Extended Gelfand-Tsetlin graph, its q-boundary, and q-B-splines

Abstract

A continuation of the joint work by Vadim Gorin and the author, J. Funct. Anal. 270 (2016), 375-418; arXiv:1504.06832. The extended Gelfand-Tsetlin graph, introduced in that paper, is a novel combinatorial object. Its q-boundary is formed by infinite point configurations on a two-sided q-lattice. The q-boundary carries a continuous family of probability measures that are a q-analogue of the so-called zw-measures, which originated in the problem of harmonic analysis on the infinite-dimensional unitary group. In the present paper, it is proved that certain transition Markov kernels, linked to the extended Gelfand-Tsetlin graph and its q-boundary, possess the Feller property. This property is needed for constructing a model of Markov dynamics on the q-boundary. A connection with the classical B-splines and their q-analogues is discussed. Some results of the paper arXiv:1504.06832 are rederived in another way.