Disjointness of representations arising in harmonic analysis on the infinite-dimensional unitary group

We prove the pairwise disjointness of representations $T_{z,w}$ of the infinite-dimensional unitary group. These representations are a natural generalization of the regular representation to the “big” group $U(\infty)$. They were introduced and studied by G. Olshanski and A. Borodin. The disjointness of these representations reduces to that of certain probability measures on the space of paths in the Gelfand–Tsetlin graph. We prove the latter disjointness using probabilistic and combinatorial methods.