Harmonic analysis on the infinite symmetric group, and the Whittaker kernel (Q2761454)

Let \(S(\infty)\) be the group of finite permutations of the natural numbers. \textit{S. Kerov, G. Olshanski}, and \textit{A. Vershik} [C. R. Acad. Sci. Paris, Sér. I 316, 773-778 (1993; Zbl 0796.43005)] defined a family of generalized regular representations of \(S(\infty)\) depending on a complex parameter. These representations can be seen as deformations of an analog of the regular representation which is irreducible (such a phenomenon is possible only for ``big'' groups). NEWLINENEWLINENEWLINEIn the paper under review the author gives a description of spectral measures for representations from the above family. The spectral measures are defined over an infinite-dimensional simplex, and their description is given in terms of certain stochastic point processes associated with them. The correlation functions of the processes are found; they are expressed explicitly via the Whittaker functions.