2 1. THE OBJECTS: KAC-MOODY GROUPS, ROOT DATA AND TITS BUILDINGS

Let D = (I, A, Λ, (ci)i∈I , (hi)i∈I ) be a Kac-Moody root datum. Let TΛ is the

split torus scheme, i.e. TΛ is the group functor on the category of rings defined

by TΛ(R) = Homgr(Λ,

R×).

With the datum D, J. Tits [Tit87b, §3.6] associates a system F =

(G, (ϕi)i∈I , η) consisting of:

• a group functor G on the category of rings,

• a collection (ϕi)i∈I of morphisms of functors ϕi : SL2 → G,

• a morphism of functors η : TΛ → G,

which satisfies the following conditions, where

rhi

denotes the element λ → r

λ,hi

of TΛ:

(KMG1): if K is a field, G(K) is generated by the images of ϕi(K) and

η(K);

(KMG2): for every ring R, the homomorphism η(R) : TΛ(R) → G(R) is

injective;

(KMG3): for i ∈ I and r ∈ R×, one has ϕi

r 0

0

r−1

= η(rhi );

(KMG4): if ι is an injective homomorphism of a ring R in a field K, then

G(ι) : G(R) → G(K) is injective;

(KMG5): there is a homomorphism Ad : G(C) → Aut(gA) whose kernel is

contained in η(TΛ(C)), such that, for c ∈ C,

Ad ϕi

1 c

0 1

= exp ad cei,

Ad ϕi

1 0

c 1

= exp ad (−cfi),

and, for t ∈ TΛ(C),

Ad(η(t))(ei) = t(ci) · ei, Ad(η(t))(fi) = t(−ci) · fi.

The group functor G as above is called a Tits functor of type D and of

basis F. By definition, a (split) Kac-Moody group of type D over a field K is

the value on K of a Tits functor of type D. The main result of [Tit87b] asserts

that the restriction of G to the category of fields is completely characterized by the

conditions (KMG1)–(KMG5) modulo some additional non-degeneracy condition on

the images of the ϕi’s (see [Tit87b, Theorem 1] for a precise statement).

1.1.4. An alternative construction in the 2-spherical case. Let A be

a generalized Cartan matrix over a (finite) set I. For each subset J ⊂ I, we set

AJ := (Aij )i,j∈J . The matrix A is called 2-spherical if for every 2-subset J of I

the matrix AJ is a (classical) Cartan matrix. Equivalently, A is 2-spherical if and

only if Aij Aji ≤ 3 for all i = j ∈ I.

In this section we present an explicit construction of Kac-Moody

groups of type D, where D = (I, A, Λ, (ci)i∈I , (hi)i∈I ) is the simply connected

Kac-Moody root datum associated with a 2-spherical generalized Cartan matrix A.

Let K be a field and assume that K is of cardinality at least 3 (resp. at least

4) if Aij = −2 (resp. Aij = −3) for some i, j ∈ I. For each i ∈ I, let Xi be a copy

of SL2(K) and for each 2-subset J = {i, j} of I, let Xi,j be a copy of the universal

Chevalley group of type AJ over K. Let also ϕi,j : Xi → Xi,j be the canonical

monomorphism corresponding to the inclusion of Cartan matrices A{i} → A{i,j}.

The direct limit of the inductive system formed by the groups Xi and Xi,j along