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

TL;DR

This paper proves an irreducibility criterion for the Koopman representations of the inductive limit of general linear groups acting on a space with Gaussian measure, extending previous results from two to three rows.

Contribution

It extends the irreducibility criterion for Koopman representations from two to three infinite rows, providing more comprehensive understanding of these group actions.

Findings

Established irreducibility criterion for m=3

Extended previous m≤2 results to m=3

Involved complex proof techniques

Abstract

Consider the inductive limit of the general linear groups ${\rm GL}_0(2\infty,{\mathbb R})$ $= \varinjlim_{n}{\rm GL}(2n-1,{\mathbb R})$, 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.