x INTRODUCTION
for some x
K×.
An automorphism of SL2(K) is called diagonal-by-sign if it is
either diagonal or the composite of a diagonal automorphism with a transpose-
inverse.
Our main result is the following.
Theorem A. Let (A, G, (ϕi)i∈I ) and (A , G , (ϕi)i∈I ) be systems as above and
let K, K be fields. Let ϕ : G(K) G (K ) be an isomorphism.
Suppose that |K|≥ 4 and G(K) is infinite. Then there exist an inner automor-
phism α of G(K), a bijection π : I I and, for each i I, a field isomorphism
ζi : K K , a diagonal-by-sign automorphism δi of SL2(K) such that the diagram
SL2(K)
SL2(ζi−−−−)
SL2(K )
ϕi◦δi




ϕπ(i)
G(K)
−ϕ◦α
−−− G (K )
commutes for every i I. Furthermore, if K is infinite then
Aij Aji = Aπ(i)π(j)Aπ(j)π(i)
for all i, j I and if char(K) = 0 then Aij = Aπ(i)π(j) for all i, j I.
It follows in particular that the isomorphism ϕ induces an isomorphism of the
respective Weyl groups of G(K) and G (K ) which preserves the set of canonical
generators.
Theorem A can be used to characterize automorphisms of the Kac-Moody group
G(K). Denoting by (Uα)α∈Φ the system of root groups of G(K), it follows that, un-
der the hypotheses of the theorem, every automorphism of G(K) leaves the union
of the conjugacy classes of U+ and U− invariant, where U+ := Uα| α Φ+ and
U− := Uα| α Φ+ . This fact in turn yields naturally a decomposition of any
automorphism of G(K) as a product of an inner automorphism, a sign automor-
phism, a diagonal automorphism, a field automorphism and a graph automorphism
(see Theorem 4.2 below). In the case where char(K) = 0 or char(K) is prime to ev-
ery off-diagonal entry of the generalized Cartan matrix A, this provides a complete
description of the group Aut(G(K)).
The proof of Theorem A combines the use of the two main available tools to
explore the structure of a Kac-Moody group G(K). The first one is the strongly
transitive action of G(K) on a twin building B, constructed by M. Ronan and J. Tits
and described in [Tit90], [Tit92]. Such a twin building B consists of the product
of two thick buildings, say B+ × B−, each corresponding to a BN-pair of G(K).
Both BN-pairs have the same Weyl group; actually, the strong link which relates
these two BN-pairs is an opposition relation between their respective Borel groups.
This opposition relation yields an opposition relation between the chambers of B+
and B−, which is called a twinning. The existence of such a twinning invariant
under the diagonal G(K)-action makes these structures rather rigid, as we will see
in the sequel.
The second tool is of more algebraic nature: It is the adjoint representation
of the Kac-Moody group G(K) on a K-vector space obtained by tensoring up a
Z-form2
of the universal enveloping algebra of the Kac-Moody algebra gA of type
2This
Z-form was constructed by Tits [Tit87b] and plays a fundamental role in the construc-
tion of Tits functors.
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