Vershik, A. M.; Gel’fand, I. M.; Graev, M. I. A commutative model of presentation of the group of flows \(SL(2,R)^ X\) that is connected with a unipotent subgroup. (English. Russian original) Zbl 0536.22008 Funct. Anal. Appl. 17, 137-139 (1983); translation from Funkts. Anal. Prilozh. 17, No. 2, 70-72 (1983). Let G be a metrizable topological group, \(G_ 0\) a commutative subgroup and \(\Pi\) a unitary representation of the group G. If the restriction of \(\Pi\) on \(G_ 0\) is cyclic, then the family of operators \(\{\Pi(a),a\in G_ 0\}\) can be reduced to diagonal form by an isometric operator. The corresponding realization of the representation of G will be called a commutative model of the representation \(\Pi\) with respect to \(G_ 0\). The group of restricted measurable functions on a manifold X with smooth measure with values in SL(2,R) is defined as the group of flows \(SL(2,R)^ X\). In [Usp. Mat. Nauk, 28, No.5, 82-128 (1973; Zbl 0288.22005)] a unitary representation of the group \(SL(2,R)^ X\) is described by the authors. Now the commutative model of this representation with respect to the unipotent subgroup \(\left\{ \begin{pmatrix} 1 & 0 \\ \gamma(\cdot) & 1 \end{pmatrix} \right\}\) of \(SL(2,R)^ X\) is constructed. In three propositions theoretical foundations are given, the construction of a representation is pointed out. It is proved that this representation is the wanted model. Reviewer: K.Riives Cited in 1 ReviewCited in 14 Documents MSC: 22A25 Representations of general topological groups and semigroups Keywords:group of flows; commutative model of fundamental representation; commutative subgroup; unitary representation; diagonal form; commutative model; SL(2,R); unipotent subgroup Citations:Zbl 0288.22005 PDFBibTeX XMLCite \textit{A. M. Vershik} et al., Funct. Anal. Appl. 17, 137--139 (1983; Zbl 0536.22008); translation from Funkts. Anal. Prilozh. 17, No. 2, 70--72 (1983) Full Text: DOI References: [1] A. M. Vershik, I. M. Gel’fand, and M. I. Graev, ”Representations of the group SL(2, R), where R is a ring of functions,” Usp. Mat. Nauk,28, No. 5, 83-128 (1973). · Zbl 0288.22005 [2] I. M. Gel’fand and N. Ya. Vilenkin, Applications of Harmonic Analysis, Academic Press (1964). [3] I. M. Gel’fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions [in Russian], Nauka, Moscow (1966). [4] A. V. Skorokhod, Random Processes with Independent Increments [in Russian], Nauka, Moscow (1964). · Zbl 0132.12504 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.