Finite and ane Kac-Moody symmetric spaces

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Finite and ane Kac-Moody symmetric spaces
Lecture notes by Walter Freyn
June 24,2011
Abstract
These lecture notes give an introduction to nite and innite dimensional symmet-
ric spaces.Emphasis is put on a detailed description of the geometry.It is explained
that nite dimensional Riemannian symmetric spaces and innite dimensional ane
Kac-Moody symmetric spaces are closely related sharing most fundamental properties
of their geometry and classication.
1 Introduction
These lecture notes on symmetric spaces were written as a supplement to a mini course
given by the author at the Rutgers University,New Jersey in March/April 2011.The
principal aimof the course was to introduce students to the most important concepts of the
geometry of nite and innite dimensional symmetric spaces.In this course we wanted to
emphasize the crucial fact that nite dimensional Riemannian symmetric spaces and ane
Kac-Moody symmetric spaces share the same structure and especially the same geometric
building blocks,which can be described quite easily in terms of rank 1-symmetric spaces
and the structure of ats respective in terms of their rank 1-and rank-2 building blocks.In
this way we can understand the geometry of those spaces in a very clear intuitive way.Due
to a lack of time,there was no chance to prove the important structure theorems about
symmetric spaces.The interested student is encouraged to look for more information in the
books [Hel01] or [Loo69b,Loo69a].Hence those notes are not assumed to be a complete
course on symmetric spaces,but give an introductory overview and should be read together
with complementary texts.Numerous references and a guide to the literature (appendix
A) are designed to help the student to nd suitable texts.My special thanks go to Lisa
Carbone (Rutgers university) who made this mini course possible.
1
Contents
1 Introduction 1
2 Symmetric spaces 3
2.1 Introduction....................................3
2.2 Foundations....................................6
2.3 The A
1
-family - an extended example.....................9
2.4 Towards the classication............................13
3 The geometric structure: ats,Weyl groups,local models and blow-up 15
4 The classication of Riemannian symmetric spaces 23
4.1 The easy classical spaces.............................23
4.2 Other classical spaces..............................23
4.3 The easy exceptional spaces...........................23
4.4 The other exceptional spaces..........................24
5 Ane Kac-Moody symmetric spaces 25
5.1 Kac-Moody groups and their Lie algebras...................25
5.1.1 Geometric ane Kac-Moody algebras.................25
5.1.2 Ane Kac-Moody groups........................29
5.2 Kac-Moody symmetric spaces..........................33
6 The geometry of Kac-Moody symmetric spaces 38
6.1 A rst example:
d
MSU
n
.............................38
6.2 The noncompact dual
d
MSL(n;C=
d
MSU(n)..................43
6.3 The general case.................................43
A Guide to the literature 46
A.1 Geometry and nite dimensional symmetric spaces..............46
A.2 Kac-Moody algebras and loop groups......................47
A.3 Functional analysis................................47
A.4 Kac-Moody geometry..............................48
2
2 Symmetric spaces
2.1 Introduction
Riemannian symmetric spaces are fundamental objects in nite dimensional dierential
geometry,displaying numerous connections with Lie theory,physics,and analysis.The
search for innite dimensional symmetric spaces associated to ane Kac-Moody algebras
has been an open problem for 20 years [Ter95].The author has given a complete solution
to this problem in his thesis.In these lecture notes we focus on an easy to go introduc-
tion,emphasizing the unifying structure properties of nite dimensional Riemannian and
innite dimensional Kac-Moody symmetric spaces.The fundamental observation is,that
vital structure properties and geometric structures from nite dimensional Riemannian
symmetric spaces extend to this setting.Important aspects of the geometry can be de-
scribed in both cases by studying only subspaces of rank 1 and rank 2 respective\ ats"
and subspaces of rank 1.
Symmetric spaces are homogeneous spaces;their symmetry groups are semisimple Lie
groups,the Euclidean group and ane Kac-Moody groups.The investigation of these
groups has a long history.Finite dimensional Lie groups and Lie algebras where known
since the 19'th century;important investigations were performed by S.Lie,F.Engel,
W.Killing and

E.Cartan.The quest for innite dimensional Lie algebras and Lie groups
was supported by

E.Cartan,who published several papers dedicated to their investigation.
The innite dimensional Lie algebras,which are nowadays called Kac-Moody algebras,
were introduced and rst studied in the 60's independently by V.G.Kac [Kac68],R.V.
Moody [Moo69],I.L.Kantor [Kan70] and D.-N.Verma (unpublished) as a generalization
of semisimple Lie algebras.The wide class of Kac-Moody algebras contains various sub-
classes of independent interest.For example,in the framework of Kac-Moody algebras,
the class of nite-dimensional semisimple Lie algebras corresponds just to Kac-Moody al-
gebras of so-called spherical type.Kac-Moody algebras of spherical type are precisely the
nite dimensional examples.The most studied innite dimensional subclass is the one
of ane Kac-Moody algebras resp.ane Kac-Moody groups.Ane Kac-Moody groups
can be described as certain torus extensions
b
L(G;) of (possibly twisted) loop groups
L(G;).Correspondingly,ane Kac-Moody algebras are 2-dimensional extensions of
(possibly twisted) loop algebras L(g;).In this notation G denotes a compact or complex
simple Lie group,g its Lie algebra and  2 Aut(G) a diagram automorphism,dening the
\twist".Depending on the regularity assumptions on the loops (e.g.of Sobolev class H
k
),
one gets families of completions of the minimal (=algebraic) ane Kac-Moody groups
\
L
alg
G

.The minimal algebraic loop group just consists of Laurent polynomials.Fol-
lowing Jacques Tits,completions dened by imposing regularity conditions on the loops
are called\analytic completions"in contrast to the more algebraic formal completion.
Various analytic completions and objects closely related to them play an important role
in dierent branches of mathematics and physics,especially quantum eld theory,inte-
grable systems and dierential geometry.In most cases their use is motivated by the
requirement to use functional analytic methods or by the need to work with manifolds
and Lie groups,see [PS86],[Gue97],[SW85],[Tsv03],[Pop05],[PT88],[Kob11],[KW09],
[HPTT95],[HL99],[Hei06] and references therein.Usually properties of the application in
question naturally leads to a well-dened type of completion.
From their construction,ane Kac-Moody algebras and simple Lie algebras share
important parts of their structure properties.The aim of Kac-Moody geometry is,to
investigate to which extend these similarities extend to geometric objects whose structure
is determined by simple Lie algebras (groups) resp.ane Kac-Moody algebras (groups).
3
The geometric objects associated to nite dimensional simple Lie groups were known for
a long time.Ane Kac-Moody geometry claims the existence of innite dimensional
counterparts to these nite dimensional dierential geometric objects,whose symmetries
are now described by real or complex simple Lie groups [Hei06,Fre12].The symmetries of
those corresponding innite dimensional objects are the analogs of complex or real ane
Kac-Moody groups.Let us remark,that the existence of similar ojects associated to all
Kac-Moody groups is an important open conjecture.
We want to sketch this picture in a few lines;to this end,we start with the description
of the nite dimensional blueprint,that is the world of geometric objects,determined by
simple Lie groups.
In this nite dimensional situation symmetry groups are (semi-)simple complex or real
Lie groups.A simple complex Lie group has up to conjugation a unique compact real form.
All further real forms are non-compact.The most important (and historically rst) class
of objects attached to simple Lie groups are symmetric spaces.A Riemannian symmetric
space is a Riemannian manifold (M;g),which allows for each point p 2 M an isometry

p
such that d
p
= Id on T
p
M.The classication of Riemannian symmetric spaces
is achieved by associating simply connected Riemannian symmetric spaces to orthogonal
symmetric Lie algebras (OSLA).
Attached to any Riemannian symmetric space,there is a spherical building [AB08].
From a combinatorial point of view,a building is described as a simplicial complex satisfy-
ing several axioms.Geometrically the building corresponding to a symmetric space M has
a clear intuitive description [BH99]:suppose the dimension k of maximal ats (maximal
at subspaces),called the rank,satises k  2 (otherwise the building degenerates to a
set of points).Choose now a point p 2 M.The building can be seen in the tangent space
T
p
M as follows:Consider rst all k-dimensional tangent spaces to maximal ats through
the point p.For a compact simple Lie group choose p = Id the identity,then T
p
M can be
identied with the Lie algebra,maximal ats through the identity are maximal tori and
the tangent spaces to maximal ats are just Abelian subalgebras of the Lie algebra g.One
then removes all intersection points (the so-called singular points).Then each at is parti-
tioned into chambers and the set of all those chambers for all ats through p corresponds
exactly to the chambers of the building.All other cells of the building can be seen in the
set of singular points.Chambers belonging to one at form an apartment.Nevertheless,
in this description,one does not see all apartments.To see all apartments,one has to
project the construction to the sphere at innity,to get a construction independent of the
choice of a base point.
Via the geometric description of the building in the tangent space the connections
between the building and polar actions become manifest:Let M be a symmetric space
and p 2 M.The isotropy representation of M is the representation of K
p
,the group of
all isometries of M xing p on T
p
M:
K:T
p
M !T
p
M;
The easiest class of examples are compact Lie groups:For them the isotropy represen-
tation coincides with the Adjoint representation.
The maximal orbits of the isotropy representation are called\principal orbits".They
contain most of the structure information of the symmetric space.In particular,they have
the following properties:
1.The principal orbits meet the tangent spaces to the ats orthogonally and have com-
plementary dimension.In particular,the normal spaces to the orbits are integrable.
4
Actions with this property are called polar.Conversely,any polar action is orbit
equivalent to the isotropy representation of a symmetric space [BCO03].
2.The geometry of the principal orbits as submanifolds of Euclidean space is par-
ticularly nice and simple.They are so-called isoparametric submanifolds [PT88].
Conversely,by a theorem of Thorbergsson [Tho91],isoparametric submanifolds are
principal orbits of polar representations if they are full,irreducible,and the codi-
mension is not 2.
To summarize the nite dimensional blueprint:
- Polar representations correspond to symmetric spaces,isoparametric submanifolds
correspond roughly to polar representations.
- Chambers in buildings correspond to points in isoparametric submanifolds.
- Buildings describe the structure at innity of symmetric spaces of non-compact type.
- The geometric realization of buildings can be equivariantly embedded via the isotropy
representation into the tangent space of a symmetric space.
Let us now turn to the innite dimensional counterpart:
Ane Kac-Moody groups are distinguished among other classes of innite dimensional Lie
groups by two important properties:
- They share the most important structure properties of nite dimensional simple Lie
groups,i.e.they have BN-pairs,Iwasawa and Bruhat decompositions and highest
weight representations [PS86],[Kac90],[Kum02],[MP95].The main dierences can
be traced back to the Weyl group being nite for Lie groups and innite for Kac-
Moody groups.
- They have good explicit linear realizations in terms of 2-dimensional extensions of
loop algebras and loop groups;thus they allow the use of functional analytic methods
and the denition of manifold structures.The resulting Kac-Moody algebras and
Kac-Moody groups are called of\analytic"type,in contrast for example to formal
completions [Tit84].Depending on the type of completions,analytic Kac-Moody
algebras (groups) are Hilbert-,Banach-,Frechet-,etc.Lie algebras (groups) [PS86].
Work done during the last 25 years by various authors,notably Ernst Heintze and
Chuu-Lian Terng shows that analytic completions of ane Kac-Moody groups describe the
symmetries of various interesting objects;many of these objects have nite dimensional
counterparts,thus conrming the philosophy of Kac-Moody geometry [PS86],[Hei06],
[Fre12],[KW09],[Ter89],[Ter95],[SW85],[Pop06],[HPTT95],[HL99],[Fre09].
The fundamental global objects of this theory are ane Kac-Moody symmetric spaces,
which were introduced by the author in his thesis [Fre09].Their structure theory parallels
closely the theory of nite dimensional Riemannian symmetric spaces.For example they
contain nite dimensional ats,that come equipped with some action of an (ane) Weyl
group.Furthermore there is a theory of polar actions on Hilbert spaces [Ter95] and
proper Fredholm isoparametric submanifolds in Hilbert spaces [Ter89].Principal orbits of
polar actions on Hilbert spaces are proper Fredholmisoparametric submanifolds in Hilbert
spaces.Thus a broad generalization of the nite dimensional blueprint appears.We nd
the same classes of objects,satisfying similar relations among each other and sharing
similar structure properties [Hei06,Fre12].
5
In these notes,we focus exclusively on the symmetric spaces aspect of the theory.We
introduce nite dimensional symmetric spaces,study some of their structure properties
and turn than to innite dimensional Kac-Moody symmetric spaces.We emphasis the
structural properties common to all classes of objects in Kac-Moody geometry.
2.2 Foundations
Denition 1
A (pseudo)-Riemannian manifold is a manifold M together with a metric g:TMTM !
R.A metric is a denite symmetric bilinear form.At each point p 2 M the metric has the
explicit form g
p
:T
p
MT
p
M !R.In a basis a metric can be described by a symmetric
matrix.There is a basis in which the metric is diagonal.If all eigenvalues are positive,the
metric is called Riemannian.If exactly one eigenvalue is negative and all other eigenvalues
are positive the metric is called Lorentzian.
The space of (pseudo)-Riemannian manifolds is huge.It is a very dicult problem to
get a good explicit description of a generic Riemannian manifold.We give an example:
Example 2
Let M = R
n
.The tangential space T
p
M at any point p 2 R
n
is isomorphic to R
n
.Dene
a metric by g:TR
n
TR
n
!R be a matrix
g
p
(~x;~y) = ~x
t
O(p)diag

e
iv
1
(p)
;:::;e
iv
n
(p)

O(p)
t
~y:
Here diag(e
iv
1
(p)
;:::;e
iv
n
(p)) describes a diagonal matrix with positive entries,the eigen-
values of the metric at p.O(p) describes the rotation of the standard basis in the eigenbasis.
Hence the metric is dened by a map'
1
:R
n
!O(n) and by a map'
2
:R
n
!R
n
,
describing the eigenvalues of the metric at each point p 2 R
n
(remark,that this description
is not unique as the eigenvalues may be permuted by some orthogonal matrix).Depending
on our requirements the maps'
1
and'
2
have to satisfy regularity conditions,in example
to be smooth.Now try to describe the resulting geometry for generic choices of'
1
and
'
2
....Good luck!!
Hence it is dicult a dicult problem,to nd examples,one can clearly describe and
extract results about the structure.This is the situation where symmetric spaces spaces
come into play:Because of their huge symmetry group they are very special and relatively
easy to describe.Thus they are a good starting point to get acquainted with dierential
geometry.But of course one has now to ask:Are they really useful or are they too special
to be of any use - especially do they capture essential features of Riemannian manifolds?
The answer two this question is not a theorem but is more philosophical.It turns out to
be twofold:
1.In the space of all Riemannian manifolds symmetric spaces are usually at\special"
points.They prove hence to be important objects in the study of spaces of Rie-
mannian manifolds.They provide examples for many special cases,i.e.holonomy
groups.
2.Most Riemannian manifolds that come fromspecic applications have more structure
and symmetry then an arbitrary Riemannian manifold and a surprising number
of them are symmetric spaces (take coset models in physics,Grassmannians,Real
structures on complex space,complex structures on real space etc.....) Those
explicit examples are reason enough to study symmetric spaces.
6
Hence Riemannian symmetric spaces are on the one hand suciently special to allow
for a good structure theory,a good geometric understanding and a complete classication;
on the other hand they are suciently general to include objects interesting in their own
right and showing prototypical examples for a large variety of geometric situations.
So let us give the denition:
Denition 3
A (pseudo-)Riemannian symmetric space (M;g) is a pseudo-Riemannian manifold M such
that for each m 2 M there is an isometry 
m
:M !M such that 
m
(m) = m and
d
m
j
T
m
M
= Id.
Let us give some examples:
Example 4 (Euclidean space (R
n
;Eucl))
The easiest (but also the only nontypical) example is Euclidean n-space (R
n
;Eucl).Geodesics
are straight lines.The isometry 
0
at the point 0 2 R
n
is dened by 
0
:x !x.Clearly
d
0
j
T
0
R
n = Id.The isometry 
p
at any point p 2 R
n
is dened by the concatenation
of 
0
(x) with the translation x 7!x + p.Hence 
p
(x) = p  (x  p) = 2p  x.Again
we calculate d
p
j
T
p
R
n = Id.The group of isometries is the Euclidean isometry group
E(n)

= R
n
o O(n).Here R
n
denotes the group of translations while O(n) denotes the
group of rotations.
Example 5 (The n-sphere S
n
)
The n-sphere S
n
:= fx 2 R
n+1
jjxj = 1g.The Lie group SO(n + 1) acts transitively on
S
n
.The stabilizer of a point p called the isotropy group Isot(p),is isomorphic to SO(n).
Choose for example the point p = (0;:::;0
|
{z
}
n
;1) 2 S
n
.The stabilizer SO(n)
p
of p can be
embedded into the the group SO(n +1) via
:SO(n),!SO(n +1)
A 7!

A 0
0 1

The isotropy subgroup of any other point q 2 S
n
is isomorphic to SO(n).It can be
realized as a subgroup of SO(n +1) conjugate to the isotropy group of p by any element
r
pq
2 SO(n +1),mapping p to q.As a consequence we can write S
n
as a quotient
S
n
= SO(n +1)=SO(n)
Let us just check the dimensions:As dim(SO(n)) =
1
2
n(n  1) we nd:dim(S
n
) =
dim(So(n +1)) dim(So(n) =
1
2
((n +1)n n(n 1)) = n.The curvature of the sphere
is +1.Hence SO(n) is the typical model space of positive curvature.
Example 6 (The n-hyperbolic space H
n
)
We study R
n;1
,that is (R
n+1
;g
Lorentz
) where g
Lorentz
is dened by
g
Lorentz
=
0
B
B
B
B
@
1 0:::0
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1 0
0:::0 1
1
C
C
C
C
A
7
Then we dene hyperbolic n-space by
H
n
:= fx 2 R
n1
jjxj = 1;x
n+1
> 0g
Hence hyperbolic n-space is one connected component of the sphere of radius 1 in R
n;1
.
Similarly to the sphere of radius k in Euclidean space which has sectional curvature
1
k
2
,the
sphere of radius k has sectional curvature 
1
k
2
.Hence hyperbolic n-space is the prototype
of a negatively curved manifold similarly to the sphere being the prototype of a positively
curved manifold.Similarly to the sphere we have a description as a quotient:
H
n
:= SO(n;1)=SO(n)
where SO(n;1) denotes the group of isometries of R
n+1
xing the origin.
Remark 7 (An exceptional isomorphism and H
2
)
In complex analysis the 2-dimensional hyperbolic space is often introduced as the upper
halfplane:
H
2
:= fz 2 Cj=(z) > 0g
together with the metric g
hyp
(z) =
1
=(z)
2
g
Eucl
.Then one checks that the group SL(2;R) acts
transitively on H
2
;in the upper-half-space model the action is by Moebius transformations.
For any point p 2 H
2
the group of (orientation preserving) isometries xing p is isomorphic
to SO(2).Hence we deduce a description:
H
2
= SL(2;R=SO(2)
The two descriptions are related by the fact that
SL(2;R)

=
SO(2;1)
Attention:This is one of the exceptional isomorphisms between low dimensional Lie
groups.A similar statement is not in general true.SL(n;R) has NOTHING to do with
hyperbolic spaces!!
Regarding the quotient descriptions of the n-sphere and hyperbolic n-space there is a
curious thing:In both cases the quotient is by the same subgroup SO(n).As this subgroup
is in both cases the group,xing a point p,it acts on the tangential space T
p
at p.We
formalize this in the following denition:
Denition 8 (Isotropy group and isotropy representation)
Let M be a symmetric space,I(M) its isometry group.
1.Let I
m
(M)  I(M) denote the group of all isometries that x m.I
m
(M) is called
the isotropy group of m denoted I
m
(M) or Isot
m
(M).
2.The action I
m
M:T
m
M !T
m
M is called the isotropy representation of M.
Let us give two short remarks:
1.A symmetric space M has a description
M

= I(M)=I
m
(M)
2.The isotropy representation is independent of the point m 2 M as the isotropy
groups of dierent points are conjugate.
8
Some easy consequences of the denition of a symmetric space are the following:
1.Symmetric spaces are complete homogeneous spaces.
2.Each geodesic is innitely long.
3.Let g
m
denote the metric on T
m
M.Clearly I
m
(M)  O(g
m
).Hence for a Rieman-
nian symmetric space,we nd that I(M)
m
is a closed subgroup of an orthogonal
group,especially I
m
(M) is compact.This compactness result reappears in the de-
nition of symmetric pairs in section 2.4.
As the isotropy groups of the n-sphere and the hyperbolic n-space are the same,also
their isotropy representations are the same.
Denition 9 (Duality)
Symmetric spaces with isomorphic isotropy representations are called dual.
They have a closely related structure;the\only"dierence is that the curvature tensor
changes sign.
As the isotropy representation is a very important object let us note some further
details:
Denition 10 (polar representation)
A representation of a Lie group G:V !V is called polar if there exists a subspace
  V,called a section that meets each orbit G v;v 2 V orthogonally.
It is easy to check that the isotropy representations of symmetric spaces are polar.
Sections are\ ats"i.e.subspaces isometric to Euclidean R
n
.We meet ats in section 3.
Let us just note that the section (hence the at) denes the structure of the polar action
completely.The action of the Weyl group of the symmetric space on the at is recovered
in the geometric structure of the polar action.
Conversely we have the following result by Jiri Dadok:
Theorem 11
Every polar representation is orbit equivalent to the isotropy representation of a symmetric
space.
Hence polar representations and symmetric spaces are in a certain sense two dierent
ways of looking at the same\absolute"geometric facts.Similar observations are true
for buildings and isoparametric submanifolds - but because of time constraints a detailed
investigation is beyond the scope of this course.The interested reader may consult [Bro02]
for an introduction to buildings and [AB08] for a thorough treatment.
2.3 The A
1
-family - an extended example
All nite dimensional Riemannian symmetric with the only exception of R
n
are related to
semisimple Lie groups,their real forms and involutions.SL(2;C) is the easiest of those
groups and the fundamental building block appearing in the construction of all other simple
Lie groups respective Kac-Moody groups.Similarly on the geometric level the symmetric
spaces associated to this group are the\local building blocks"of all other nite or ane
symmetric spaces.We investigate this further in section 3
9
Denition 12
The complex simple Lie group SL(2;C) is dened by
SL(2;C):=

a b
c d

j a;b;c;d 2 C;ad bc = 1

:
Its Lie algebra is dened by
sl(2;C) =

a b
c d

j a;b;c;d 2 C;a +d = 0

:
As the equation ad bc = 1 is complex-linear it reduces the complex dimension by 1.
Hence SL(2;C) and sl(2;C) are 3-C-dimensional or equivalently 6-R-dimensional.
The exponential function of SL(2;C) is the matrix exponential function,dened by
exp:sl(2;C) !SL(2;C)
A 7!
1
X
n=0
A
n
n!
:
To verify that Im(exp(sl(2;C)))  SL(2;C) we need the easy to check identity
exp(trace(A)) = det (exp(A)) for any matrix A:
Let us rst investigate the topology of SL(2;C):Using the projection onto the rst
factor

a b
c d

7!

a
c

:
we deduce (as a and c can't be both 0) that this map is surjective onto the space
C
2
nf0g

= S
3
 R.The second row of the matrix forms a C-bundle over this space.
As C
2
nf0g

=
S
3
 R is simply connected this C-bundle is topologically trivial.As a
consequence we get the structure result:
SL(2;C)

= S
3
R
3
:
Remark that this is a (topological) direct product of a 3-dimensional nonnegatively
curved compact space and a 3-dimensional nonpositively curved non-compact space.
Now let us study the real forms:Up to conjugation there is exactly one compact real
form:
Example 13 (SU(2))
The special unitary group of rank 1 is dened by
SU(2):=

 




j ; 2 C;jj
2
+jj
2
= 1

:
Hence SU(2) can be described as the 3-dimensional unit sphere in C
2
- this corresponds
exactly to the compact factor of our description of SL(2;C).SU(2) is directly related to
SL(2;C) via the Iwasawa decomposition:
Theorem 14 (Iwasawa decomposition)
SL(2;C) has a decomposition:
SL(2;C) = SU(2) AN
such that A:=

a 0
0 a
1

j a 2 R
+

and N:=

1 n
0 1

j n 2 C

.
10
AN

=
R
3
.In other words the (nontrivial) topology of the complex Lie group SL(2;C)
is completely contained in its compact real form.The rest of the group SL(2;C) is just a
(topologically) direct R
n
-factor.
The second real form is the (split) real form:
Denition 15
The noncompact real form is dened by
SL(2;R) =

a b
c d

j a;b;c;d 2 R;ad bc = 1

:
Having introduced some simple Lie groups and described their structure properties,
let us now look for Riemannian symmetric spaces.To this end we need to nd a suitable
a metric.The common approach to the denition of a metric is in two steps:First we
dene the unique Ad-invariant scalar product on the Lie algebra (resp.the tangent space
of the symmetric space).Then in a second step we extend the description of the metric
to the whole space by left translation.
Any Ad-invariant scalar product on a simple Lie algebra is a multiple of the Cartan-
Killing form:
Denition 16
The Cartan-Killing form B(X;Y ) of a simple Lie algebra is dened by
B(X;Y ) = trace (ad(X)  ad(Y )):
For a Lie algebra in matrix representation we can use up to a constant factor the easier
to evaluate expression
B(X;Y ) = c  trace (XY ):
where c depends on the type of the Lie algebra.
Let us note another important formula:For a Lie group the sectional curvature can
be calculated by an easy formula using the Lie bracket:IF X;Y are orthonormal then we
have
K(X;Y ) =
1
4
k[X;Y ]
2
k:
Doing the calculations for SU(2) shows that SU(2) together with its Ad-invariant
metric is a 3-sphere.As a coset space we can describe this space as
SU(3)

=
SU(2) SU(2)=(SU(2))

=
S
3

=
SO(4)=SO(3):
There is furthermore the quotient space RP(2) = S
2
=fIdg.
In a similar way to SU(2) we nd that the quotient space
H
3
= SL(2;C)=SU(2):
is a symmetric space,namely 3-dimensional hyperbolic space.
Nevertheless besides those three candidates there are still further symmetric spaces in
the SL(2;C) family:Namely there are the spaces of the formG=K where G is a real form.
Then K is up to a nite group the xed point group of an involution.Hence we have
to study involutions of real forms.Again a rigidity property comes to our help:It turns
out that it is enough to study involutions of SU(2).All involutions of complex simple
Lie groups and noncompact real forms may be reduced to the ones of the compact real
11
form.To get a feeling for the structure of these involutions,let us focus on the special
case SU(n):
Let I
n
denote the n-dimensional identity matrix and dene
I
p;q
=

I
p
0
0 I
q

and J
n
=

0 I
n
I
n
0

:
Denote furthermore complex conjugation by :SU(n) !SU(n);g 7!(g) =
g and
let Int(g)(h) = ghg
1
denote conjugation.
Theorem 17
Every involutive automorphism of SU(n) is conjugate to one of the following:
A
R
q
 (n = q +1),
A
C;q
n1
Ad(I
p;q
) (1  q 

n
2

;p +q = n),
A
H
2q+1
  Ad(J
q+1
) n=2(q +1).
A proof can be found in the book [Loo69a].Let us specialize this result to SU(2):
Lemma 18
For SU(2),we have
Int (iJ
1
)   = Int (iJ
1
) = Ad(I
1;1
)
  Ad(J
1
) = Id
Proof.Let g =

 




2 SU(2).
The rst identity follows as
Int (iJ
1
)  

 




=

0 
i 0



 

0 i
i 0

=
=

 



=

1 0
0 1

 




1 0
0 1

= Ad(I
1;1
)

 




:
The second identity follows as
  Ad(J
1
)

 




() =  

0 1
1 0

 




0 1
1 0

=

 




:
Hence,SU(2) has up to conjugation exactly one single nontrivial automorphisms,
namely Ad(I
1;1
).The xed point group of Ad(I
1;1
) consists of all elements
g =

 




2 SU(2)
such that  =
 = 0,hence the abelian group
SO(2):=

 0
0


j jj = 1

:
We thus nd two additional simply connected compact symmetric spaces namely S
2
=
SU(2)=SO(2) and H
2
= SL(2;R)=SO(2).
Collecting these observations we nd the following symmetric spaces associated to the
root systems of type A
1
:
There are two simply connected compact symmetric spaces:
12
Type A
1
The space of type A
1
is isomorphic to SU(2),equipped with its (unique up to
scalar multiple) Ad-invariant metric.
M

=
SU(2)

=
SO(4)=SO(3)

=
S
3
:
Type AI
1
The space of type AI
1
is isomorphic to SU(2)=SO(2),equipped with its nat-
ural Ad-invariant metric.
M

= SU(2)=SO(2)

= S
2
:
Furthermore we have two nonsimply connected compact symmetric spaces (subscript
ns for non-simply):
Type A
1;ns
The space of type A
1
is isomorphic to SO(3),equipped with its (unique up
to scalar multiple) Ad-invariant metric.
M

=
SO(3)

=
RP
3
:
Type AI
1;ns
The space of type AI
1;ns
is isomorphic to SO(3)=SO(2),equipped with its
natural Ad-invariant metric.
M

=
SO(3)=SO(2)

=
RP
2
:
On the other hand we have two noncompact symmetric spaces:
Type A
1
The space of type A
1
is isomorphic to SL(2;C=SU(2),equipped with its (unique
up to scalar multiple) Ad-invariant metric.
M

=
SL(2;C)=SU(2)

=
H
3
:
Type AI
1
The space of type AI
1
is isomorphic to Sl(2;R)=SO(2),equipped with its
natural Ad-invariant metric.
M

= SL(2;R)=SO(2)

= H
2
:
2.4 Towards the classication
The SL(2;C)-family is typical for the whole classication:We have always pairs of Lie
groups (G;K) such that K is a compact subgroup and G is either compact,complex or
noncompact.Formalize those concepts is the rst step on the way to a classication of
symmetric spaces:
Denition 19 (Symmetric pair)
Let G be a connected Lie group,H a closed subgroup.The pair (G;H) is called a symmetric
pair if there exists an involutive analytic automorphism :G !G such that (H

)
0

H  H

.Here H

denotes the xed points of  and (H

)
0
its identity component.If
Ad
G
(H) is compact,it is said to be Riemannian symmetric.
Each symmetric space denes a symmetric pair.Conversely each symmetric pair describes
a symmetric space [Hel01].
Denition 20 (OSLA)
An orthogonal symmetric Lie algebra is a pair (g;s) such that
13
1.g is a Lie algebra over R,
2.s is an involutive automorphism of g,
3.the set of xed points of s,denoted k,is a compactly embedded subalgebra.
Clearly each Riemannian symmetric pair denes an OSLA.The converse is true up to
coverings.
Hence to give a classication of Riemannian symmetric spaces,we just have to classify
OSLA's.
We focus now our attention to the Riemannian case:Call a symmetric space irreducible
if it is not the direct product of two symmetric spaces.The most important result is the
following:
Theorem 21
Let M be an irreducible Riemannian symmetric space.Then either its isometry group is
semisimple or M = R.
In the non-Riemannian case this is no longer true.While the pseudo-Riemannian
symmetric spaces with semisimple isometry group are completely classied [Ber57] recent
results of Ines Kath and Martin Olbricht [KO04],[KO06] show that for pseudo-Riemannian
symmetric spaces with a non-semisimple isometry group a classication needs a complete
classication of solvable Lie algebras,which is out of reach.
There are two classes of irreducible Riemannian symmetric spaces with semisimple
isometry group:Spaces of compact type and spaces of noncompact type.
Denition 22
A Riemannian symmetric space is of compact type i its isometry group is a compact
semisimple Lie group.It is called of noncompact type if its isometry group is a noncompact
semisimple Lie group.
Theorem 23
Let (g;s) be an orthogonal symmetric Lie algebra and (L;U) a Riemannian symmetric
pair associated to (g;s):
i) If (L;U) is of the compact type,then L=U has sectional curvature  0.
ii) If (L;U) is of the noncompact type,then L=U has sectional curvature  0.
iii) If (L;U) is of the Euclidean type,then L=U has sectional curvature = 0.
The Cartan-Hadamard theorem tells us that symmetric spaces of noncompact type
are dieomorphic to a vector space.Hence for every orthogonal symmetric Lie algebra of
noncompact type,there is exactly one symmetric space of noncompact type.In contrast
the topology of symmetric spaces of compact type is nontrivial.As the fundamental group
need not be trivial for one orthogonal symmetric Lie algebra of the compact type there
may be dierent symmetric spaces.There is always a simply connected one which is the
universal cover of all the others.
Symmetric spaces of compact type and noncompact type appear in duality:For every
simply connected,irreducible symmetric space of the compact type,there is exactly one
of the noncompact type and vice versa.
14
Example 24
Let us study (l;u):= (so(n +1);so(n)).A Riemannian symmetric pair associated to (l;u)
is (SO(n +1);SO(n)).The corresponding symmetric space is isomorphic to the quotient
L=U,hence is a sphere.Another symmetric space associated to (so(n +1);so(n)) is the
projective space RP(n).The noncompact dual symmetric space is the hyperbolic space
H
n
= SO(n;1)=SO(n).
Example 25 (The SL(2;C)-family)
For the four simply connected symmetric spaces of the SL(2;C)-family the symmetric pairs
are as follows:
- For the symmetric space SU(2) we have (l;u):= (su(2) su(2);su(2)) and (L;U) =
(SU(2) SU(2);SU(2)).
- For the symmetric space SU(2)=SO(2) we have (l;u):= (su(2);so(2)) and (L;U) =
(SU(2);SO(2)).
- For the symmetric space SL(2;C)=SU(2) we have (l;u):= (sl(2;C);su(2)) and
(L;U) = (SL(2);SU(2)).
- For the symmetric space SL(2;R)=SO(2) we have (l;u):= (sl(2;R);so(2)) and
(L;U) = (SL(2;R);SO(2)).
Besides at R
n
there are four classes of Riemannian symmetric spaces,two classes of
spaces of compact type and two classes of spaces of noncompact type
1.Type I consists of spaces of the form G=K,where G is a compact simple Lie group
and K is a compact subgroup satisfying Fix()
0
 K  Fix().
2.Type II consists of compact simple Lie groups G equipped with their biinvariant
metric.(L;U) = (GG;),where  is the diagonal subgroup.
3.Type III consists of spaces G=K where G is a noncompact,real simple Lie group
and K a maximal compact subgroup.
4.Type IV consists of spaces G
C
=G where G
C
is a complex simple Lie group and G
a compact real form.
Types I and III are in duality as are types II and IV.
3 The geometric structure: ats,Weyl groups,local models
and blow-up
Let us start with a somewhat vague philosophical claim which we try to justify by describ-
ing various examples later in this section:
Symmetric spaces,simple Lie groups and their innite dimensional counterparts have
a structure that can be described by two fundamental objects:rst there are ats:that is
at subspaces with actions of the Weyl group;this might be called the\absolute geometry".
They describe the particular structure of this space in its class.In these ats there are
distinguished re ection hyperplanes xed by various elements of the Weyl group.Second
there are\residues",that is local structures around those singular planes that contain
various ats.The structure of those local building blocks\local residues"is dened by the
15
class of an objects (compact,complex,noncompact real Lie group,symmetric space over
R or C or H,compact or noncompact).
Let us looks at these ideas in more detail To this end we study the geometry of the
following space:
Example 26 (Positive denite symmetric matrices)
The space of positive denite symmetric matrices is dened by
P
R
(V
n
) = fA 2 Mat(n n;F) j A = A
t
and v
t
Av > 0g:
We can understand this space as the space of metrics on an n-dimensional vector space
over R.
The choice of the eld (in our case R) is not important for the ideas.We could
have used C as well,but then we would have had to look at hermitian matrices.The
same philosophy applies to\symmetric spaces"and groups of semi-simple Lie resp Kac-
Moody type over any other eld.More geometrically we can think of P
R
(V
n
) as the space
of n  1-dimensional eggs (ellipsoids) by identifying an element A 2 P
R
(V
n
) with the
hypersurface v
t
Av = 1  R
n
.Using the geometric intuition coming from this idea we can
make a certain series of claims about the structure this space should have:
Claim 1 As each ellipsoid has its principal axis we can nd a basis in which the describing
matrix is in diagonal form (\principal axis transformation").As a consequence each
element is diagonalizable.
Claim 2 We have a global scaling factor by just multiplying each matrix with an arbitrary
real number.Hence the space P
R
(V
n
) should split into a product P
R;1
(V
n
) R of
P
R;1
(V
n
):= fx 2 P
R;1
(V
n
) j det(X) = 1g and the scaling factor R.
Claim 3 There should be n-dimensional at subspaces corresponding to rescaling the
principal axes without rotating them (call them\ ats".
Claim 4 All n-principal axes are equivalent.Hence for any compatible choice of principal
axes there should be a Sym(n)-symmetry permuting them.
Claim 5 If two principal axes have the same length then the choice of principal axes is
not unique.In view of claim 3,rescaling of those elements are only well dened after
a CHOICE of one particular set of principal axes.As a consequence those elements
lie in more ats,each corresponding to another specication of principal axes.
Claim 5 At the sphere S
n1
(corresponding to the identity matrix),there is a symmetry
mapping
P
R
(V
n
) !P
R
(V
n
);
(a
1
;:::a
n
) 7!

1
a
1
;:::;
1
a
n

:
So let us now try to justify those assertions by calculations:
We start with the denition of the tangent space:
Denition 27
The tangent space at any point p 2 P
R
(V
n
) consists of the space of symmetric matrices
S
R
(V
n
) =

A 2 Mat(n n;R) j A = A
t

:
16
The matrix exponential function denes a homeomorphism between S
R
(n) and P
R
:
exp:S
R
(n) !P
R
(n);
X 7!
1
X
k=0
X
n
n!
 1 +X +
X
2
2
+
X
3
6
+::::
The splitting of P
R
(V
n
) into P
R;1
(V
n
)Rgives on the tangent space the decomposition
S
R
(V
n
) = S
R;0
(V
n
) R
into the space S
R;0
(V
n
) of matrices with vanishing trace and the trace R in the second
factor.
To make this space into a symmetric space we have to specify the metric
Denition 28
The metric at a point p 2 P
R
(n) is dened as follows:For x;y 2 T
p
P
R
(n) we put
hX;Y i = trace(p
1
Xp
1
Y ):
If p = Id we just recover
hX;Y i = trace(XY ):
As a last piece of information we need the action of the general linear group GL(n;R)
on P
R
(n):
':GL(n;R) P
R
(n) !P
R
(n);
(g;p) 7!gpg
t
:
We can now state the main theorem of this section (also proven in the book [BH99]):
Theorem 29
P
F
(n) satises the following properties:
1.The action'of GL(n;R) is transitive,
2.The action is by isometries,
3.The isotropy subgroup at the identity is the orthogonal group:I(Id) = SO(n),
4.fIdg  GL(n;R) acts trivially,the quotient GL(n;R)=fIdg acts eectively,
5.P(n;R) is a symmetric space.The symmetry at a point p 2 P(n) is given by

p
(q) = pq
1
q:
Let us prove this:
Proof.1.It is enough to nd for any p 2 P
R
(n) nd an element g
p
2 GL(n;R) such
that p = g
p
g
t
p
= g
p
Idg
t
p
.This amount to a transformation mapping Id onto p.By
concatenation we can construct a transformation mapping any point q onto any
point p.To nd this element g
p
we use that p is diagonalizable.Hence there are an
orthogonal matrix O and a diagonal matrix D such that p = ODO
t
.We put now
g
p
= O
p
DO
t
where
p
D is to be understood pointwise.Then a straight forward
calculation shows:
g
p
g
t
p
= O
p
DO
t
O
p
DO
t
= ODO = p
17
2.Let X;Y 2 T
p
M.Then
hdg(X);dg(Y )i
gp
= trace

(g
t
)
1
p
1
g
1
gXg
t
(g
t
)
1
p
1
g
1
gY g
t

=
= trace

(g
t
)
1
p
1
Xp
1
Y g
t

=
= trace

p
1
Xp
1
Y

= hX;Y i
p
:
3.This is the denition of the orthogonal group:
SO(n):= fx 2 Mat(n n;R)jxx
t
= Idg:
The stabilizer of a point p is the group I
p
(P
R
(n)) = p  O(n)  p
1
.
4.clear.
5.We have to check three things:
(a) 
p
(p) = p,
(b) 
p
is an isometry.
(c) d
p
j
T
p
M
= Id
This amounts to three small calculations:
(a) 
p
(p) = pp
1
p = p.
(b) The conjugation formula 
q
= g
pq
 
Id
= g
pq

Id
q
1
shows that 
p
is a concate-
nation of the application of a group element g
pq
and 
Id
,it is enough to check,
that each of those two factors separately is an isometry.For g
pq
we did thins
in the second part of the proof.Hence we need to check it only for 
Id
:Let
X;Y 2 T
q
1M.Then
hd
Id
(X);d
Id
(Y )i
q
1 = trace

qq
1
Xq
1
qq
1
Y q
1

=
= trace

Xq
1
Y q
1

=
= trace

q
1
Xq
1
Y

= hX;Y i
q
:
(c) 
p
is conjugate to 
Id
.We know from (1) that the action of GL(n;R) is by
isometries.Hence it is enough to check the assertion for one point.We choose
the identity;the assertion follows now as
d
dt

Id
(X) = X for X 2 T
Id
P
n
R.
Having established the most important properties of P
n
(R),let us now study its ge-
ometry in more detail:
Denition 30
A at is a subspace F  P
R;1
(n) which is isometric to Euclidean space R
k
.The maximal
dimension of ats in a symmetric space is called its rank.
By the formula K(X;Y ) =
1
4
j[X;Y ]j
2
we see that in the matrix representation ats
correspond exactly to exponential images of subalgebras of commuting matrices.As com-
muting matrices are simultaneously diagonalizable, ats correspond to diagonal subgroups.
For example the subspace F
0
:= fdiag(a
1
;:::;a
n
)j
Q
a
i
= 1;a
i
> 0g is a at in P
R;1
(n).
It is even a maximal at.Any other at corresponds to any subspace of simultaneously
18
diagonalizable matrices.Those spaces are at most n-dimensional.The additional restric-
tion to the determinant means that a at has maximal dimension n 1.Hence the rank
of P
R;1
(n) is n 1.As any symmetric matrix is diagonalizable,any point in P
R;1
(n) lies
in a maximal at.
The Weyl group of this symmetric space is the symmetric group in n-letters.It can be
realized on any maximal at as the permutation group of the entries.
Sym(n) F
0
!F
0
;
(;(a
1
;:::;a
n
)) 7!(a
(1)
;:::;a
(n)
):
Singular elements are exactly elements with at least two identical entries a
i
= a
j
.Let
A
ij
be a point in F
0
such that a
i
= a
j
.The permutation 
ij
2 Sym(n) xes A
ij
.More
generally there is a 1-codimensional subspace (hence a hyperplane) F
ij
 F
0
stabilizes by

ij
dened by
F
ij
= diagfa
1
;:::;a
n
j
Y
a
k
= 1;a
k
> 0;a
i
= a
j
g:
Without loss of generality suppose ij = 12.We want to study the space of all maximal
ats containing F
12
.
Using the fact that P
R;1
(n) = SL(n;R)=SO(n) we dene the group
SL
12
(n;R):=
8
>
>
>
<
>
>
>
:
0
B
B
B
@
a
1
A
0 0 0
0
a
3
0 0
0
0
.
.
.
0
0
0 0 a
n
1
C
C
C
A
j A 2 SL(2;R);(a
1
)
2
Y
k3
a
k
= 1
9
>
>
>
=
>
>
>
;
:
We furthermore dene SO
12
(n):=
8
>
>
>
<
>
>
>
:
0
B
B
B
@
A
0 0 0
0
1 0 0
0
0
.
.
.
0
0
0 0 1
1
C
C
C
A
j A 2 SO(2)
9
>
>
>
=
>
>
>
;
.
Clearly we have the relations:SL
12
(n;R)  SL(n;R) and SO
12
(n)  SO(n).
Furthermore the groups satisfy:SO
12
(n) = SL
12
(n)\SO(n).Hence we get on the
level of quotient spaces the following embedding:
P
R;1
(n)
12
= SL
12
(n;R)=SO
12
(n),!P
R;1
(n):
We want to study the subspace P
R;1
(n)
12
.This space can be described as positive
denite symmetric matrices having entries only at very few positions:
P(n;R)
12
:=
8
>
>
>
>
<
>
>
>
>
:
0
B
B
B
B
B
@
a
1
a a
1
b
0 0 0
a
1
b a
1
d
0 0 0
0 0
a
3
0 0
0 0
0
.
.
.
0
0 0
0 0 a
n
1
C
C
C
C
C
A
j

a b
b d

is positive denite and a
i
> 0
9
>
>
>
>
=
>
>
>
>
;
:
In terms of Lie groups we have in the upper left corner the space
H
2
= a
1
 SL(2;R)=SO(2):
19
which we KNOW to be geometrically hyperbolic space,scaled by a factor a
1
.Hence
we can write our space also in this way:
P(n;R)
12
:=
8
>
>
>
<
>
>
>
:
0
B
B
B
@
a
1
H
2
0 0 0
0
a
3
0 0
0
0
.
.
.
0
0
0 0 a
n
:
1
C
C
C
A
j a
i
> 0
9
>
>
>
=
>
>
>
;
Written in this form we can describe explicitly the geometric structure of P(n;R)
12
:
It is the direct product of a 2-dimensional hyperbolic space H
2
with an n2-dimensional
abelian subspace.Maximal abelian subspaces in H
2
are 1-dimensional and elements in
H
2
are orthogonal to our at.Hence maximal ats in P(n;R)
12
are n  1-dimensional.
Especially the at F
0
is a maximal at in P(n;R)
12
.Flats in this space corresponding to
ats containing our singular hyperplane F
12
are exactly those that pass through the point
x 2 H
2
that is stabilized by the group SO(2) which we used as our quotient group.The
space of geodesics through one point in hyperbolic 2-space is exactly RP(1),1-dimensional
projective space.As a consequence we note:
Theorem 31
The space of maximal ats containing F
12
(or more generally F
ij
) is isomorphic to RP(1).
This is an important structure element of this space,distinguishing it from other
symmetric spaces.So we should try to include this information into the Dynkin diagram
associated to it.We do the following:We give each node a superscript denoting the
dimension of the projective space of ats around the corresponding singular element:We
get the following diagram:
d d d d d d
...
1 1 1 1 1 1
To see that this idea is meaningful let us study a closely related symmetric space,
namely the space
M = SL(n;C)=SU(n)
of positive denite hermitian metrics on the complex vector space C
n
.
This space has the explicit description
Example 32 (Positive denite hermitian matrices)
The space of positive denite hermitian matrices is dened by
P
C
(V
n
) = fA 2 Mat(n n;C) j A =
A
t
and v
t
A
v > 0g:
Similarly as to its real counterpart we can prove that this space similarly
G
0
:= fdiag(a
1
;:::;a
n
)j
Y
a
i
= 1;a
i
> 0g:
The condition G
0
=
G
0
means that G
0
is a real subspace.Hence G
0
has exactly the
same structure as F
0
and the same action of the Weyl group,which is again the symmetric
group.Let us now study again the space of all ats containing the hyperplane G
12
dened
as the subspace of G
0
xed by 
12
2 Sym(n).Dene
SL
12
(n;C):=
8
>
>
>
<
>
>
>
:
0
B
B
B
@
a
1
A
0 0 0
0
a
3
0 0
0
0
.
.
.
0
0
0 0 a
n
1
C
C
C
A
j A 2 SL(2;C)a
2
1
Y
k3
a
k
= 1
9
>
>
>
=
>
>
>
;
:
20
We furthermore dene SU
12
(n):=
8
>
>
>
<
>
>
>
:
0
B
B
B
@
A
0 0 0
0
1 0 0
0
0
.
.
.
0
0
0 0 1
1
C
C
C
A
j A 2 SU(2)
9
>
>
>
=
>
>
>
;
.
Clearly we have the relations:SL
12
(n;C)  SL(n;C) and SU
12
(n)  SU(n).
Furthermore the groups satisfy:SU
12
(n) = SL
12
(n;C)\SS(n).Hence we get on the
level of quotient spaces the following embedding
P
C;1
(n)
12
= SL
12
(n;C)=SU
12
(n),!P
C;1
(n):
We want to study the subspace P
R;1
(n)
12
.This space can be described as positive
denite hermitian matrices having entries only at very few positions:
P(n;C)
12
:=
8
>
>
>
>
<
>
>
>
>
:
0
B
B
B
B
B
@
a
1
a a
1
b
0 0 0
a
1
b a
1
d
0 0 0
0 0
a
3
0 0
0 0
0
.
.
.
0
0 0
0 0 a
n
1
C
C
C
C
C
A
j a
i
> 0 and

a b
b d

pos.def.hermitian
9
>
>
>
>
=
>
>
>
>
;
:
In terms of Lie groups we have in the upper left corner the space H
3
= SL(2;C)=SU(2)
which we KNOWto be geometrically 3-dimensional hyperbolic space.Hence we can write
our space also in this way:
P(n;C)
12
:=
8
>
>
>
<
>
>
>
:
0
B
B
B
@
a
1
H
3
0 0 0
0
a
3
0 0
0
0
.
.
.
0
0
0 0 a
n
:
1
C
C
C
A
j a
i
> 0
9
>
>
>
=
>
>
>
;
Written in this form we can describe explicitly the geometric structure of P(n;C)
12
:
It is the direct product of a 3-dimensional hyperbolic space H
3
with an n2-dimensional
abelian subspace.Maximal abelian subspaces in H
3
are 1-dimensional and elements in
H
3
are orthogonal to our at.Hence maximal ats in P(n;C)
12
are n  1-dimensional.
Especially the at G
0
is a maximal at in P(n;C)
12
.Flats in this space corresponding to
ats containing our singular hyperplane G
12
are exactly those that pass through the point
x 2 H
3
that is stabilized by the group SU(2) which we used as our quotient group.The
space of geodesics through one point in hyperbolic 3-space is exactly RP(2),2-dimensional
projective space.As a consequence we note:
Theorem 33
The space of maximal ats containing G
12
(or more generally G
ij
) is isomorphic to RP(2).
This is an important structure element of this space,distinguishing it from other
symmetric spaces having the same Weyl group.So include this information into the
Dynkin diagram associated to it:As the dimension of the projective space is 2,we get the
following diagram:
d d d d d d
...
2 2 2 2 2 2
Hence we see that,to understand a symmetric space,we have to understand
1.The structure of the\ at"subspaces carrying a Weyl group action,
2.The structure how those ats are pieces together around singular elements.
21
This is typical for all geometric objects associated to Weyl groups,especially Lie groups
and Kac-Moody groups,nite and innite dimensional symmetric spaces,buildings,polar
actions etc.etc.etc.
Let us now turn to some compact spaces:
We just take the space M = SU(n) of unitary n  n-matrices.This space has a
maximal at H
0
of purely imaginary diagonal matrices:
H
0
:= fdiag(a
1
;:::;a
n
)ja
k
= e
ix
k
and
Y
a
k
= 1)g:
As the Weyl group we get again the symmetric group acting as the permutation group
of the entries of the at:Dene H
12
as the xed torus of the permutation 
12
 Sym(n).
We perform the same steps as before:Start by dening
SU
12
(n):=
8
>
>
>
<
>
>
>
:
0
B
B
B
@
aA
0 0 0
0
a
3
0 0
0
0
.
.
.
0
0
0 0 a
n
1
C
C
C
A
j A 2 SU(2)a
Y
k3
a
k
= 1
9
>
>
>
=
>
>
>
;
:
We get the embedding (well-known from Lie theory):
SU
12
(n;R),!SU(n):
We want to study the subspace SU
12
(n;R).
We KNOWthat SU(2)

=
S
3
,the 3-dimensional sphere.Hence we can write our space
also in this way:
SU
12
(n) =
8
>
>
>
<
>
>
>
:
0
B
B
B
@
a
1
S
3
0 0 0
0
a
3
0 0
0
0
.
.
.
0
0
0 0 a
n
:
1
C
C
C
A
9
>
>
>
=
>
>
>
;
:
Written in this formwe can describe explicitly the geometric structure of SU
12
(n):It is
the direct product of a 3-dimensional sphere S
3
with an n2-dimensional abelian subspace.
Maximal abelian subspaces in S
3
are 1-dimensional and elements in S
3
are orthogonal to
our at.Hence maximal ats in SU
12
(n) are n  1-dimensional.Especially the at H
0
is a maximal at in SU
12
(n).Flats in this space corresponding to ats containing our
singular hyperplane H
12
are exactly those that pass through the identity x 2 SU(2) = S
3
.
The space of geodesics through one point in a 3-sphere is exactly RP(2),2-dimensional
projective space.As a consequence we note:
Theorem 34
The space of maximal ats containing H
12
(or more generally H
ij
) is isomorphic to RP(2).
Hence the space of ats around a singular hyperplane is the same for SU(n) and
SL(n;C)=SU(n).The geometry of the local building blocks changes just by a sign.While
the sphere has positive sectional curvature K = 1,the hyperbolic space has negative
sectional curvature K = 1 Those spaces have thus the same Dynkin diagram:
d d d d d d
...
2 2 2 2 2 2
22
4 The classication of Riemannian symmetric spaces
Using the intuition and the examples we developed in the last section we can list now all
irreducible simply connected nite dimensional Riemannian symmetric spaces:We just
give the names,the name of the root system and the marked Dynkin diagram.
4.1 The easy classical spaces
Those spaces correspond to simple Lie groups:They are either the compact real forms or
they are quotients of the complex Lie group by the maximal compact real form.Hence
they are symmetric spaces of types II and IV of the classical types.
type
compact noncompact
isotr
root system
marked Dynkin diagram
A
n
su(n) su(n) sl(n;C)
su(n)
A
n
over C
d d d d d d
...
2 2 2 2 2 2
B
n
so(2n +1) so(2n +1) so(2n +1;C)
so(2n +1)
B
n
over C
d d d d d d
...
=)
2 2 2 2 2 2
C
n
sp(n) sp(n) sp(2n;C)
sp(2n)
C
n
over C
d d d d d d
...
(=
2 2 2 2 2 2
D
n
so(2n) so(2n) so(2n;C)
so(2n)
D
n
over C
d d d d
d
d
d
2 2 2 2
2
2


S
S
:::
4.2 Other classical spaces
We just choose some spaces,omitting details in some special cases.The reader interested
in a complete classication is encouraged to check in the books [Hel01] and [Loo69a].
type
compact noncompact
Isot
root system
marked Dynkin diagram
A I
su(n) sl(n;R)
so(n)
A
n
over R
d d d d d d
...
1 1 1 1 1 1
A II
su(2n) su  (2n)
sp(n)
A
n
over H
d d d d d d
...
4 4 4 4 4 4
A III
su(p +q) su(p;q)
S(u
p
u
q
)
depends on the values of p;q
B I
C I
sp(n) sp(n;R)
u(n)
C
n
over R
d d d d d d
...
(=
1 1 1 1 1 1
C II
sp(p +q) sp(p;q)
sp(p) sp(q)
depends on the values of p;q
D I
D III
4.3 The easy exceptional spaces
The following spaces are symmetric spaces of types II and IV corresponding directly to
exceptional Lie groups.Their marked Dynkin diagrams are the ones of the Lie groups
of the same types together with the marking 2 at every node.The associated symmetric
spaces are the ones of type II (hence compact Lie groups) and type IV (hence quotients
of complex Lie groups by their maximal compact forms).Here h
l
denotes the compact
real form,(h
l
)
C
the complex Lie algebra.
23
type
compact noncompact
Isot
root system
marked Dynkin diagram
G
2
g
2
g
2
(G
2
)
C
g
2
G
2
d d
2 2
(
F
4
f
4
f
4
I (F
4
)
C
f
4
F
4
d d d d
2 2 2 2
(
E
6
e
6
e
6
(E
6
)
C
e
6
E
6
d d d d d
2 2 2 2 2
2
d
E
7
e
7
e
7
(E
7
)
C
e
7
E
7
d d d d d d
2 2 2 2 2 2
2
d
E
8
e
8
e
8
(E
8
)
C
e
8
E
8
d d d d d d d
2 2 2 2 2 2 2
2
d
4.4 The other exceptional spaces
The following table describes symmetric spaces associated to the exceptional Lie groups
that are of type I and III,hence quotients of real forms of the Lie groups by xed point
groups of involutions.
type
compact noncompact
Isot
root system
marked Dynkin diagram
G
g
2(14)
g
2(2)
su(2) su(2)
G
2
d d
1 1
W
F I
f
4(52)
f
4(4)
sp(3) su(2)
F
4
d d d d
1 1 1 1
(
F II
f
4(52)
f
4(20)
so
9
BC
1
dg
8[7]
E I
e
6(78)
e
6(6)
sp(4)
E
6
over R
d d d d d
1 1 1 1 1
1
d
E II
e
6(78)
e
6(2)
su(6) +su(2)
F
4
d d d d
2 2 1 1
(
E III
e
6(78)
e
6(14)
so(10) R
BC
2
d dg
6 8[1]
=)
E IV
e
6(78)
e
6(26)
f
4
A
2
over O
d d
8 8
E V
e
7(133)
e
7(7)
su(8)
E
7
over R
d d d d d d
1 1 1 1 1 1
1
d
E V I
e
7(133)
e
7(5)
so(12) su(2)
F
4
over H
d d d d
4 4 1 1
(
E V II
e
7(133)
e
7;(25)
e
6
R
C
2
over O
d d d
8 8
(
1
E V III
e
8(248)
e
8;(8)
so(16)
E
8
over R
d d d d d d d
1 1 1 1 1 1 1
1
d
E IX
e
8(248)
e
8(24)
e
7
+su(2)
F
4
over O
d d d d
8 8 1 1
(
24
5 Ane Kac-Moody symmetric spaces
5.1 Kac-Moody groups and their Lie algebras
5.1.1 Geometric ane Kac-Moody algebras
The classical references for (algebraic) Kac-Moody algebras are the books [Kac90] and
[MP95] written by the two founders of the subject.The book [Car02] gives a detailed
overview of nite dimensional simple Lie algebras and ane Kac-Moody algebras.For our
purposes,this book is by far sucient.In case of ambiguity we use the term algebraic
Kac-Moody algebra for the classical Kac-Moody algebras:
Simple Lie algebras and ane Kac-Moody algebras share a simlar structure theory.
They can be dened starting by a set of 3n generators subject to certain relations,which
depend on a special integer matrix,called the Cartan matrix.Let us give a short denition:
Denition 35 ((spherical) Cartan matrix)
An Cartan matrix A
nn
is a square matrix with integer coecients,such that
1.a
ii
= 2 and a
i6=j
 0.
2.a
ij
= 0,a
ji
= 0.
3.For any vector v > 0 (component wise) such that Av > 0.
Denition 36 (ane Cartan matrix)
An ane Cartan matrix A
nn
is a square matrix with integer coecients,such that
1.a
ii
= 2 and a
i6=j
 0.
2.a
ij
= 0,a
ji
= 0.
3.There is a vector v > 0 (component wise) such that Av = 0.
Example 37 (2 2-ane Cartan matrices)
There are { up to equivalence { two dierent 2-dimensional ane Cartan matrices:

2 2
2 2

;

2 1
4 2

:
They correspond to the non-twisted algebra
~
A
1
and the twisted algebra
~
A
0
1
.
1.The indecomposable (spherical) Cartan matrices are
A
n
;B
n
;C
n
;D
n
;E
6
;E
7
;E
8
;F
4
;G
2
:
2.The indecomposable non-twisted ane Cartan matrices are
e
A
n
;
e
B
n
;
e
C
n
;
e
D
n
;
e
E
6
;
e
E
7
;
e
E
8
;
e
F
4
;
e
G
2
:
Every non-twisted ane Cartan matrix
e
X
l
can be constructed from a (spherical)
Cartan matrix X
l
by the addition of a further line and column.Hence Cartan
matrices and nontwisted aen Cartan matrices are in bijection.The denomination
as\non-twisted"points to the explicit construction as loop algebras.
25
3.The indecomposable twisted ane Cartan matrices are
e
A
0
1
;
e
C
0
l
;
e
B
t
l
;
e
C
t
l
;
e
F
t
4
;
e
G
t
2
:
The Kac-Moody algebras associated to them can be constructed as xed point al-
gebras of certain automorphisms  of a non-twisted Kac-Moody algebra X.This
construction suggests an alternative notation describing a twisted Kac-Moody alge-
bra by the order of  and the type of X.This yields the following equivalences:
e
A
0
1
2
e
A
2
e
C
0
l
2
e
A
2l
;l  2
e
B
t
l
2
e
A
2l1
;l  3
e
C
t
l
2
e
D
l+1
;l  2
e
F
t
4
2
e
E
6
e
G
t
2
3
e
D
4
As in the nite dimensional case to all those classes of Cartan matrices one can associate
their Lie algebra realizations.Those correspond exactly to the ane Kac-Moody algebras.
To describe the loop algebra approach to Kac-Moody algebras we follow the terminol-
ogy of the article [HG09].
Let g be a nite dimensional reductive Lie algebra over F = R or C.Hence g is a direct
product of a semisimple Lie algebra g
s
with an Abelian Lie algebra g
a
.Let furthermore
 2 Aut(g
s
) denote an automorphism of nite order of g
s
such that j
g
a
= Id.If g
s
is a
Lie algebra over R we suppose it to be of compact type.
L(g;):= ff:R !g jf(t +2) = f(t);f satises some regularity conditiong:
We use the notation L(g;) to describe in a unied way algebraic constructions that apply
to various explicit Lie algebra realizations constructions of loop algebras satisfying sundry
regularity conditions | i.e.smooth,real analytic,(after complexication) holomorphic
or algebraic loops.If we discuss loop algebras of a xed regularity we use other precise
notations:Mg for holomorphic loops on C

,L
alg
g for algebraic,A
n
g for holomorphic loops
on the annulus A
n
= fz 2 Cje
n
 jzj  e
n
g.
Denition 38 (Geometric ane Kac-Moody algebra)
The geometric ane Kac-Moody algebra associated to a pair (g;) is the algebra:
b
L(g;):= L(g;) Fc Fd;
equipped with the lie bracket dened by:
[d;f]:= f
0
;[c;c] = [c;d] = [c;f] = [d;d] = 0;
[f;g]:= [f;g]
0
+!(f;g)c:
Here f 2 L(g;) and!is a certain antisymmetric 2-form on Mg satisfying the cocycle
condition.
Let us remark that in contrast to the usual Kac-Moody theory,g has not to be simple,
but may be reductive,i.e.a product of semisimple Lie algebra with an Abelian one.The
algebra
e
L(G;):= L(g;) Fc is called the derived algebra.
We give some further denitions:
26
Denition 39
A real form of a complex geometric ane Kac-Moody algebra
b
L(g
C
;) is the xed point
set of a conjugate linear involution.
Involutions of a geometric ane Kac-Moody algebra restrict to involutions of irre-
ducible factors of the loop algebra.Hence the invariant subalgebras are direct products of
invariant subalgebras in those factors together with the appropriate torus extension.
Denition 40 (compact real ane Kac-Moody algebra)
A compact real form of a complex ane Kac-Moody algebra
b
L(g
C
;) is a real form which
is isomorphic to
b
L(g
R
;) where g
R
is a compact real form of g
C
.
Remark 41
A semisimple Lie algebra is called of\compact type"i it integrates to a compact semisim-
ple Lie group.The innite dimensional generalization of compact Lie groups are loop
groups of compact Lie groups and their Kac-Moody groups,constructed as extensions of
those loop groups (see section 5.1.2).Thus the denomination is justied by the fact that
\compact"ane Kac-Moody algebras integrate to\compact"Kac-Moody groups.
To dene a loop group of the compact type we use an innite dimensional version of
the Cartan-Killing form:
Denition 42 (Cartan-Killing form)
The Cartan-Killing form of a loop algebra L(g
C
;) is dened by
B
(g
C
;)
(f;g) =
Z
2
0
B(f(t);g(t)) dt:
Denition 43 (compact loop algebra)
A loop algebra of compact type is a subalgebra of L(g
C
;) such that its Cartan-Killing form
is negative denite.
Lemma 44
Let g
R
be a compact semisimple Lie algebra.Then the loop algebra L(g
R
;) is of compact
type.
Proof.The Cartan-Killing form on g
R
is negative denite.Hence B
(g
C
;)
(f;g) is negative
denite.
To nd noncompact real forms,we need the following result of Ernst Heintze and
Christian Gro (Corollary 7.7.of [HG09]):
Theorem 45
Let G be an irreducible complex geometric ane Kac-Moody algebra i.e.G =
b
L(g;) with g
simple,U a real form of compact type.The conjugacy classes of real forms of noncompact
type of G are in bijection with the conjugacy classes of involutions on U.The correspon-
dence is given by U = K  P 7!K iP where K and P are the 1-eigenspaces of the
involution.
Thus to nd noncompact real forms,we have to study automorphism of order 2 of
a geometric ane Kac-Moody algebra of the compact type.From now on we restrict to
involutions b'of type 2,that is b'(c) = c.
A careful examination of the construction of a geometric ane Kac-Moody algebra of
a non simple Lie algebra allows,to extend this result to the broader class of geometric
ane Kac-Moody algebras [Fre09]:
27
Theorem 46
Let G be a complex geometric ane Kac-Moody algebra,U a real formof compact type.The
conjugacy classes of real forms of noncompact type of G are in bijection with the conjugacy
classes of involutions on U.The correspondence is given by U = KP 7!KiP where
K and P are the 1-eigenspaces of the involution.Furthermore every real form is either
of compact type or of noncompact type.A mixed type is not possible.
Lemma 47
Let g be semisimple and
b
L(g;)
D
be a real form of the noncompact type.Let
b
L(g;)
D
=
KP be a Cartan decomposition.The Cartan Killing form is negative denite on K and
positive denite on P
Proof.Suppose rst  is the identity.Let'be an automorphism.Then without loss of
generality'(f) ='
0
(f(t)) [HG09].Let g = k  p be the decomposition of g into the
1-eigenspaces of'
0
.Then f 2 Fix(') i its Taylor expansion satises
X
n
a
n
e
int
=
X
'
0
(a
n
)e
int
:
Let a
n
= k
n
p
n
be the decomposition of a
n
into the 1 eigenspaces with respect to'
0
.
Hence
f(t) =
X
n
k
n
cos(nt) +
X
n
p
n
sin(nt):
Then using bilinearity and the fact that fcos(nt);sin(nt)g are orthonormal we can calculate
B
g
:
B
g
=
Z
2
0
X
n
cos
2
(nt)B(k
n
;k
n
) 
Z
2
0
X
n
sin
2
(nt)B(p
n
;p
n
):
Hence B
g
is negative denite on Fix(').Analogously one calculates that it is positive
denite on the 1-eigenspace of'.If  6= Id then one gets the same result by embedding
L(g;) into an algebra L(h;id) which is always possible [Kac90].
Lemma 48
Let g be Abelian.The Cartan-Killing form of L(g) is trivial.
Proof.Direct calculation.
Now we can dene OSAKAs:
Denition 49 (Orthogonal symmetric Kac-Moody algebra)
An orthogonal symmetric ane Kac-Moody algebra (OSAKA) is a pair

b
L(g;);b

such
that
1.
b
L(g;) is a real form of an ane geometric Kac-Moody algebra,
2.b is an involutive automorphism of
b
L(g;) of the second kind,
3.Fix(b) is a compact real form.
Following Helgason,we dene 3 types of OSAKAs:
Denition 50 (Types of OSAKAs)
Let

b
L(g;);b

be an OSAKA.Let
b
L(g;) = KP be the decomposition of
b
L(g;) into
the eigenspaces of
b
L(g;) of eigenvalue +1 resp.1.
28
1.If
b
L(g;) is a compact real ane Kac-Moody algebra,it is said to be of the compact
type.
2.If
b
L(g;) is a noncompact real ane Kac-Moody algebra,
b
L(g;) = U  P is a
Cartan decomposition of
b
L(g;).
3.If L(g;) is Abelian it is said to be of Euclidean type.
Denition 51 (irreducible OSAKA)
An OSAKA

b
L(g;);b

is called irreducible i its derived algebra has no non-trivial de-
rived Kac-Moody subalgebra invariant under b.
Thus we can describe the dierent classes of irreducible OSAKAs of compact type.
1.The rst class consists of pairs consisting of compact real forms
d
Mg,where g is
a simple Lie algebra together with an involution of the second kind.Complete
classications are available [Hei08].This are irreducible factors of type I.They
correspond to Kac-Moody symmetric spaces of type I.
2.Let g
R
be a simple real Lie algebra of the compact type.The second class consists
of pairs of an ane Kac-Moody algebra
\
M(g
R
g
R
) together with an involution
of the second kind,interchanging the factors.Those algebras correspond to Kac-
Moody symmetric spaces that are compact Kac-Moody groups equipped with their
Ad-invariant metrics (type II).
Dualizing OSAKAs of the compact type,we get the OSAKAs of the noncompact type.
1.Let g
C
be a complex semisimple Lie algebra,and
b
L(g
C
;) the associated ane
Kac-Moody algebra.This class consists of real forms of the noncompact type that
are described as xed point sets of involutions of type 2 together with a special
involution,called Cartan involution.This is the unique involution on G,such that
the decomposition into its 1 eigenspaces K and P yields:KiP is a real form of
compact type of
b
L(g
C
;).Those orthogonal symmetric Lie algebras correspond to
Kac-Moody symmetric spaces of type III.
2.Let g
C
be a complex semisimple Lie algebra.The fourth class consists of
b
L(g
C
;)
with the involution given by the complex conjugation b
0
with respect to a compact
real form,i.e.
b
L(g
R
;).Those algebras correspond to Kac-Moody symmetric spaces
of type IV.
The derived algebras of the last class of OSAKAs |the ones of Euclidean type |are
Heisenberg algebras [PS86].The maximal subgroup of compact type is trivial.
5.1.2 Ane Kac-Moody groups
There are several dierent approaches to ane Kac-Moody groups.The usual algebraic
approach follows the denition of algebraic groups via a functor.Tits denes a group
functor from the category of rings into the category of groups,whose evaluation on the
category of elds yields Kac-Moody groups.Various completions of the groups dened
this way are possible.Nevertheless,restricting to ane Kac-Moody groups,there is a
second much more down to earth approach which consists in the denition of Kac-Moody
groups as special extensions of loop groups - for those two and other approaches compare
29
the article [Tit84].This second approach for ane Kac-Moody groups relies on a curi-
ous identication:Let G denote an ane algebraic group scheme and study the group
G(k[t;t
1
]),where k is a eld.Either,dening a torus extension of G(k[t;t
1
]) we get
a Kac-Moody group over the eld k or tensoring with the quotient eld k(t) of k[t;t
1
]
we get an algebraic group over k(t).This hints to a close connection between algebraic
groups and ane algebraic Kac-Moody groups,with a loop group as the intermediate
object.Now\analytic"completions can be dened just by completing the ring k[t;t
1
]
with respect to some norm.
In this section we focus on the loop group approach to Kac-Moody groups.Our
presentation follows the book [PS86].
We use again the regularity-independent notation L(G
C
;) for the complex loop group
and L(G;) for its real form of compact type.To dene groups of polynomial maps,we
use the fact,that every compact Lie group is isomorphic to a subgroup of some unitary
group.Hence we can identify it with a matrix group.Similarly,the complexication can
be identied with a subgroup of some general linear group [PS86].
Kac-Moody groups are constructed in two steps.
1.The rst step consists in the construction of an S
1
-bundle in the real case (resp.a
C

-bundle in the complex case) over L(G;) that corresponds via the exponential
map to the derived algebra.The ber corresponds to the central term Rc (resp.Cc)
of the Kac-Moody algebra.
2.In the second step we construct a semidirect product with S
1
(resp.C

).This
corresponds via the exponential map to the Rd- (resp.Cd-) term
Study rst the extension of L(G;) with the short exact sequence:
1 !S
1
!X !L(G;) !1:
There are various groups X that t into this sequence.We need to nd a group
e
L(G;)
such that its tangential Lie algebra at e 2
e
L(G;) is isomorphic to
e
L(g;).
As described in [PS86] this S
1
-bundle is best represented by triples (g;p;z) where g is an
element in the loop group,p a path connecting the identity to g and z 2 S
1
(respective C

)
subject to the relation of equivalence:(g
1
;p
1
;z
1
)  (g
2
;p
2
;z
2
) i g
1
= g
2
and z
1
=
C
!
(p
2
 p
1
1
)z
2
.Here C
!
(p
2
 p
1
1
) = e
R
S(p
2
p
1
1
)
!
where S(p
2
 p
1
1
) is a surface bounded
by the closed curve p
2
 p
1
1
and!denotes the 2-form used to dene the central extension
of L(g;).The term z
1
= C
!
(p
2
 p
1
1
)z
2
denes a twist of the bundle.The law of
composition is dened by
(g
1
;p
1
;z
1
)  (g
2
;p
2
;z
2
) = (g
1
g
2
;p
1
 g
1
(p
2
);z
1
z
2
):
If G is simply connected and!integral (which is the case in our situation),it can be
shown that this object is a well dened group [PS86],theorem 4.4.1.If G is not simply
connected,the situation is a little more complicated:Let G = H=Z where H is a simply
connected Lie group and Z = 
1
(G).Let (LG)
0
denote the identity component of LG.
We can describe the extension using the short exact sequence:
1 !S
1
!
g
LH=Z !(LG)
0
!1
[PS86],section 4.6.
In case of complex loop groups,the S
1
-bundle is complexied to a C

-bundle.
The second much easier extension yields now Kac-Moody groups:
30
Denition 52 (Kac-Moody group)
1.Let G be a compact real Lie group.The compact real Kac-Moody group
b
L(G;) is
the semidirect product of S
1
with the S
1
-bundle
e
L(G;).
2.Let G
C
be a complex simple Lie group.The complex Kac-Moody group
b
L(G
C
;) is
the semidirect product of C

with the C

-bundle
e
L(G
C
;).It is the complexication
of
b
L(G;).
The action of the semidirect S
1
-(resp.C

-) factor is in both cases dened to be shift of
the argument:
C

3 w:MG!MG:f(z) 7!f(z  w):
Remark that in the complex case for this shift to be well dened we need holomorphic
loops on C

.Hence this is the maximal possible realization of a loop group admitting
complexication.
Let us describe some subgroups of
b
L(G;):
Example 53 (the nite dimensional Lie group G)
The easiest important subgroup is the nite dimensional simple Lie group G.It can be
embedded into
b
L(G;) as the subgroup of constant loops:
':G,!
b
L(G;)
g 7!g(t)  g:
This is clearly a subgroup as group multiplication of two elements g
_
f is a constant function.
Let T  G be a maximal torus.Obviously'(T) 
b
T 
b
L(G;) is mapped into a torus of
b
L(G;) such that
b
T\G = T.Nevertheless
b
T is greater than T:
b
T = T S
1
c
S
1
d
where the rst S
1
-factor corresponds to the c-central extension and the second one corre-
sponds to the d-semidirect factor.Over the subgroup of constant loops those two extensions
are direct products:
1.The action of an element e
it
d
 S
1
d
by shift of argument is trivial on constant func-
tions.
2.The S
1
-bundle of the central extension is trivial as the antisymmetric 2-form!
vanishes over constant loops because of the derivatives vanishing.
Similarly the normalizer N of T embeds via'into
b
N 
b
L(G;).This embedding is
such that
N =
b
N\G
Nevertheless
b
N is by far greater then N:While N=T is the nite spherical Weyl group
W
S
we have that
b
N=
b
T is the innite ane Weyl group W
a
.With the embedding'we get
also for every root of the nite dimensional Lie group G a subgroup SU(2) 
b
L(G;).
Let us now look for subgroups of
b
L(G;) that are not contained in the subgroup G of
constant loops.
31
Example 54 (Conjugations)
Let
b
f 2
b
L(G;).Then
'
f
:G,!
b
L(G;)
g 7!
b
f'(g)
b
f
1
where G is again the nite dimensional simple Lie group and'denotes the embedding of
G into the group of constant loops denes a subgroup.For this group we can again study
the tori,their normalizers or the SU(2)-subgroups.
Example 55 (SU(2))
Let G = SU(2) and let T  SU(2) be the standard torus.There is a subgroup of
b
L(SU(2);Id) isomorphic to SU(2) of the form
SU(2)
z
2:=

a z
2
b
1
z