Point Processes and the Infinite Symmetric Group. Part IV: Matrix Whittaker kernel

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.