0. General Introduction

In this introduction I will describe some general conjectures about the existence

and uniqueness of invariant measures for unitary groups associated to symmetriz-

able Kac-Moody Lie algebras. In the body of the paper, I will prove the existence

parts of the conjectures for affine algebras, i.e. for infinite classical algebras and

loop algebras, and also the Virasoro algebra.

I will use the book [Kacl] and the survey article [Kac2] as general references

for the basics of the theory of Kac-Moody algebras and associated groups. For

readers principally interested in loop groups or Diff(S1)1 familiarity with [PS]

should suffice (the basic terminology concerning Kac-Moody algebras is presented

in chapter 5, and motivation for much of what follows is presented in §14.5).

§0.1. Some conjectures on existence of invariant measures.

Let A denote an indecomposable generalized n x n Cartan matrix which is

symmetrizable. Let G(A) denote the complex algebraic group associated to A,

in the sense introduced by Kac and Peterson ([KP1]), and let K(A) denote the

unitary real form of G(A). If A is of finite type, i.e. if n is finite and A is a classical

Cartan matrix, then in particular K(A) is a compact topological group, so that

there exists a unique K(A)-biinvariant probability measure supported on K(A),

the Haar measure of K(A). Conversely the existence of a (left or right) K(A)-

invariant measure class supported on K(A) implies that K(A) is locally compact,

by Mackey and Weil's converse to Haar's theorem, hence that A is of finite type.

To go beyond the finite type case we are guided by the fact that the most

elemental functions which we would like to be able to integrate, namely the matrix

coefficients of integrable highest weight representations of G(A), naturally define

regular functions on a space which we will call the formal completion of G(A), and

which we will denote either by Q or G(A)forrnai. This space is modelled on the

linear space

nformal X

Vformal

X nformal'

where xi^ormal is the direct product of the root spaces corresponding to positive

roots (resp., negative roots), and rjformal is the direct product of the coroot spaces

(if n oo, then 1)formal = f})- The transition functions arise from continuously

extending the translational action of G(A) on itself, expressed in terms of the

coordinates provided by the top stratum of the Birkhoff decomposition,

N~ x H x

N+

— • G(A) : (I, diag, u) — g = I • diag • u

(for readers not familiar with the Birkhoff decomposition, see page 190 of [Kac2],

or chapter 8 of [PS] for the loop group case). The two-sided translational action

of G(A) on itself extends continuously to a holomorphic two-sided action of G(A)

l