标题： 点过程与无限对称群。 第四部分：矩阵Whittaker核 显示英文标题

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

摘要： 我们研究了一类定义在穿孔实线（随机点配置的点集中在零附近）上的计数点配置空间上的双参数概率测度族。 这些测度（或等价地，点过程）在第二部分（A. Borodin, math.RT/9804087）中被引入，与无限对称群上的调和分析问题有关。 本文的主要结果是给出了相关函数的一个行列式公式。 该公式涉及一个被称为矩阵Whittaker核的核。 其两个对角块分别控制了该过程在两条半直线之一上的投影；在半直线上对应的核在第三部分（A. Borodin 和 G. Olshanski, math/RT/9804088）中被研究过。 虽然矩阵Whittaker核的对角块是对称的，但整个核却具有$J$-对称性，即关于自然不定内积对称。 我们还讨论了我们的过程与B. Eynard 和 M. L. Mehta（cond-mat/9710230）最近关于耦合随机矩阵特征值相关性的研究之间相当令人惊讶的联系。 显示英文摘要

摘要： 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.