Criteria of irreducibility of the Koopman representations for the group ${\mathrm{GL}}_0(2\infty ,\mathbb{R})$

Abstract

Our aim is to find the irreducibility criteria for the Koopman representation, when the group acts on some space with a measure (Conjecture 1.5). Some general necessary conditions of the irreducibility of this representation are established. In the particular case of the group ${\mathrm{GL}}_0(2\infty ,\mathbb{R})$ $={\underrightarrow{\lim }}_n\mathrm{GL}(2n-1,\mathbb{R})$, the inductive limit of the general linear groups we prove that these conditions are also the necessary ones. The corresponding measure is infinite tensor products of one-dimensional arbitrary Gaussian non-centered measures. The corresponding $G$-space $X_m$ is a subspace of the space $\mathrm{Mat}(2\infty ,\mathbb{R})$ of infinite in both directions real matrices. In fact, $X_m$ is a collection of $m$ infinite in both directions rows. This result was announced in [20]. We give the proof only for $m\le 2$. The general case will be studied later.