ID: 2307.11198

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

July 20, 2023

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Alexandre Kosyak, Pieter Moree
Mathematics
Representation Theory
Functional Analysis

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.

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