Point processes and the infinite symmetric group

The central theme of noncommutative harmonic analysis is decomposition of natural unitary representations T into elementary ones (i.e., into irreducible or factor representations). When T is endowed with a distinguished cyclic vector, its decomposition is governed by a measure, called the spectral or Plancherel measure. For instance, one of the achievements of the classical representation theory is the explicit calculation by Gindikin and Karpelevich of the spectral measure for the natural representation in the L 2 space on an arbitrary Riemannian symmetric space.