Contents

General Introduction 1

Some conjectures on existence of invariant measures - Basic intuition

and compactness - Comments on contents and organization.

Part I. General Theory

Chapter 1. The Formal Completions of G(A) and G(A)/B 9

The Kac-Peterson group G(A) - Birkhoff decompositions - The spaces

Q = G(A)formal and T — Q/B+ - The infinitesimal actions of Q(A)

on Q and T - Formal completions and the Kac-Peterson version of the

Peter-Weyl and Borel-Weil theorems.

Chapter 2. Measures on the Formal Flag Space 18

The top stratum has full measure - On uniqueness of K(A)-invariant

measures on T - Compactness results for measures.

Part II. Infinite Classical Groups

Chapter 0. Introduction for Part II 27

Kac-Peterson completions - Segal's theory of Hilbert-Schmidt restricted

unitary groups - Olshansky's theory of unitary representations.

Chapter 1. Measures on the Formal Flag Space 29

Existence and Uniqueness.

Chapter 2. The Case g = s/(oo, C) 30

Classification of biinvariant measures on Q - Plancherel formula - Diag-

onal distributions.

Chapter 3. The Case g = sl(2oc, C) 34

Existence and explicit expressions for biinvariant measures on Q/C -

Finite dimensional approximations and alternative proofs of existence

- Nonexistence of biinvariant measures on Q - Continuity properties of

unitary representations.

Chapter 4. The Cases g = o(2oo, C), o(2oo + 1, C), sp(oc,C) 42

Realizations - Biinvariant measures and diagonal distributions for finite

dimensional groups - Existence of biinvariant measures on Q/C.

Part III. Loop Groups

Chapter 0. Introduction for Part III 53