On the regular representation of solvable Lie groups with open coadjoint quasi-orbits (Q6583663)

The paper studies the regular representations of solvable Lie groups, specifically those with open coadjoint quasi-orbits and presents a Lie-theoretic characterization of these groups. The concept of coadjoint quasi-orbits, a key tool used in this study, was first introduced by \textit{L. Pukanszky} [Ann. Sci. Éc. Norm. Supér. (4) 4, 457--608 (1971; Zbl 0238.22010)]. Pukanszky's theory provides foundational methods for analyzing the quasi-orbits and has been pivotal in the approach of this paper to solvable Lie groups. The fundamental focus is on identifying connected, simply connected solvable Lie groups for which the von Neumann algebras isomorphic to the hyperfinite specifically type II\(_{\infty}\) factors. This characteristic enables a detailed understanding of these groups' algebraic structure, extending previous results on group representations. The paper has four sections with the structure is logical, moving from theoretical background to specific applications and examples. Section 1 introduces the main problem, presents background on factor representations and outlines the known results for discrete and Lie groups. Section 2 reviews elements of Pukanszky theory, laying a mathematical foundation essential for the main results. The section covers the concept of quasi-orbits and related representation techniques. Section 3 introduces the main results of the paper. The first result is Theorem 3.5, which proves that an intrinsic characterization of solvable Lie groups with open coadjoint quasi-orbits whose regular representation is a factor representation. The second result is Corollary 3.6 which proves that when the group's regular representation is a factor, the von Neumann algebra is isomorphic to the hyperfinite type II\(_{\infty}\) factor. The last result is Corollary 3.7 which shows that for considered groups, all Casimir functions are constant, underscoring the rigidity in their structure. Section 4 explores the complementary case of solvable Lie groups with open coadjoint orbits. It shows that in these cases, the von Neumann algebra generated by the regular representation is always of type I (Corollary 4.2) and includes examples to illustrate this property. This study significantly extends the understanding of connected, simply connected solvable Lie groups and their representations, providing valuable insights into both the algebraic structure of these groups and their applications in mathematical physics.