Irreducibility of the Koopman representations for the group \rm GL₀(2∞,\mathbb R) acting on three infinite rows

2023/07/20 by Kosyak, Alexandre, Moree, Pieter
#22E65 (28C20 #43A80 #58D20) #FOS: Mathematics #Functional Analysis (math.FA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2307.11198

Abstract

Consider the inductive limit of the general linear groups \rm GL₀(2∞,\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≤ 2. Our proof is in the same spirit, but the details are far more involved.

Related

• Criteria of irreducibility of the Koopman representations for the group \rm GL₀(2∞,\mathbb R)
• A central limit theorem for the characters of the infinite symmetric group and of the infinite Hecke algebra
• The Behavior of Laplace Transform of the Invariant Measure on the Hypersphere of High Dimension
• Asymptotics of Plancherel measures for GL(n,q)
• Irreducible Koopman representations for nonsingular actions on boundaries of rooted trees