Regular, Quasi-regular and Induced Representations of Infinite-dimensional Groups

Almost all harmonic analysis on locally compact groups is based on the existence (and uniqueness) of a Haar measure. Therefore, it is very natural to attempt a similar construction for non-locally compact groups. The essential idea is to replace the non-existing Haar measure on an infinite-dimensional group by a suitable quasi-invariant measure on an appropriate completion of the initial group or on the completion of a homogeneous space. The aim of the book is a systematic development, by example, of noncommutative harmonic analysis on infinite-dimensional (non-locally compact) matrix groups. We generalize the notion of regular, quasi-regular and induced representations for arbitrary infinite-dimensional groups. The central idea to verify the irreducibility is the Ismagilov conjecture. We also extend the Kirillov orbit method for the group of upper triangular matrices of infinite order. In order to make the content accessible to a wide audience of nonspecialists, the exposition is essentially self-contained and very few prerequisites are needed. The book is aimed at graduate and advanced undergraduate students, as well as mathematicians who wish an introduction to representations of infinite-dimensional groups. Keywords: Quasi-invariant measure on infinite-dimensional group, ergodic measure, Hilbert–Lie group, unitary, irreducible, regular, quasi-regular, induced representations, Ismagilov conjecture, Schur–Weil duality, Kirillov orbit method, von Neumann algebras, factor, type of factors, C*-group algebras, finite field