  We study a 2-parametric family of probability measures on the space of
countable point configurations on the punctured real line (the points of the
random configuration are concentrated near zero). These measures (or,
equivalently, point processes) have been introduced in Part II (A. Borodin,
math.RT/9804087) in connection with the problem of harmonic analysis on the
infinite symmetric group. The main result of the present paper is a
determinantal formula for the correlation functions.
  The formula involves a kernel called the matrix Whittaker kernel. Each of its
two diagonal blocks governs the projection of the process on one of the two
half-lines; the corresponding kernel on the half-line was studied in Part III
(A. Borodin and G. Olshanski, math/RT/9804088).
  While the diagonal blocks of the matrix Whitaker kernel are symmetric, the
whole kernel turns out to be $J$-symmetric, i.e., symmetric with respect to a
natural indefinite inner product.
  We also discuss a rather surprising connection of our processes with the
recent work by B. Eynard and M. L. Mehta (cond-mat/9710230) on correlations of
eigenvalues of coupled random matrices.
