CHAPTER 1

The Formal Completions of G(A) and G(A)/B

§1.1 The Kac-Moody group G(A).

Given an n x n generalized Cartan matrix A, the associated Kac-Moody algebra

g = g'{A) is a certain complex Lie algebra generated by 3n letters e*, fi, and hi

(1 i n + 1 ) . The possibility n — oo is allowed. These letters satisfy the

Chevalley-Serre relations

[hi, hj] = 0, [eu fi } = hi, [ei ,fj]=Q,i^j

[tli , 6j\ = ClijCj, \P'i')Jj\ — Q'ijJj')

(adfe))1"^-

= 0,

{adUi)f'aiih

= W j .

We will always assume that A is irreducible and symmetrizable. In this case it is

known that these relations define the Lie algebra g (see Theorem 9.11 of [Kacl]).

The Lie algebra g has a triangular decomposition

g = n~ 0 f ) 0 n +

where n~ is generated by the {/j}, f ) by the {hj}, and n + by the {ej}. As a

Lie algebra g is graded by its root space decomposition; if n = oo, then we will

view 0o = fy as the direct sum 0C/ij, so that the graded pieces will all be finite

dimensional. It is also graded as a Lie algebra by height, where the heights of fj,

hj, and ej are -1, 0, and +1, respectively. We will write heights(x) k (resp, k)

if each of the nonzero homogeneous components of x has height k (resp, k).

For each 1 j n + 1 , the Lie subalgebra g^ — Cf3; © Ch3; 0 Cej is isomorphic

to 5/(2, C). In each integrable module of g, the action by g^ can be integrated to

an action by a copy G^ of SL(2, C). The group G(A), the minimal possible group

associated to g and A, is the group generated by these n copies of SL(2, C), subject

to the relations imposed by considering all integrable modules of g ([KP1] or [Kac2]).

Each G(j) is faithfully represented in G(A). Below we will use the fact that 0L(A^),

the sum of the fundamental highest weight modules, is a faithful G(A)— module.

If A is a classical Cartan matrix, then G(A) is a simply connected complex

Lie group. In general G(A) is not a Lie group. If A is an infinite classical Cartan

matrix, then G(A) is an inductive limit of finite dimensional groups (see Part II); if

A is the affine Cartan matrix associated to a Cartan matrix A, then G(A) (modulo

its center) is the polynomial loop group associated to G(A) (see Part III); in both

these cases one can complete G(A) so that the completion is a Lie group. In general

it is unknown whether (and seems unlikely that) there exists a Lie group completion.

§1.2. Formal completions, proalgebraic manifolds.

Given a countable direct sum of finite dimensional complex vector spaces, V =

0Vfc, the formal completion of V, denoted Vformal, is the topological vector space

Y\ Vk, where the topology is the product topology.

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