Quasi-Invariant Measures and Irreducible Representations of the Inductive Limit of Special Linear Groups

Unitary representations of the group $G=\operatorname{SL}_0(2\infty,\mathbb{R})=\varinjlim_{n}\operatorname{SL}(2n-1,\mathbb{R})$ are constructed. The construction uses $G$-quasi-invariant measures on some $G$-spaces that are subspaces of the space $\operatorname{Mat}(2\infty,\mathbb{R})$ of two-way infinite real matrices. We give a criterion for the irreducibility of these representations.