Quasiregular representations of the infinite-dimensional Borel group
The notion of quasiregular or geometric representation is well known for locally compact groups [\textit{A. A. Kirillov}, Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften 220 (Springer, Berlin) (1976; Zbl 0342.22001); Introduction to the theory of representations and noncommutative harmonic analysis, Representation theory and noncommutative harmonic analysis I, Encycl. Math. Sci. 22 (Springer, Berlin) 1--156 (1994; Zbl 0805.43004); \textit{D. P. Zhelobenko} and \textit{A. I. Shtern}, Representations of Lie Groups (Nauka, Moscow) (1983; Zbl 0521.22006)]. In this paper, the authors construct an analog of the quasiregular representation for the solvable infinite-dimensional Borel group and present a criterion of irreducibility of the constructed representations. This construction uses G-quasi-invariant Gaussian measures on some G-spaces and extends the method used in [\textit{A. V. Kosyak}, Funkts. Anal. Prilozh. 37, 78--81 (2003; Zbl 1022.22020)] for the construction of the quasiregular representations as applied to the nilpotent infinite-dimensional group.
scientific article; zbMATH DE number 2136768
scientific article