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
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