An introduction to harmonic analysis on the infinite symmetric group

Abstract

The aim of the present survey paper is to provide an accessible introduction to a new chapter of representation theory - harmonic analysis for noncommutative groups with infinite-dimensional dual space. I omitted detailed proofs but tried to explain the main ideas of the theory and its connections with other fields. The fact that irreducible representations of the groups in question depend on infinitely many parameters leads to a number of new effects which never occurred in conventional noncommutative harmonic analysis. A link with stochastic point processes is especially emphasized. The exposition focuses on a single group, the infinite symmetric group. The reason is that presently this particular example is worked out the most. Furthermore, the infinite symmetric group can serve as a very good model for more complicated groups like the infinite-dimensional unitary group.