INTRODUCTION xiii
on the cardinality of the
field3;
however, non-linearity results for Kac-Moody groups
over finite fields seem to be much harder to prove. In particular, the techniques
developed by B. emy to tackle this problem are considerably more elaborated
than the ones we use to prove Theorem E. Actually, according to B. emy’s work,
the algebraic group point of view on Kac-Moody groups should be replaced by
a discrete group point in the case of finite ground fields4; this allows to combine
classical arguments from the theory of algebraic groups with tools from dynamics
and ergodic theory (see [R´ em03] for a survey).
Although the Kac-Moody groups considered in Theorem A are split, it is prob-
able that some of the ideas developed in this context can also be used to study the
isomorphism problem in the non-split case (see [R´ em02b] for the relative theory
of Kac-Moody groups). As an illustration of this possibility, we include the solu-
tion of the isomorphism problem for unitary forms of complex Kac-Moody groups
(see Theorem 8.2). These unitary forms were defined and studied by V. Kac and
D. Peterson (see [KP87] and references therein). In the finite-dimensional case,
they coincide with the compact semisimple Lie groups. In the affine case, they co-
incide (up to a central extension by a copy of
S1)
with the so-called ‘algebraic’ loop
groups of the compact semisimple Lie groups. In the indefinite type case, no such
convenient description is known. However, in all cases, unitary forms of complex
Kac-Moody groups carry a natural structure of connected Hausdorff topological
group, and it follows from our result that any epimorphism with central kernel
between two such unitary forms is continuous. This is of course well known in the
finite-dimensional case.
We now come to the organization of the text. The first chapter collects stan-
dard prerequisites on Kac-Moody groups, their root data and twin buildings. The
second chapter is devoted to CAT(0) geometry. After reviewing some standard def-
initions, we recall Bruhat-Tits fixed point theorem and mention some consequences
for groups with bounded generation. The third chapter reviews B. R´emy’s construc-
tion of the adjoint representation of Kac-Moody groups, as well as the relationship
between the adjoint action and the action on the twin building. These first three
chapters are essentially preliminary and contain nothing new (although the Jor-
dan decomposition of bounded elements of Kac-Moody groups over field of positive
characteristic does not seem to appear in the literature). The next three chapters
are devoted to the isomorphism problem for Kac-Moody groups; they constitute
the heart of this work. Chapter 4 contains a statement of the isomorphism theorem,
and proceeds next to a detailed study of diagonalizable and completely reducible
subgroups of Kac-Moody groups. The rest of the proof of the isomorphism theorem
is divided up among Chapter 5 and Chapter 6, which correspond respectively to
the case of characteristic 0 and positive characteristic. Chapter 5 also contains the
3More
precisely, one expects that Kac-Moody groups of type A over finite fields are non-linear
as soon as the Coxeter diagram M (A) associated with A is non-spherical and non-affine. There
are however generalized Cartan matrices A of indefinite type such that M (A) is a Coxeter diagram
of type
˜
A 1. Over finite fields, the corresponding Kac-Moody groups should be considered with
special care, see §4.1.2.
4A
Kac-Moody group over a finite field is finitely generated and embeds as a lattice in the
product of the automorphism groups of its two buildings; this automorphism group is canonically
endowed with a locally compact totally disconnected topology. These facts fail for Kac-Moody
groups over infinite fields.
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