Irreducibility of the Koopman representations for the group ${\mathrm{GL}}_0(2\infty ,\mathbb{R})$ acting on three infinite rows

Abstract

Consider the inductive limit of the general linear groups ${\mathrm{GL}}_0(2\infty ,\mathbb{R})$ $={\underrightarrow{\lim }}_n\mathrm{GL}(2n-1,\mathbb{R})$, acting on the space $X_m$ of $m$ rows, infinite in both directions, with Gaussian measure. This measure is the infinite tensor product of one-dimensional arbitrary Gaussian non-centered measures. In this article we prove an irreducibility criterion for $m=3$. In 2019, the first author [28] established a criterion for $m\le 2$. Our proof is in the same spirit, but the details are far more involved.

Small changing in Section 8.3