Irreducibility Criterion for Quasiregular Representations of the Group of Finite Upper Triangular Matrices

An analog of the quasiregular representation is defined for the group of infinite-order finite upper triangular matrices. It uses $G$-quasi-invariant measures on some $G$-spaces. The criterion for the irreducibility and equivalence of the constructed representations is given. This criterion allows us to generalize Ismagilov's conjecture on the irreducibility of an analog of regular representations of infinite-dimensional groups.