Onishchik, A. L. Lectures On Real Semisimple Lie Algebras and Ther Rerpesentations
Onishchik, A. L. Lectures On Real Semisimple Lie Algebras and Ther Rerpesentations
Onishchik, A. L. Lectures On Real Semisimple Lie Algebras and Ther Rerpesentations
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ESI_Onishchik_titelei.qxd 30.01.2004 9:33 Uhr Seite 3
Arkady L. Onishchik
Lectures on
Real Semisimple
Lie Algebras
and Their
Representations
S E
M M
E S
S E
M
E S
M
European Mathematical Society
ESI_Onishchik_titelei.qxd 30.01.2004 9:33 Uhr Seite 4
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
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Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Complexication 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. Schrodinger Institute for Mathematical Physics (Vi-
enna, Austria) during the autumn semesters 2001 and 2002, respectively. The
main goal was an exposition of the theory of nite dimensional representations of
real semisimple Lie algebras.
The foundation of this theory was laid in E. Cartan [3]. Iwahori [13] gave an
updated exposition of the E. Cartans work (see also [10], Ch. 7). This theory
reduces the classication 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 classication for all simple real Lie algebras g0 was given
(without proof) in the tables of Tits [24].
The aim proclaimed by E. Cartan in [2, 3] was to nd all irreducible subalgebras
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 simplied (and somewhat extended and corrected) exposition of
the main part of [15] and to relate it to the theory developped in [10, 13].
The rst 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 xed
there. In the main body of the lecture notes (2 8) the exposition is given with
detailed proofs.
2 deals with general facts on complexication and realication, real and com-
plex 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
Lie algebra g and involutive automorphisms of g discovered by E. Cartan [4]
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
cation 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
exploited in the original proof of E. Cartan).
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 = 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 homo-
morphisms. 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 = is an irreducible representation g sln (C). First we consider
the relation = in the case when is an outer automorphism of sln (C), i.e.,
Ad g = de g Aut 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)
2 1. Preliminaries
k
g= gi ,
i=1
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],
invariant integral of a function f is denoted by G f (g)dg; we
Ch. VIII). The
suppose that G dg = 1. The following invariance conditions are satised:
1
f (gh)dg = f (hg)dg = f (g )dg = f (g)dg , h G.
G G 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.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 Aut g be involutive, i.e.,
satises 2 = e. Then is diagonalizable with eigenvalues 1, and hence we have
the eigenspace decomposition
g = g+ g , where g = {x g | x = x} . (2)
[g+ , g+ ] g+ , [g+ , g ] g , [g , g ] g+ .
g = kp, (3)
(u , h) = (h) , h t ; (6)
(, ) := (u , u ) = (u ) = (u ) , , t .
1. Preliminaries 5
This form is real and positive denite on t(R) . We also dene the vector
2
h = u (7)
(, )
E e , H h , F f .
() = ,
(t(R)) = t(R) ,
(g ) = g( )1 .
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 is called simple (with respect to
D) if = + , where , + . Let + denote the subset of simple roots.
We will write = {1 , . . . , l }.
(II.17) Any + has the form = i1 + . . .+ ir , where i1 + . . .+ ik +
for each k = 1, . . . , r.
(II.18) The set is a basis of t(R) . In particular, l = dimR t(R) = dimC t is
the rank of g. Any can be uniquely written in the form = li=1 ki i ,
where ki Z and either all ki 0 (for ) or all ki 0 (for ).
+
The property (II.18) suggests to call the set of simple roots 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 is simple with respect to an
appropriate Weyl chamber. Given a xed subset of simple roots , the Weyl
group W is generated by r , .
Let us x a Weyl chamber D and consider the corresponding subsets +
of positive and simple roots. Dene the linear form
1
= . (10)
2 +
r () = , (h ) = 1 f or any .
hi = hi , ei = ei , fi = fi , i = 1, . . . , l , (11)
where the root vectors ei and fi are chosen as in (II.11). Introducing the integers
2(i , j )
aij = i (hj ) =
(j , j )
( (x))(v) = ((x)v) , x g, V , v V ,
or, equivalently,
(x) = (x) , x g. (13)
(x) = (x) , x g.
A special case of the weight space decomposition (for = ad) is the root space
decomposition (4); we have ad = {0}.
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 t (R) we have = i=1 i i where i = (hi ).
The irreducible representation with the highest weight can be described by
writing coecients i 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 : g gl(V ) be an irreducible representation, and be its highest weight.
Then the highest vector v V generates V over g. More precisely, let us denote
v = v ,
vi1 ,... ,ik = (fi1 ) . . . (fik )v , 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:
(hi )v = i v ,
(hi )vi1 ,... ,ik = (i ai1 ,i . . . aik ,i )vi1 ,... ,ik ,
(fi )vi1 ,... ,ik = vi,i1 ,... ,ik ,
(ei )v = 0 ,
(ei )vi1 ,... ,ik = (ii1 (hi ) + (fi1 )(ei ))vi2 ,... ,ik .
= i1 . . . ik ,
where k 0.
Let gi , i = 1, 2, be two complex semisimple Lie algebras. Consider g = g1 g2 .
If i : gi gl(Vi ), i = 1, 2, are linear representations, then one can dene the
representation = 1 2 : g gl(V1 V2 ) by
S(u + iv) = u iv , u, v V ,
V0 = V S = {v V | Sv = v} . (1)
All these general notions can be introduced in the category of Lie algebras.
Given a real Lie algebra g0 , we endow the complexication g = g0 (C) of the vector
space g0 with the bracket extending the bracket of g0 , i.e., given by
Clearly, g is a complex Lie algebra. On the other hand, for any complex Lie algebra
g, the realication gR of the vector space g is a real Lie algebra.
Now, if g = g0 (C) is the complexication of a real Lie algebra g0 , then the
complex conjugation , dened by
(x + iy) = x iy , x, y g0 , (2)
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 realication
is g0 .
If g is a complex Lie algebra, then the complex conjugate vector space g, en-
dowed with the same bracket as g, is a complex Lie algebra as well. It is called
14 2. Complexification and real forms
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
: X = (xij ) X = (xij ).
The restriction of 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
One proves easily that 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
,
B D
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 identied
with the quaternion structure J in C2m given by
z1 w1
.. ..
. .
z wm
J m=
w1 z1
. .
.. ..
wm zm
or
J(v) = Sm v , v C2m ,
where
0 Im
Sm = . (5)
Im 0
Consider the Lie algebra glm (H) of endomorphisms of the vector space Hm (or of
quaternion n n-matrices). Clearly, it is identied with the real subalgebra {X
gl2m (C) | XJ = JX} of gl2m (C). Now, we dene the mapping : gl2m (C)
gl2m (C) by (X) = JXJ 1 = Sm XSm . This is an antiinvolution determining
the real form gl2m (C) = glm (H) of the Lie algebra gl2m (C). The elements of this
real form are block matrices of the form
A B
,
B A
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
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 dene the desired complex structure J
by J(x + y) = ix iy, x, y a. Clearly, J 2 = e, and we only have to verify
the condition (3). For z1 = x1 + y1 and z2 = x2 + y2 , where xi , yi a, we get
[z1 , z2 ] = [x1 , x2 ] + [y1 , y2 ], whence
[z1 , Jz2 ] = [x1 , ix2 ] [y1 , iy2 ] = i[x1 , x2 ] i[y1 , y2 ] = J[z1 , z2 ] .
It follows that g0 is the realication 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 sub-
algebra 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 g of g
is the root subspace of g corresponding to the root t . Thus, the system of
= { | }, where is the system of roots of g. It follows
roots of g is
that the roots of g determine the same real form t(R) of t as the roots of g do,
and |t(R) = |t(R), . Identifying with its restriction to t(R), we
get = . By Proposition 2 (ii) and (II.5), the Killing forms of g and of g in-
duce 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 = {1 , . . . , l } 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 : g g such that
(hi ) = hi , (ei ) = ei , (fi ) = fi , i = 1, . . . , l . (6)
2 2
Then is an automorphism of g leaving invariant our generators. Thus, = id,
and so is a real structure in g. The corresponding real form g 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 ) , x1 + . . . + xn = 0 .
In particular, rk sln (C) = n 1.
18 2. Complexification and real forms
Let Eij denote the matrix with (i, j)-entry 1, all other entries being 0. Then
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 x the Weyl chamber
Then
+ = {xi xj | i < j} ,
= {1 , . . . , n1 } , where i = xi xi+1 , i = 1, . . . , n 1 .
It follows that the normal real form is sln (R), and the corresponding real structure
is given by (X) = X.
To get another real form, we rst construct an involutive automorphism of g
which will be important in the sequel. With the Weyl chamber D xed above we
can associate the subset D t(R) which is, clearly, a Weyl chamber, too. The
corresponding system of simple roots is = {1 , . . . , l }, and its Cartan
matrix coincides with that of . As a canonical system of generators of g, we
may choose hi = hi , fi , ei , i = 1, . . . , l. By (II.22), we get a unique
automorphism Aut g such that
= = (9)
= 1 .
Proposition 2.
(i) The function h is a Hermitian form coinciding with kg on the real form g .
(ii) h (x, y) = h (x, y), x, y g, for any Aut g satisfying = .
(iii) h ([x, u], v) = h (u, [x, v]) for x g , u, v g.
(iv) The real form g is compact if and only if the Hermitian form h is positive
denite.
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 g constructed in
2 is compact. We will use the notation of 2. The real structure was dened
by (2.9), in terms of the canonical generators ei , fi , hi corresponding to a simple
root system = {1 , . . . , l }. We note some properties of .
Proof. The rst relation follows from the denition of . To prove the second one,
we apply to the equation
fact, we can dene the inverse mapping log sending a positive denite self-adjoint
operator with eigenvalues i > 0 to the linear operator admitting the same
eigenspace decomposition, but with the eigenvalues log i R.
Any P(E) can be included into a unique real one-parameter subgroup
(t), t R, of the Lie group GL(E), lying in P(E) and satisfying (1) = . In
fact, if (t) is a one-parameter subgroup with these properties, then (0) S(E)
and = (1) = exp (0). Thus, (0) = log is uniquely determined by . On
the other hand, the formula (t) = exp(t log ) gives, clearly, the desired subgroup.
If t Z, then
(t) = t .
The same notation will be used for arbitrary t R. This is suggested by the
following fact: if i > 0 are the eigenvalues of , then the operator (t) can be
given by the same eigenspace decomposition as with the eigenvalues ti .
If , P(E) satisfy = 1 , where GL(E), then log = (log) 1 and
= t 1 for any t R. In the complex case the same is true for any invertible
t
for any t R. It follows that t 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 in a
given complex semisimple Lie algebra g (existing by Theorem 1) is xed. We will
study the correspondence = 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 ( , ) = h (see (1)).
Lemma 2.
(i) For any antiinvolution , we have = S(g), and thus 2 P(g)Aut g.
(ii) For any involution Aut g, we have = ( )2 P(g) Aut g.
Proof. (i) To prove that = , we get, using (I.3),
h (x, x) = k(x, x) = k( x, x) = k( x, x)
= k( x, x) = (( )x, ( )x) > 0 for any x = 0 .
Given a complex Lie algebra g and two real structures , in g, we say that
real forms g , g are compatible if = .
Proposition 5. Let g be a complex semisimple Lie algebra. If two compact real
forms of g are compatible, then they coincide.
Proof. Let , be two compact real structures in g, such that = . We want
to prove that = . To do this, consider the involution = . We have to
show that = id (this is equivalent to = 1 = ). Since is an automorphism
and g is a real form of g, it is sucient to prove that x = x for each x g .
By Proposition 1 (ii), g is invariant under , and hence we have the eigenspace
decomposition g = g+ g , where g = {x g | (x) = x} (see (1.2)). To
show that the second summand is 0, we use the positive denite Hermitian forms
h and h given by (1). If there is a non-zero x g , then h (x, x) > 0 and
h (x, x) > 0. But h (x, x) = k(x, x), while h (x, x) = k(x, x) = k(x, x). This
gives a contradiction.
Proposition 6. For any real structure in g, there exists Int g such that
= 1 satises = , i.e., the real forms g = (g ) and g are
1
compatible. We may choose = 4 , where = ( )2 .
Proof. Consider the automorphism = Aut g. By Lemma 2, = 2
P(g) Aut g. By Lemma 1, we get the one-parameter subgroup t Int g, t R.
Using Proposition 1 (i), we see that = 1 . This implies, by Lemma 2, that
t = t , t R. Clearly, we also have 1 = , whence t 1 = t , t R.
We will search the desired inner automorphism in the form = t for a certain
t R. We can write
(1 ) = t t = t t = t t = 2t ,
(1 ) = t t = t t = t 1 t = 2t 1 .
Corollary. For any two compact real structures , in g there exists Int g such
that = 1 . Thus, any complex semisimple Lie algebra admits a unique, up
to conjugacy by inner automorphisms, compact real structure. The automorphism
can be chosen in the form described in Proposition 6.
Proof. Apply Proposition 6 to the real structure and use Proposition 5.
Now we prove a similar proposition, where instead of the antiinvolution an
involution is considered.
Proposition 7. Given an involution Aut g, there exists Int g such that
1
1 commutes with . We may choose = 4 , where = ( )2 .
Proof. The proof is similar to that of Proposition 6. One considers the auto-
morphism = ( )2 P(g) Aut g (see Lemma 2 (ii)). Then one veries that
1
= 4 is the desired inner automorphism.
Now we dene the desired bijection between the conjugacy classes of antiin-
volutions and involutions of a complex semisimple Lie algebra g. As above, we
assume that a compact real structure is xed. For any real structure , we
can choose a real structure = 1 such that = , where Int g is
the inner automorphism constructed in Proposition 6. Then we assign to the
automorphism = , which is involutive due to Proposition 1 (ii).
Theorem 2. Let g be a complex semisimple Lie algebra. The mapping
described above determines a bijection between the conjugacy classes of antiinvolu-
tions and involutions by inner automorphisms (or by automorphisms) of g. This
bijection does not depend on the choice of the compact real structure in g.
Proof. First we shall prove that our mapping actually determines a mapping of
conjugacy classes. Suppose that we have two antiinvolutions , 1 which are con-
jugate by an inner automorphism. Then the corresponding antiinvolutions , 1
commuting with are also conjugate by an inner automorphism, i.e., 1 = 1 ,
where Int g. Clearly, 1 commutes with and 1 . By Corollary of Propo-
sition 6, there exists Int g commuting with 1 , such that 1 = 1 .
Then = 1 commutes with and satises 1 = 1 . Clearly, for the
corresponding involutions = and 1 = 1 we have 1 = 1 . Thus, we
have a well dened mapping of conjugacy classes.
Now we show that this mapping is bijective. First we prove the surjectivity.
Let an involution Aut g be given. By Proposition 7, there exists Int g
such that = 1 satises = . Then = is an antiinvolution. Since
= , our mapping sends the conjugacy class of to that of .
Next we show the injectivity. Suppose that two antiinvolutions 1 , 2 are given
and that 1 = 1 and 2 = 2 satisfy 2 = 1 1 for a certain Aut g.
Here i are antiinvolutions conjugate to i and commuting with . Clearly, we
may assume that i = i and i = i , i = 1, 2. Then 2 commutes with
and 1 . By Corollary of Proposition 6, there exists Int g commuting with
2 , such that 1 = 1 . Then = 1 commutes with and satises
2 = 1 1 . Clearly, 2 = 1 1 .
26 3. Real forms and involutive automorphisms
Finally, let us consider another compact real structure 1 in g and the corre-
sponding bijection between conjugacy classes of antiinvolutions and involutions.
We claim that it coincides with the original bijection determined by . In fact,
by Corollary of Proposition 6, we have 1 = 1 , where Int g. Let
= , where is conjugate to and commutes with , be the
mapping giving the original bijection. Then, clearly, the new bijection is deter-
mined by the mapping 1 1 = ( 1 )1 = 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 = e = id corresponds. Now, if is the normal real
structure given by (2.5), then the corresponding involutive automorphism is the
Weyl involution = given by (2.7).
Example 3. In Example 2.1 we described certain real structures in sln (C). All
these structures commute with the compact real structure (X) = X (see
Example 1). We list the involutive automorphisms = corresponding to various
real structures .
For the normal real structure (X) = X we get (X) = (X) = X .
1
For (X) = Sm XSm , n = 2m, we get (X) = Sm X Sm
1
= (AdSm )(X T ).
For (X) = Ip,q X Ip,q we get (X) = Ip,q XIp,q .
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
given by (2.9). Clearly, is a compact real structure on g g commuting
with . Thus, the involution, corresponding to the real form gR by Theorem 2, is
= ( ) : (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 com-
plex 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 Aut g leaving t invariant, the transformation of t (R) maps onto
itself. Let us also choose a Weyl chamber in t(R), and let denote the
corresponding subset of simple roots. Consider the subgroups of Aut g dened by
Proof. We only have to prove the assertion concerning Ker . Clearly, exp(ad h)
acts trivially on t for any h t. Conversely, suppose that Aut(g, t, ) satises
28 4. Automorphisms of complex semisimple Lie algebras
Then (II.16) implies that Int g (Aut ) = {e}. Hence, the latter decomposition
is semidirect.
Corollary 1. We have
Aut g/ Int g
Aut .
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 (Aut ) ,
Thus, C(C )1 should commute with all X sln (C). This is true precisely in the
case, when C(C )1 = In , where C , which can be rewritten as C = C .
Transposing this relation, we get C = C. It follows that C = 2 C, whence
= 1. Thus, we get two dierent cases.
I. If = 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
Using this classication, 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 Aut g, dened 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, Aut(g, t), |t(R) = id, and so (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 Aut . Then we get
the diagram automorphism which is involutive as well. Following the proof of
Theorem 1, we can write
= ,
where N satises |t(R) = w0 . Our goal is to describe the automorphism
explicitly.
Consider the element
h= h t(R) . (4)
+
r (h ) = hr () , ,
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 dene the elements e, f g by
l
l
e= ri ei , f= ri fi . (6)
i=1 i=1
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
0 1 0 0 1 0
E= e , F = f , H= h (7)
0 0 1 0 0 1
Using (1), (6) and (8), we get s(e) = e, s(f ) = f . Thus, s|s = id.
In the Lie group SL2 (C), take the element
0 1
g0 = exp (E F ) = .
2 1 0
One easily veries that (Ad g0 )(X) = g0 Xg01 = X , X sl2 (C). Thus, 0 =
Ad g0 = exp(ad 2 (EF )) is the Weyl involution of sl2 (C) (see Example 2.3). Using
the principal three-dimensional subalgebra, we can dene a similar automorphism
Int g by
= exp(ad (e f )) . (9)
2
To relate it to , 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 Int h we have f = f , where Int g. If = exp(ad x), x
h, then we may take = 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 = Ad g0 , g0
H, then we may take = Ad F (g0 ).
Proof. If = Ad g0 , where g0 H, then we may write F (g0 (h)) = F (g0 hg01 ) =
F (g0 )F (h)F (g0 )1 = F (g0 ) (F (h)), h G, i.e., F g0 = F (g0 ) F . Dierentiating
this relation, we get f = (Ad F (g0 ))f , and so we may set = Ad F (g). In
32 4. Automorphisms of complex semisimple Lie algebras
q0 = q . (11)
r(i) = ri , i = 1, . . . , l . (12)
l
l
l
( ri ei ) = ri ci f(i) = ri fi .
i=1 i=1 i=1
4. Automorphisms of complex semisimple Lie algebras 33
Then
w0 (diag(x1 , . . . , xl , x1 , . . . , xl ))
= diag(x1 , . . . , xl1 , xl , x1 , . . . , xl1 , xl ) .
Proof. (i) First we show the existence of a maximal toral subalgebra t g which
is invariant under . By Proposition 3.7, there exists a -invariant compact form
u of g. Since 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}
t0 = t u = zu (a) .
Let us denote
k = (g0 )+ , p = (g0 ) .
Since = coincides with on g0 , we have k = u+ , p = iu . Thus, the
decompositions (1) have the form
g0 = k p , (2)
u = k ip . (3)
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
Clearly, b is the restriction of the positive denite Hermitian form h and hence
is a scalar product (a positive denite symmetric bilinear form). It follows that
Now let us start with a real semisimple Lie algebra g0 . A direct sum decompo-
sition
g0 = k p (7)
is called a Cartan decomposition if it is a Z2 -grading and if the Killing form k = kg0
satises (6). As we have seen above, the decomposition (2) of g0 constructed with
the help of a compact real form u of the complexication 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
The projection (x, (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) = ( )1 for any Aut g0 . In particular, = .
(ii) ad (x) = (ad x) for any x g0 .
Proof.
(i) (1 x, y) = k(1 x, y)
= k(x, y) = (x, y)
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 denite
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 suces to prove that d0 exp : S(E) S(E) is
an isomorphism for any 0 S(E). Take S(E) such that (d0 exp) = 0.
5. Cartan decompositions and maximal compact subgroups 39
Denoting (t) = 0 + t and (t) = exp (t), we have (t)(t) = (t)(t) for all
t R. Dierentiating this relation at t = 0, we get
exp 0 + 0 (0) = (0)0 + (exp 0 ).
Since (0) = (d0 exp) = 0, this implies that commutes with exp 0 , and
hence with 0 . Therefore (t) = (exp 0 )(exp t), whence 0 = (0) = (exp 0 ),
and so = 0.
Proposition 2. Let E be an euclidean vector space. Then the mapping : O(E)
P(E) GL(E) given by (k, p) = kp and the mapping (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 = pu1 , whence p2 = exp 2s = a a.
It follows that
1 1 1
s = log(a a), p = exp( log(a a)) , u = a exp( log(a a)) . (10)
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 decom-
position (7). Consider the involutive automorphism 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 = { Aut g0 |
= }, and ad k ad g0 is the Lie subalgebra corresponding to K.
The multiplication mapping 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 : K P Aut g0 and (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 = { Int g0 | = } .
Proof. To prove our claim concerning Aut g0 , it suces 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 (z) = z,
i.e., z p.
Since the mapping (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 (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 dene an analytic action T of the
linear Lie group G = Aut g0 on the manifold P given by
n
r(, ) = i gii , P(E) . (14)
i=1
Proof. We x P(E) and use the expression (14). Since the set of all orthonor-
mal bases (i.e., O(E)) and are compact, there exists a constant b > 0 such that
gii b, i = 1, . . . , n, for all and all orthonormal bases of E. Then
n
r(, ) b i b , P(E) , .
i=1
f (b) f (a) bt ta
f (t) < f (a) + (t a) = f (a) + f (b) , a < t < b. (16)
ba ba ba
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
bt ta bt ta
f (t) = F (t, s(t)) < F (a, s(t)) + F (b, s(t)) f (a) + f (b) .
ba ba ba ba
We need this lemma to establish the following important fact.
Lemma 5. Choose an P(E) and a compact P(E). Then the function
f (t) = ( t ), t R,
is strictly convex.
Proof. Using an appropriate orthonormal basis of E, we can write, due to (14),
n
n
r( t , ) = gii ti = gii et log i ,
i=1 i=1
where i > 0 are the eigenvalues of . Since gii > 0, the analytic function
t r( t , ) is strictly convex for any xed P(E). By Lemma 4, f (t) =
max r( t , ) 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
on P given by (13), where is a compact subset of P . Applying Lemma 3 to
the subset F = P , we see that there exists 0 P such that (0 ) () for
all P . We want to prove that 0 is the only minimum point of .
One sees immediately from (11) that r is invariant under the action T of
G = Aut g0 on P , i.e.,
It follows that
(T ()) = T1 () () , G. (17)
Since the action is transitive, we may assume that 0 = e. Suppose that P is
another minimum point of . Consider the function
f (t) = ( 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 the orbit L(e) = { | L} of L under the action (11).
This is an L-invariant compact subset of P . By (17), the corresponding function
is L-invariant. Thus, L(0 ) = 0 , and the existence of the point xed 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 nish, we note that the following generalization of Theorem 3 is true: any two
maximal compact subgroups of a Lie group G with a nite 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 al-
gebras under homomorphisms of these algebras. More precisely, suppose a homo-
morphism f : g h of complex semisimple Lie algebras and a real form g0 of g be
xed. 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 and be real structures in g and h, respectively. Then f (g ) h if and
only if f = f . In fact, if the latter relation holds, then we get immediately
f (x) = (f (x)) for any x g . Conversely, suppose that f (g ) h . Then f
and 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 and be real structures in g and h, respectively. Then f (g )
h if and only if f .
Now we note certain simple properties of the extension relation, assuming that
f : g h is xed.
Proposition 2.
(i) If a a and b b , then aa bb .
1
(ii) If a a and a and a are invertible, then a1 a .
(iii) An operator a in h extends idg , i.e., idg a if and only if a |f (g) = id.
(iv) For any Int g, there exists Int h such that . If = exp(ad x),
where x g, then we may take = exp(ad f (x)).
(v) If idg , where P(h) relative to a compact real structure in h, then
idg 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 = Ad g, where g G,
then we may write F (g (h)) = F (ghg 1 ) = F (g)F (h)F (g)1 = F (g) (F (h)), h
G, i.e., F g = F (g) F . Dierentiating this relation, we get f = (Ad F (g))f , and
so we may set = Ad F (g). In the case when = exp(ad x) = Ad(exp x), x g,
we have = Ad F (exp x) = exp(ad f (x)).
To prove (v), we note that, by (iii), idg means |f (g) = id. Thus, f (g) lies
in the eigenspace of corresponding to the eigenvalue 1. Clearly, all t , t R,
have the only eigenvalue 1 in this space. Hence idg t , t R.
6. Homomorphisms and involutions 45
Proof. Let us denote by and the real structures determining g0 and h0 , re-
spectively. By Proposition 6.3, there exists such an Int g that 1 = 1
commutes with . Then = 1 is an involution corresponding to g0 .
First suppose that f (g0 ) h0 . Then . By Proposition 2 (iv), ,
where Int h, and hence 1 1 . Therefore we may assume that 1 = ,
and = . Consider now = ( )2 Aut h P(h) (see Lemma 3.2 (i)).
1
By Proposition 3.6, 1 = 1 , where = 4 , commutes with , and then
= 1 is an involution corresponding to h0 . But by Proposition 2 (i) e = 2 ,
and, by Proposition 2 (v), this yields e . Using Proposition 2 again, we see
that 1 , and hence .
Conversely, suppose that extends to an involution corresponding to h0 .
Consider = ( )2 Aut h P(h) (see Lemma 3.2 (ii)). By Proposition 3.7,
1
1 = 1 , where = 4 , commutes with . By Proposition 2 (i), e =
2 , which implies that e . It follows that 1 , and hence 1 1 =
1 . By Proposition 2 (iv), , where Int h. Therefore 1 1 ,
1
and hence f (g0 ) (h1 ). On the other hand, corresponds to both real
structures and 1 , and Theorem 3.2 implies that h1 is conjugate to h0 by an
inner automorphism.
matrix form : g sln (C) relative to a basis of V . If we change the basis, then
the new matrix form will be x C (x)C 1 = (Ad C)(x), x g, where the
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 .
Proposition 1. Let : 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 : g sln (C) the following conditions are satised:
0 ; (2)
0 ; (3)
(x), x t , are diagonal matrices. (4)
0 . (5)
Proof. First note that the equivalence of (2) and (6) and of (3) and (7) follows
from Proposition 6.1. Clearly, (6) means that (g ) 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 is irreducible. As was men-
tioned above, it follows from (III.11) that V admits a basis in which all (x), x
g , are expressed by real matrices. Then the corresponding matrix form of sat-
ises (6), and hence (2). Now, it follows from Proposition 6.4 that there exists
a unique compact real structure on sln (C) which extends . Since = ,
we see from the same proposition that 0 = 0 . But the standard compact
real structure 0 also commutes with 0 . By Proposition 3.6, = 0 1 , where
Int sln (C) and 0 = 0 . Then the matrix representation 1 = 1 satises
both (2) and (3). In fact, we have
1 = 1 = 1 0 = 0 1 = 0 1 ,
1 = 1 = 1 = 0 1 = 0 1 .
these (x) are symmetric real matrices. Since they form a commutative family,
there exists a matrix D On such that all the matrices from D(t(R))D1 are
real diagonal ones. Going over to the new basis with transition matrix D1 , 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 satises 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 is irreducible.
Proposition 2. Let : g sl(V ) be an irreducible linear representation of a com-
plex semisimple Lie algebra g. Suppose that we x a basis of V such that (2) and
(3) are satised. 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 repre-
sentation = 1 : g sln (C), where = Ad C. Clearly, the relations 0
and 0 imply, respectively, 0 1 and 0 1 . It follows from
Proposition 6.4 that 0 1 = 0 . Thus, = 0 1 commutes with 0 , and
we get an involution = 0 . Clearly, . Since = 0 1 , (5) and
Theorem 6.2 imply that = 0 . Hence, = 0 . Thus, commutes with 0 , 0
and 0 .
This allows to deduce our assertion, using simple calculations with matrices.
For any X sln (C), we have
C XC 1 = CXC 1 = C X C 1 ,
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 of a complex semisimple Lie algebra
g we have
,
where Aut is the automorphism of the Dynkin diagram of g described in
Proposition 4.4. If is irreducible with highest weight , then the highest weights
of is ().
Proof. The condition (5) means that ((x)) = (x) . Thus, in the basis, chosen
in Proposition 1, has the same matrix form as (see (III.1)). It follows that
7. Inclusions between real forms 53
Thus, a = ei(t) .
1
(iii) It follows from (4) that the matrix A = e 2 i(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 nd the outer automorphisms of sln (C) to which extends
by an irreducible representation and to deduce a formula expressing the index
j(g, , ) in terms of the highest weight of . Assuming that is of the form (8),
we denote
s0 = s = s (10)
(see Proposition 4.4 (iii)). Clearly, s20 = e.
Theorem 1. Let : g sln (C) be an irreducible representation of a complex
semisimple Lie algebra g with highest weight , and let denote an involutive
automorphism of g of the form (8). An outer involutive automorphism of sln (C)
satisfying exists if and only if
s0 () = . (11)
= (Ad R(z))s = 0 .
Hence there exists B GLn (C) satisfying = (Ad B)0 . We show that the
outer automorphism = (Ad B)0 is involutive. Clearly, e = 2 . But
2
2 2
Int sln (C). Since is irreducible, = 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, s commutes with and , and hence the left side of (12)
satises the conditions (2) and (3). Clearly, the same is true for 0 . Therefore,
by Proposition 2, we may assume that B On . Under this assumption, we clearly
have
B 2 = jIn . (14)
Now, by (5), 0 = . Using Proposition 4 (iii) and Proposition 4.3 (v), we
get from (13) the following relation:
(Ad A)s = (Ad B) exp(ad (e f )) .
2
s = Ad(ABR(g)) , (15)
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 v of , we get
since
l
(h) = ri i Z , (17)
i=1
by (4.5).
One more relation we can obtain if we apply the rst and the last terms of (12)
to u = 2 it g. We get (u) = (Ad B)((u)) = B(u)B 1 . By exponentiating,
one gets R(z) = BR(z)1 B 1 . Using (9), we get BAB 1 = ei(t) A. It follows
that ei(t) = 1. Hence (t) Z and
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 s, s
Aut , acts on s trivially. It follows from (14) that the matrix
C = ABR(g)
56 7. Inclusions between real forms
commutes with any (x), x s. Hence C leaves invariant any weight subspace
of |s. The weights of |s are restrictions of the weights of to hR and may
be identied with the corresponding eigenvalues of the operator (h). Thus, for
l
any we identify |hR with the integer (h) = i=1 ri (hi ), where ri
are dened by (4.5). In particular, to the highest weight of there corresponds
the non-negative integer m = (h) given by (17). By (III.11), any other weight
of has the form = i1 . . . ik , k > 0, and, due to Lemma 4.1, the
corresponding weight of |s is identied with (h) = m 2k < m. Thus, the
highest vector v Cn is the only, up to a scalar factor, weight vector of |s
corresponding to the weight m. It follows that Cv = cv 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 . But v , up to a complex factor, should enter into this basis, since dim V = 1.
Thus, we may assume that v is one of the basic vectors and, in particular, that
v Rn . Thus, v 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
Comparing this with the above result, we see that j(1)(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(ih) and z 2 = exp(it)
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
Note that the integer (h) can be calculated by the formula (17). Then (12) can
be expressed in the following form:
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 rst kind:
1. g0 = so2k,2(lk) , = Ad diag(Ik , Ilk , Ik , Ilk ), where k = 0, . . . , l 2.
2. g0 = ul (H), = Ad Sl .
Since s = e, we see from Proposition 4.4 that for l = 2m the condition (11) is
always satised, while for l = 2m + 1 it is satised, whenever 2m = 2m+1 .
Now, in the case 1 = exp(ad 2 it), where t t(R) is given by k (t) = 2, i (t) =
0 for i = k. One gets from (12) and Table 4 the formula
(1)(m+k)(2m1 +2m ) if l = 2m,
j(g, , ) =
1 if l = 2m + 1.
One also has the following real forms of the second kind:
3. g0 = so2k+1,2(lk)1 , = Ad diag(Ik , Ilk , Ik+1 , Ilk1 ), where k =
0, . . . , l 2.
If l = 2m, then s0 = s = e, and (11) is satised, whenever 2m1 = 2m . If
l = 2m + 1, then s0 = e, and (11) is always satised.
58 7. Inclusions between real forms
To calculate the index, we should present in the form (8). It is easy to verify
that the non-trivial diagram automorphism s is given by s = 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 is conjugate to exp(ad 2 it)s, where
t t(R) is given by k (t) = 2, i (t) = 0 for i = k. Then (12) and Table 4 imply
the following result:
1 if l = 2m,
j(g, , ) =
(1)(m+k)(2m +2m+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 ma-
trices
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
= {1 , . . . , l }, where i = xi xi+1 , i = 1, . . . , l 1, l = 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 rst kind):
g0 = so2k,2(lk)+1 , = Ad diag(Ik , Ilk , Ik , Ilk+1 ), where k = 0, . . . , l.
The condition (11) is always satised.
We can write = exp(ad 2 it), where t t(R) is given by k (t) = 2, i (t) = 0
for i = k. Using (12) and Table 4, we get
1
j(g, , ) = (1)(k+ 2 l(l+1))l .
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 ma-
trices
H = diag(x1 , . . . , xl , x1 , . . . , xl ).
The system of roots is = {xi xj (i = j), 2xi }. We will use the system of
simple roots
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 rst kind):
1. g0 = spp,lp , = Ad diag(Ip , Ilp , Ip , Ilp ), p = 0, . . . , l 1.
2. g0 = sp(R), = 0 |g.
The condition (11) is always satised.
In the case 1, we have = exp(ad 2 it), where t t(R) is given by p (t) =
2, i (t) = 0 for i = p. Using (12) and Table 4, we get
1 +3 +...+2[ 1 (l+1)]1
j(g, , ) = (1) 2 .
j(g, , ) = 1.
Example 5. Let G be a simply connected complex semisimple Lie group and g its
Lie algebra. Let be an irreducible representation of g satisfying the condition
(11). Due to Remark 1, the Karpelevich index of 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, , ) = 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), 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 : gg sl(V ). Consider the involution : (x, y) (y, x) of gg
which corresponds to the real form of this algebra isomorphic to gR (see Example
3.4). Using Theorem 1, one can nd the condition, under which extends to an
outer involutive automorphism of sl(V ), and calculate the index.
Any irreducible representation of g g has the form = 1 2 , where
i , 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 is = (1 , 2 ), where i is the highest
weight of i , i = 1, 2. Clearly, = s is a canonical automorphism determined
by the transposition s of two copies of the system of simple roots of g, and
s() = (2 , 1 ). By (10) and Proposition 4.4 (ii), s0 () = ((2 ), (1 )). It
follows that the condition (11) has the form 2 = (1 ) or, equivalently, 2 1.
Using (12), we see that j(g g, s, 1 1 ) = (1)
1 (h)+(1 )(h)
= (1)21 (h) = 1.
Thus, under the above condition, 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 be an irreducible representation of g0 (C). The second case is studied
in Example 6. In the rst case, the involution s0 and the index j(g, , ) 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(lk)
so2(lk),2k+1 , and therefore the index formula for g0 = sop,2l+1p ,
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, k an
involutive automorphism of gk and s0k Aut k the corresponding involution of
the system of simple roots k of gk (see (10)). Then the involution s0 induced by
s01 and s02 on the system of simple roots = 1 2 of g = g1 g2 corresponds
to the automorphism = 1 2 of g. If k is an irreducible linear representation
of gk , k = 1, 2, then
Next, we apply Theorem 1 to study of bilinear invariants of an irreducible
representation. Let : 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 if
b(u, v) = (u)(v) , u, v V,
is a bilinear form on V , and any bilinear form is obtained in this way. Clearly, non-
degenerate bilinear forms correspond to isomorphisms V V . Further, bilinear
forms invariant under correspond to the mappings satisfying
(x) = (x) , x g.
7. Inclusions between real forms 61
In the case when det B = 0, the relation (21) can also be written in the form
= (Ad B 1 )0 ,
R (g) = tr R(g) , g G.
One of these formulas also exploits the invariant integration over a maximal com-
pact subgroup of G (see 1 I).
62 7. Inclusions between real forms
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 = g is a compact
real form of g invariant under , and the corresponding maximal compact subgroup
U G is invariant under . 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
Therefore, B0 = qp
n In . Since B = B 1 , we get from (26)
since they act in distinct complex vector spaces and, as we shall see later, are not
necessarily equivalent. At the same time, ()R = R .
Let us nd 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
(x), x g0 . Then (x) is expressed in the same basis, regarded as a basis of V ,
by the complex conjugate matrix C(x). Thus, (x) has the following matrix form:
= 0 , (1)
where by 0 the real structure X X in the Lie algebra gln (C) is denoted.
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 : g0 gl(V ), we have
(R )C + .
Proof. Consider the complex vector space W = V V and dene 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 + . 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 0 induced on Wd onto R . Thus, (R )C C 0 = + .
Theorem 1. Any irreducible real representation : g0 gl(V0 ) of a real Lie algebra
g0 satises precisely one of the following two conditions:
(i) C is an irreducible complex representation;
(ii) = R , where is an irreducible complex representation admitting no in-
variant real structures.
Conversely, any real representation satisfying (i) or (ii) is irreducible.
Proof. Suppose that is irreducible, but C is reducible. We want to prove that
V0 admits a complex structure invariant under . Fix a non-zero complex vector
subspace W V = V0 (C) invariant under C . Using the complex conjugation with
respect to V0 , we can construct the complex vector subspaces W , Y = W + W
and Z = W W of V invariant under C . We have Y = Y, Z = Z, whence
Y = (Y V0 )(C) and Z = (Z V0 )(C). Since is irreducible, this implies
that Y V0 = V0 and Z V0 = {0}. It follows that V = Y = W W , and
V0 = {u + u | u W }. Then we can dene a complex structure J on V0 by
J(u + u) = i(u u), u W . Clearly, we have
the Schur Lemma implies that S1 S 1 = de, where d C . We get S1 = dS, and
hence S12 = (dS)(dS) = (dd)S 2 = (|d|2 c)e. Clearly, sgn(|d|2 c) = sgn c.
(iii) By
(ii), we 1have S1 = S22 = e implies |d| = 1. If x V S2 ,
= dS2 , and S12
then S1 ( d x) = d S1 x = d x, and thus d x V S1 .
As was shown in the proof of this theorem, we may assume that B On satises
(7.13). One deduces from (3) that
0 (x) = S0 (x)S 1 , x g0 .
S 2 = j(g, , )In .
Therefore
(x, x)Z = 1 (x)Z + Z1 (x) , 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 : g gl(V ) be an irreducible complex representation of a
complex semisimple Lie algebra g and dim V > 1. Then the real representation
R 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 . For any x g,
we have (ix)S = i(x)S = iS(x). But, on the other hand, (ix)S = S(ix) =
S(i(x)) = iS(x). Thus, is trivial, and dim V = 1. This construction is
actually a special case of Example 2, since = 2 , where 2 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 0
8. Real representations of real semisimple Lie algebras 71
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 x
g0 and regard (0 ) as a function () of the highest weight of = 0 (C). By
Theorem 3, this function is dened on the subsemigroup (g0 ) of the semigroup of
all dominant weights of g = g0 (C) given by the equation s0 () = . By the same
theorem, (0 ) coincides with the Karpelevich index j(g, , ) which is expressed
by (7.12). It follows that ( + ) = ()( ), i.e., is a homomorphism of
(g0 ) into the group {1, 1}. We also note the following property of the index.
Lemma 1. For any dominant weight of g, we have ( + s0 ()) = 1.
Proof. We should prove that ( + s0 ())(t + h) is even. Using s(t) = t, we get
1 , . . . , r , r+1 , . . . , 2r , 2r+1 , . . . , l ,
where
s0 (i) = r + i , i = 1, . . . , r ; s0 (k) = k , k = 2r + 1, . . . , l .
i + r+i , i = 1, . . . , r ; k , k = 2r + 1, . . . , l ,
This formula goes back to E. Cartan [3] and was proved by Iwahori [13] (see
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 classication of real semisimple Lie algebras is pro-
vided 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 complexication 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 = s (see
(7.10)) of the Dynkin diagram of g from the Satake diagram of its real form g0
which corresponds to an involutive automorphism of g inducing the symmetry
s Aut 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 rep-
resentation 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
complexication g. The Cartan correspondence described in 3 implies that there
are the following data: a compact structure , a real structure and an involution
of g, such that = , g0 = g and , and commute pairwise. Thus,
(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 satises (t0 ) = t0 and has the form t0 = t+ 0 a, where t0 k. Moreover,
+
c = { | () = } = { | | a = 0} (1)
are called compact roots, while the roots from nc = \ c are called non-
compact ones. The system of positive roots + can be chosen in such a way that
() + for each non-compact + . To obtain such a system + , it
suces 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 rst elements constitute
a base of a. Let + be the corresponding set of simple roots, and let us
denote
c = c , nc = nc ,
c = c , nc = nc .
(+
nc ) = nc . (2)
(, )
r () = 2 , (4)
(, )
where 2 (,)
(,) Z. Clearly, r () | a = 0, and hence r () nc . By (II.20),
r () + , whenever nc . Thus, the assertion (i) is proved for w = r . By
+
9. Appendix on Satake diagrams 75
where k are non-negative integers. Applying and using Lemma 1 and (1),
we get
s0 () = () + c k ,
c c
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
A1 sl2 (C) {e}
Al , l 2 sll+1 (C) . . . Z2
D4 so8 (C) .. S3
..
Dl , l 5 so2l (C) . . . .. Z2
..
E6 Z2
E7 {e}
E8 {e}
F4 < {e}
G2 < {e}
Table 2
Relations between , C , (C) (V0 is over R)
Table 3
Relations between , R , (C) (V is over C)
Table 4
Elements h t(R) of the principal sl2 subalgebra
Type Coecients ri
l 2(l1) 3(l2) (l1)2 l
Al . . .
2l 2(2l1) 3(2l2) (l1)(l+2) l(l+1)/2
Bl , l 2 . . . >
2l1 2(2l2) 3(2l3) (l1)(l+1) ll
Cl , l 2 . . . <
(l1)l/2
yy
2l2 2(2l3) 3(2l4) (l3)(l+2)
yyy
Dl , l 4 . . . GyG(l2)(l+1)
GG
GG
l(l1)/2
16 30 42 30 16
E6
22
27 52 75 96 66 34
E7
49
Table 5
Indices of irreducible representations of simple complex Lie algebras
Real form Satake diagram with a weight s0 s Index
1 2 l1 l
sll+1 (R) e = e = e +1
slm (H) m
1 2 3 l1 l
e = e = e (1) i=1 2i1
l = 2m 1
1 2 p p+1
O O O
sup,l+1p = e = e e
1 p 2l
l l1 lp l+1p
l even +1
l = 2m 1 (1)(m+p)m
1 2 p1
sup,p O O O LLL
l = 2p 1 L = e = e
r p e +1
p2 rrr
2p 2p1 p+1
1 2 l1 l
sul+1 = e = e e
l even +1
l = 2m 1 (1)mm
sop,2l+1p 1 p p+1 l1 l
> e e e
1pl
l(l+1)
p = 2k (1)(k+ 2 )l
l(l+3)
p = 2k + 1 (1)(k+ 2 )l
1 2 l1 l l(l+1)
l
so2l+1 > e e e (1) 2
spp,lp [ l+1
2
]
1 2 3 2p 2p+1 l1 l 2i1
< e e e (1) i=1
1 p l1
2
spp,p 1 2 3 2p2 2p1 2p p
2i1
< e e e (1) i=1
l = 2p
1 2 l1 l
sp2l (R) < e e e +1
[ l+1
2
]
1 2 l1 l 2i1
spl < e e e (1) i=1
(continued)
80 Tables
Table 5 continued
Real form Satake diagram with a weight s0 s Index
l1
sop,2lp {{
{
1 2 p p+1
1pl2 EEl2
E
l
p, l even e e lp
e (1) 2 (l1 +l )
p, l odd = e = e
p even, l odd = e e
= e +1
p odd, l even e = e
O l1
1 2 l2zzz
sol1,l+1 FFF
l
l even e = e
= e +1
l odd = e e
l1
1 2 l2zzz
sol,l FFF
l
l even e e
e +1
l odd = e = e
l1
1 2 l2zzz
so2l FFF
l
l
l even e e e (1) 2 (l1 +l )
l odd e = e
= e +1
l1
ul (H) 1 2 3 l3 l2zzz m
2i1
l = 2m FFF e e e (1) i=1
O l1
ul (H) 1 2 3 l3 l2zzz m
FFF = e = e e (1) i=1 2i1
l = 2m + 1
l
(continued)
Tables 81
Table 5 continued
Real form Satake diagram with a weight s0 s Index
1 2 3 4 5
EI e = e = e +1
6
EIII x &
= e = e e +1
1 2 3 4 5
6
1 2 3 4 5
EIV e = e = e +1
6
1 2 3 4 5
compact
= e = e e +1
form of E6
6
1 2 3 4 5 6
EV e e e +1
7
1 2 3 4 5 6
EVI e e e (1)1 +3 +7
7
1 2 3 4 5 6
EVII e e e +1
7
1 2 3 4 5 6
compact
e e e (1)1 +3 +7
form of E7
7
(continued)
82 Tables
Table 5 continued
Real form Satake diagram with a weight s0 s Index
1 2 3 4 5 6 7
EVIII e e e +1
8
1 2 3 4 5 6 7
EIX e e e +1
8
1 2 3 4 5 6 7
compact
e e e +1
form of E8
8
1 2 3 4
FI < e e e +1
1 2 3 4
F II < e e e +1
compact 1 2 3 4
< e e e +1
form of F4
1 2
G < e e e +1
compact 1 2
< e e e +1
form of G2
References
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Soc. Math. 41 (1913), 5394.
[3] E. Cartan, Les groupes projectifs continus reels qui ne laissent invariante aucune multi-
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[4] E. Cartan, Les groupes reels simples, nis et continus, Ann. Ec. Norm. Sup. 31 (1914),
263355.
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semisimple Lie groups, Dokl. Akad. Nauk SSSR 76 (1951), 221224. (Russian)
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symmetric space, Dokl. Akad. Nauk SSSR 93 (1953), 401404. (Russian)
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213.
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Subject Index
Algebra, adjoint 1
of inner derivations 1
antiautomorphism 12, 13
, involutive 12
antiinvolution 12, 13
automorphism, corresponding to a real structure 20
, diagram 27
, inner of a Lie algebra 1
, group 1
, outer of a Lie algebra 1
of a system of simple roots 27
Base of a system of roots 7
Cartan decomposition 36
index 67
matrix 7
character 61
complexification 12, 64
component of a representation, irreducible 9
conjugation, complex 12
coroot 5
Dynkin diagram 8
of an irreducible representation 10
Element, regular 6
, singular 6
embedding, canonical 46
extension of a mapping 44
Form, bilinear (sesquilinear) invariant on a Lie algebra 1
, invariant under a representation 60
, linear dominant 10
, real 12, 13
, normal (split) 17
, of the first kind 48
, of the second kind 48
forms, real compatible 24
, of the same kind 47
function, strictly convex, 41
Group, linear adjoint 1
Integral, invariant 2
involution 12
Karpelevich index 53
Killing form 1
Lie algebra, compact 3
, complex conjugate 14
, reductive 2
, semisimple 2
group, semisimple 2
Operator, adjoint 22
, Hermitian, 22
Subject Index 85
, positive definite, 22
, self-adjoint, 22
, symmetric, 22
Presentation of an automorphism, canonical 34
product, scalar 2
of representations, tensor 11
Rank of a semisimple Lie algebra 4
realification 12, 64
representation of a Lie algebra 8
, adjoint 1
, basic 10
, completely reducible 9
, complex 64
, conjugate 64
, dual (contragredient) 9
, irreducible 9, 64
, matrix form 9
, orthogonal 61
, real 64
, self-conjugate 66
, self-dual 53
, symplectic 61
group, adjoint, 1
representations, equivalent (isomorphic) 8, 64
root 4
, compact 73
, negative 7
, non-compact 73
, positive 7
, simple 7
space 4
decomposition 4
vector 5
R-subalgebra 48
S-homomorphism 47
S-subalgebra 48
Satake diagram 74
signature 53
sl2 -triple 5
structure, complex 12, 13
, invariant 64
, quaternion 14
, invariant 64
, real 12, 13
, compact 23
, invariant 64
subalgebra, maximal toral 4
, principal three-dimensional 31
, regular 48
subrepresentation 9
sum of representations 9
system of generators, canonical 4
roots 4
weights 10
86 Subject Index
Theorem of Weyl 2
Vector, highest 10
space, complex conjugate 12
subspace, invariant 9
Weight, fundamental 10
, highest 10, 68
, lowest 71
space decomposition 9
subspace 10
Weyl chamber 6
group 6
involution 18
Z2 -grading 3