  We give a summary of the results from Parts I-V (math.RT/9804086,
math.RT/9804087, math.RT/9804088, math.RT/9810013, math.RT/9810014).
  Our work originated from harmonic analysis on the infinite symmetric group.
The problem of spectral decomposition for certain representations of this group
leads to a family of probability measures on an infinite-dimensional simplex,
which is a kind of dual object for the infinite symmetric group. To understand
the nature of these measures we interpret them as stochastic point processes on
the punctured real line and compute their correlation functions. The
correlation functions are given by multidimensional integrals which can be
expressed in terms of a multivariate hypergeometric series (the Lauricella
function of type B). It turns out that after a slight modification (`lifting')
of the processes the correlation functions take a common in Random Matrix
Theory (RMT) determinantal form with a certain kernel.
  The kernel is expressed through the classical Whittaker functions. It depends
on two parameters and admits a variety of degenerations. They include the
well-known in RMT sine and Bessel kernels as well as some other Bessel-type
kernels which, to our best knowledge, are new.
  The explicit knowledge of the correlation functions enables us to derive a
number of conclusions about the initial probability measures.
  We also study the structure of our kernel; this finally leads to a
constructive description of the initial measures.
  We believe that this work provides a new promising connection between RMT and
Representation Theory.
