Extended Gelfand–Tsetlin Graph, Its $q$-Boundary, and $q$-B-Splines

The boundary of the Gelfand–Tsetlin graph is an infinite-dimensional locally compact space whose points parameterize the extreme characters of the infinite-dimensional group $U(\infty)$. The problem of harmonic analysis on the group $U(\infty)$ leads to a continuous family of probability measures on the boundary—the so-called zw-measures. Recently Vadim Gorin and the author have begun to study a $q$-analogue of the zw-measures. It turned out that constructing them requires introducing a novel combinatorial object, the extended Gelfand–Tsetlin graph. In the present paper it is proved that the Markov kernels connected with the extended Gelfand–Tsetlin graph and its $q$-boundary possess the Feller property. This property is needed for constructing a Markov dynamics on the $q$-boundary. A connection with the B-splines and their $q$-analogues is also discussed.