩Int g and s ∈ Aut Π. We will interprete s as a symmetry of the Dynkin diagram or of the Satake diagram. Another important symmetry is the involution ν arising from the similar decomposition of the Weyl involution ω, see Proposition 4.3. Our aim is to describe the symmetry s0 = sν of the Satake diagrams in terms of the data given by this diagram, i.e., black and white vertices and the arrows. The set ∆c of compact roots can be interpreted as follows (see [20, 19]). Let m ⊂ g denote the centralizer of a in g. One can verify that m is reductive and that t+ = t+ 0 (C) is a maximal toral subalgebra of the semisimple subalgebra m0 = [m, m]. Any α ∈ ∆c vanishes on a(C) and hence may be identified with its restriction to t+ . In this way, ∆c is identified with the root system of m0 relative to t+ and Πc with a set of simple roots of m0 . Let Wc denote the subgroup of the Weyl group W of g generated by the reflections rα , α ∈ ∆c . One deduces from (1.8) that any w ∈ Wc acts trivially on a and leaves it+ 0 invariant. Moreover, restricting all w ∈ Wc to , we get an isomorphism of W it+ c onto the Weyl group of m0 . As in §4, consider 0 the element w0c ∈ Wc such that (w0c ) (Πc ) = −Πc and define νc = −(w0c ) | Πc . We will interprete νc as an involutive symmetry of the “black part” of the Satake diagram which is the Dynkin diagram of Πc . We shall also use the following Lemma 2. + (i) For any w ∈ Wc we have w (∆+ nc ) = ∆nc . (ii) If w ∈ Wc and α ∈ Πnc , then w (α) = α +



kαγ γ,

(3)

γ∈Πc

where kαγ are non-negative integers. Proof. If β ∈ Πc , α ∈ ∆nc , then by (II.9) we have rβ (α) = α − 2

(α, β) β, (β, β)

(4)

  where 2 (α,β) (β,β) ∈ Z. Clearly, rβ (α) | a = 0, and hence rβ (α) ∈ ∆nc . By (II.20),  + rβ (α) ∈ ∆+ , whenever α ∈ ∆nc . Thus, the assertion (i) is proved for w = rβ . By

§9. Appendix on Satake diagrams

75

(II.19), Wc is generated by reflections rβ , β ∈ Πc , which implies (i) in the general form. To prove (ii), suppose that β ∈ Πc , α ∈ Πnc . Then in (4) we have 2 (α,β) (β,β)  0. Thus, (3) is true for w = rβ . The general case can be proved by induction on the number of reflections rβ , β ∈ Πc , entering into an expression of w ∈ Wc through these generators. In fact, suppose that (3) holds for a certain w ∈ Wc and take β ∈ Πc . Then  kαγ rβ (γ). (wrβ ) (α) = rβ (w(α)) = rβ (α) + γ∈Πc

Clearly, rβ (γ) ∈ ∆c . Using (4), we get (wrβ ) (α) = α +



k˜αγ γ,

γ∈Πc

where k˜αγ ∈ Z. Then (II.18) implies that k˜αγ
0, γ ∈ Πc .



With the notation introduced above, we state the main result: Theorem 1. The involution s0 = sν of the Satake diagram leaves invariant both the subset of white vertices and the subset of black ones. On the white vertices, it induces the involution κ depicted by the arrows, while on the black ones it coincides with the involution νc . If g0 is simple, then νc = id, except the following cases: g0 = sun , suk,n−k , sok,2n−k (n − k odd), and the compact form of E6 . Proof. We start with ”improving” the involution θ by composing it with an appropriate inner automorphism ψ of g in order to get θ1 = ψθ ∈ Aut(g, t, Π). Then we will be able to see the symmetry s ∈ Aut Π induced by θ. We have to choose ψ ∈ N , and hence on the dual level we have to compose the transformation θ of the vector space t(R)∗ with an element of the Weyl group W ∨ . Note that + θ (w0c ) (∆+ ) = ∆− . In fact, by Lemma 2 (i), (w0c ) (∆+ nc ) = ∆nc , and by defini− tion (w0c ) (∆+ ) = −∆ . Now our assertion follows from (1) and (2). Clearly, it c c can also be written as θ (w0c ) (∆− ) = ∆+ , and therefore the mapping θ (w0c ) w0 leaves ∆+ invariant. Now, the elements w0c , w0 ∈ W are induced by inner automorphisms ac , a ∈ N respectively, and it follows that θ1 = ψθ ∈ Aut(g, t, Π), where ψ = aac ∈ N . Writing θ = ϕˆ s, where ϕ ∈ N, s ∈ Aut Π, we get θ1 = ϕ1 sˆ, where ϕ1 = ψϕ lies in the subgroup T = {exp ad h | h ∈ t} of Int g, due to Proposition 4.1. Then for the transposed linear transformations of t(R)∗ we have sˆ = θ1 = θ (w0c ) w0 = θ (−(w0c ) )(−w0 ), whence sˆ (−w0 ) = θ (−(w0c ) ). The left-hand side leaves Π invariant and induces s0 = sν ∈ Aut Π. Therefore the right-hand side also leaves Π invariant and s0 = (θ (−(w0c ) )) | Π.

(5)

The relation (5) gives the desired description of the action of s0 on the Satake diagram. In fact, if α ∈ Πnc , then, by Lemma 2 (ii),  −(w0c ) (α) = −α − kαγ γ, γ∈Πc

76

§9. Appendix on Satake diagrams

where kαγ are non-negative integers. Applying θ and using Lemma 1 and (1), we get   s0 (α) = κ(α) + cαγ γ − kαγ γ, γ∈Πc

γ∈Πc

where cαγ and kαγ are non-negative integers. But s0 (α) and κ(α) lie in Πnc , whence s0 (α) = κ(α). If α ∈ Πc , then (5) implies s0 (α) = (θ (−(w0c ) ))(α) = θ (νc (α)) = νc (α), by (1). The last assertion is easy to verify case-by-case using Table 5.



Tables Table 1 Dynkin diagrams and their automorphism groups Type

Complex Lie algebra

Dynkin diagram

Aut Π

A1

sl2 (C)

◦

{e}

Al , l ≥ 2

sll+1 (C)

◦

◦

◦.

. .◦

◦

Z2

Bl , l ≥ 2

so2l+1 (C)

◦

◦

◦.

. .◦ >

◦

{e}

Cl , l ≥ 2

sp2l (C)

◦

◦

◦.

. .◦