Introduction

In a general sense, a Kac-Moody group is a group attached to a Kac-Moody Lie

algebra. The initial motivation of the construction of these algebras by V. Kac and

R. Moody has been largely surpassed nowadays by the spectacular developments

and ramified applications that their theory has known since the origin. As the

complex semisimple Lie algebras coincide with the finite-dimensional Kac-Moody

algebras, it is natural and useful to ask whether one can obtain interesting groups by

‘integrating’ these algebras. Although it became quickly clear that this question had

an aﬃrmative answer, the actual construction of the corresponding groups turned

out to be a delicate problem, whose definitive solution was given by J. Tits [Tit87b].

We refer to [Tit89] for a thorough historical and comparative introduction to the

different ways of constructing a Kac-Moody group.

Given a generalized Cartan matrix A = (Aij )i,j∈I , i.e. a matrix with

integral coeﬃcients such that Aii = 2, Aij ≤ 0 and Aij = 0 ⇔ Aji = 0 for all

i = j ∈ I, Tits [Tit87b] constructs a group

functor1

G on the category of rings,

together with a family (ϕi)i∈I of morphisms of functors SL2 → G, and shows that

the restriction of G to the category of fields is completely characterized by a short

list of simple properties, one of which being the existence of a natural adjoint action

of G(C) on the Kac-Moody algebra g(A) of type A. These functors will be called

Tits functors in the sequel. By definition, a split Kac-Moody group over a field

K is a group obtained by evaluating a Tits functor on K.

The aforementioned characterization of Tits functors is inspired by the scheme-

theoretic definition of algebraic groups. It paves thereby the way for a development

of the structure theory of Kac-Moody groups which draws naturally parallels to the

rich and well known theory of algebraic groups. This program has been carried out

to a certain extent by several mathematicians among whom V. Kac, D. Peterson, G.

Rousseau and B. R´ emy (see [R´ em02b] and references therein). In this respect, the

study of automorphisms of split Kac-Moody groups, which is the central theme of

this work, is aimed to parallel the celebrated theory of “abstract” homomorphisms

of algebraic groups by A. Borel and J. Tits [BT73].

Recall that a diagonal automorphism of SL2(K) is an automorphism of the

form

a b

c d

→

a xb

x−1c

d

1Actually,

the parameter system of a Tits functor consists of a ‘Kac-Moody root datum’,

which is a richer structure than just a generalized Cartan matrix and generalizes in an appropriate

way the data which classify reductive groups (see §1.1.3). In order to avoid irrelevant technicalities

in this introduction, we only emphasize the dependence on a generalized Cartan matrix. The

results we state hold for all types of Tits functors.

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