Point processes of representation theoretic origin

Abstract

There are two parts to this thesis. In the first part we compute the correlation functions of the 4-parameter family of BC type Z-measures. The result is given explicitly in terms of Gauss's hypergeometric function. The BC type Z-measures are point processes on the punctured positive real line. They arise as interpolations of the spectral measures of a distinguished family of spherical representations of certain infinite-dimensional symmetric spaces. In representation-theoretic terms, our result solves the problem of noncommutative harmonic for the aforementioned family of representations. The second part of the text is based on joint work with Grigori Olshanski. We consider a new 5-parameter family of probability measures on the space of infinite point configurations of a discrete lattice. One of the 5 parameters is a quantization parameter and the measures in the family are closely related to the BC type Z-measures. We prove that the new measures serve as orthogonality weights for symmetric function analogues of the multivariate q-Racah polynomials. Further we show that the q-Racah symmetric functions (and their corresponding orthogonality measures) can be degenerated into symmetric function analogues of the big q-Jacobi, q-Meixner and Al-Salam-Carlitz polynomials, thus giving rise to a partial q-Askey scheme hierarchy in the algebra of symmetric functions.