Kac-Moody symmetric spaces and universal twin buildings [Elektronische Ressource] / vorgelegt von Walter Freyn

Sujets

• Mathematics

Informations

• Publié par: universitat_augsburg
• Publié le: 01 janvier 2009
• Nombre de lectures: 112
• Langue: Deutsch

Extrait

Kac-Moody symmetric spaces and
universal twin buildings
Dissertation
zur Erlangung des Doktorgrades
an der Mathematisch-Naturwissenschaftlichen Fakult at
der Universit at Augsburg
vorgelegt von
Walter Freyn
August 2009Betreuer:
Professor Dr. Ernst Heintze
Gutachter:
Professor Dr. Jost-Hinrich Eschenburg
Professor Dr. Ralf Gramlich
Termin der mundlic hen Prufun g:
10. August 2009To my parentsContents
1 Introduction 1
1.1 The origin of the problem and the state of the art . . . . . . . . . . . . . . 1
1.2 Geometry of Kac-Moody symmetric spaces . . . . . . . . . . . . . . . . . . 2
1.3 Universal twin buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Universal algebraic twin buildings . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Analytic foundations 9
2.1 Tame Frechet manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Frechet spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Tame Frechet spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.3 An implicit function theorem for tame maps . . . . . . . . . . . . . . 14
2.1.4 Some tame Frechet spaces . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Inverse limit constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Lie algebras of holomorphic maps . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Groups of maps . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.2 Manifold structures on groups of holomorphic maps . . . . . . . . . 30
2.5 Polar actions on tame Frechet spaces . . . . . . . . . . . . . . . . . . . . . . 35
3 Algebraic foundations 39
3.1 Kac-Moody algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.1 Algebraic approach to Kac-Moody algebras . . . . . . . . . . . . . . 39
3.1.2 The loop algebra approach to Kac-Moody algebras . . . . . . . . . . 42
3.2 Orthogonal symmetric Kac-Moody algebras . . . . . . . . . . . . . . . . . . 47
3.2.1 The nite dimensional blueprint . . . . . . . . . . . . . . . . . . . . 47
3.2.2 Orthogonal symmetric a ne Kac-Moody algebras . . . . . . . . . . 48
3.3 Tame structures and ILB-structures . . . . . . . . . . . . . . . . . . . . . . 51
3.4 Kac-Moody groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4.2 Adjoint action and isotropy representations . . . . . . . . . . . . . . 55
4 Kac-Moody symmetric spaces 57
4.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.1 Di erential geometry of tame Frechet manifolds . . . . . . . . . . . . 57
4.1.2 Lorentz on tame Frechet . . . . . . . . . . . . . 58
4.1.3 Tame Frechet symmetric spaces . . . . . . . . . . . . . . . . . . . . . 60
4.2 Kac-Moody symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3ody of compact type . . . . . . . . . . . . . . . . 62
iiiiv CONTENTS
4.3.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.2 Kac-Moody symmetric spaces of type II . . . . . . . . . . . . . . . . 63
4.3.3ody of type I . . . . . . . . . . . . . . . . 63
4.4 Symmetric spaces of non-compact type . . . . . . . . . . . . . . . . . . . . . 64
4.5 Kac-Moody symmetric spaces of the Euclidean type . . . . . . . . . . . . . 65
4.6 The structure of nite dimensional ats . . . . . . . . . . . . . . . . . . . . 65
4.7 Some remarks concerning the geometry . . . . . . . . . . . . . . . . . . . . 68
5 Universal twin building complexes 71
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Spherical buildings for Lie groups . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.1 Combinatorial theory of buildings . . . . . . . . . . . . . . . . . . . 72
5.2.2 Metric theory of buildings . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.3 Buildings and Lie groups . . . . . . . . . . . . . . . . . . . . . . . . 77
5.2.4 and symmetric spaces . . . . . . . . . . . . . . . . . . . . . 81
5.3 Buildings for Loop groups and Kac-Moody groups . . . . . . . . . . . . . . 83
5.3.1 Some remarks and notations . . . . . . . . . . . . . . . . . . . . . . 83
5.3.2 Algebraic theory: twin BN-pairs and twin buildings . . . . . . . . . 83
5.4 Universal geometric twin buildings . . . . . . . . . . . . . . . . . . . . . . . 88
5.5 Geometric twin buildings and Kac-Moody algebras . . . . . . . . . . . . . . 92
5.6 Topology and geometry of B . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.7 The spherical building at in nity . . . . . . . . . . . . . . . . . . . . . . . . 97
15.8 The Hilbert space setting of H -loops . . . . . . . . . . . . . . . . . . . . . 98
5.9 Universal algebraic twin building . . . . . . . . . . . . . . . . . . . . . . . . 99
6 Flag complexes for universal twin buildings 101
6.1 The nite dimensional blueprint: Flag complexes and buildings . . . . . . . 101
6.2 Grassmannians and periodic ag varieties . . . . . . . . . . . . . . . . . . . 103
e6.3 The special linear groups: type A . . . . . . . . . . . . . . . . . . . . . . . 107n
6.3.1 The a ne building . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3.2 The spherical building at in nity . . . . . . . . . . . . . . . . . . . . 111
6.3.3 A universal geometric twin building . . . . . . . . . . . . . . . . . . 112
e6.4 The symplectic groups C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114n
A Notation 117
B Curriculum vitae of Walter Freyn 125Chapter 1
Introduction
In this thesis we develop a theory of a ne Kac-Moody symmetric spaces and their build-
ings. There are two main parts: in the rst part we construct Kac-Moody symmetric
spaces and discuss their analytic and geometric properties. The main new feature of our
approach is the use of holomorphic loops on C . In this setting a nice complexi cation
of the Kac-Moody groups can be constructed. Therefore it allows the de nition of Kac-
Moody symmetric spaces of the non-compact type.
In the second part we describe the theory of universal geometric twin buildings asso-
ciated to Kac-Moody symmetric spaces. The new point of view is the use of functional
analytic methods. As an application we construct a completion of twin buildings which
nicely re ects the tame Frechet structure of the Kac-Moody symmetric spaces.
1.1 The origin of the problem and the state of the art
The problem of constructing Kac-Moody symmetric spaces emerged in the 90’s from the
study of isoparametric submanifolds in Hilbert spaces, P (G;H)-actions and polar actions
on Hilbert spaces as follows:
In nite dimensional di erential geometry there is a remarkable link between Riemann
symmetric spaces, polar representations, isoparametric submanifolds and spherical build-
ings: namely isotropy representations of symmetric spaces are polar; their principal orbits
are isoparametric submanifolds. Conversely Dadok proved every polar representation on
n
R to be orbit equivalent to the isotropy representation of a symmetric space. Further-
nmore a result of Thorbergsson shows any full irreducible isoparametric submanifold ofR
of rank at least three to be an orbit of some isotropy representation (see [BCO03] and
references therein). The boundary of a symmetric space of non-compact type can be iden-
ti ed with a building. In addition the building can be embedded into the unit sphere of
the representation space of the isotropy representation.
Hence, generalizing the concepts of isoparametric submanifolds and polar actions to
Hilbert spaces, Chuu-Lian Terng conjectured in her foundational article [Ter95] the exis-
tence of in nite dimensional symmetric spaces completing the generalization of the nite
dimensional blueprint. She also remarks (remark 3.4) that severe technical problems make
the rigorous de nition of those spaces di cult. The crucial point is to nd an analytic
framework that allows all algebraic constructions which are needed for the description of
the geometric theory. A recent review of the theory of isoparametric submanifolds and
polar actions on Hilbert space, which contains additional references, is given in [Hei06].
Important progress towards the construction of Kac-Moody symmetric spaces was
achieved by Bogdan Popescu only in 2005/2006 in his thesis [Pop05] where he considers
12 CHAPTER 1. INTRODUCTION
1weak Hilbert symmetric spaces, modeled as loop spaces of H -Sobolev loops, equipped
0 1 0with aH scalar product. As the di erential of H loops is only inH this approach does
not allow a convincing de nition of the torus bundle extension corresponding to the c and
d parts of the Kac-Moody algebra. To remedy this he investigates also the framework of
smooth loops, which allows the construction of symmetric Frechet spaces of the \compact
type". However there is no convincing de nition of a complexi cation for those groups.
Hence, the de nition of the dual non-compact symmetric spaces fails completely. As about
half of all symmetric spaces are of type this is a serious detriment.
In this work we overcome this problem by using holomorphic loops de ned on C =
Cnf0g. In this ansatz we have to t