Multidimensional hypergeometric distribution, and characters of the unitary group

The present paper is the publication of work notes by S. V. Kerov (1946–2000) written in 1993. The author introduces a multidimensional analog of the classical hypergeometric distribution. This is a probability measure $M_n$ on the set of Young diagrams contained in the rectangle with $n$ rows and $m$ columns. The fact that the expression for $M_n$ defines a probability measure is a nontrivial combinatorial identity, which is proved in various ways. Another combinatorial identity analyzed in the paper expresses a certain compatibility of the measures $M_n$ and $M_{n+1}$. A link with Selberg type integrals is also pointed out. The work is motivated by the problem of harmonic analysis on the infinite-dimensional unitary group.