Criteria of irreducibility of the Koopman representations for the group \rm GL₀(2∞,\mathbb R)

2016/10/15 by Alexandre Kosyak, Kosyak, Alexandre
Mathematics · #22E65 (28C20 #43A80 #58D20) #Advanced Algebra and Geometry #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1610.04710

openalex publication_date 2016/10/15 · openalex created_date 2016/10/28 · openalex updated_date 2026/07/28

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 \rm GL₀(2∞,\mathbb R) = \varinjlim_n\rm 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 \rm Mat(2∞,\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≤ 2. The general case will be studied later.

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