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

In the context of the problem of harmonic analysis on the group U(∞) a family of representations Tz,w was constructed and studied in the papers [Olsh2] and [BO]. These representations depend on two complex parameters and provide a natural generalization of the regular representation for the case of “big” group U(∞). The representation Tz,w does not change if z or w is replaced by z or w, respectively. The structure of the decomposition of Tz,w substantially depends on whether parameters are integers or not. We handle the latter case. Our aim is to prove the disjointness of the representations Tz,w. Recall that two representations T and T ′ are called disjoint if they have no equivalent nonzero subrepresentations. In this paper we prove the following theorem: