๐”– Bobbio Scriptorium
โœฆ   LIBER   โœฆ

Continuous Families of Quasi-Regular Representations of Solvable Lie Groups

โœ Scribed by D. Schueth

Publisher
Elsevier Science
Year
1995
Tongue
English
Weight
566 KB
Volume
134
Category
Article
ISSN
0022-1236

No coin nor oath required. For personal study only.

โœฆ Synopsis

We generalize a result about two-step nilpotent Lie groups by Carolyn Gordon and He Ouyang: Let (\left{\Gamma_{t}\right}{1 \geqslant 0}) be a continuous family of uniform discrete subgroups of a simply connected Lie group (G) such that the quasi-regular representations of (G) on (L^{2}\left(\Gamma{t} \backslash G\right)) are unitarily equivalent. Then if (G) is solvable with only real roots, there exists a unique continuous family (\left{\Phi_{t}\right}{t \geqslant 0}) of (\Gamma{0})-almost inner automorphisms of (G) such that (\Gamma_{t}=\Phi_{1}\left(\Gamma_{0}\right)). An even stronger result is true for exponentially solvable Lie groups (not necessarily with real roots) whose nilradical is abelian and of codimension one, at least if we assume that (\left{\Gamma_{t}\right}_{t \geqslant 0}) is not only continuous but (C^{1}) Here the (\Gamma), are related not only by almost inner, but by inner automorphisms.

' 1995 Academic Press. Inc

๐Ÿ“œ SIMILAR VOLUMES

Moyal Product and Representations of Sol
Moyal Product and Representations of Solvable Lie Groups
โœ D Arnal; J.C Cortet; J Ludwig ๐Ÿ“‚ Article ๐Ÿ“… 1995 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 640 KB

Using a parametrization for the universal covering \(c_{10}\) of any coadjoint orbit \(c\) of a solvable (connected and simply connected) Lie group \(G\), we prove that the Moyal product on \(\mathfrak{c}_{0}\) gives a realization of the unitary representations canonically associated to the orbit (

Holomorphically Induced Representations
Holomorphically Induced Representations of Some Solvable Lie Groups
โœ Junko Inoue ๐Ÿ“‚ Article ๐Ÿ“… 2001 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 508 KB

We study holomorphically induced representations r of Lie groups G=exp g from weak polarizations h at f ยฅ g\*. When G is a connected and simply connected Lie group whose Lie algebra is a normal j-algebra, we obtain a sufficient condition for non-vanishing of r and the decomposition of r into irreduc

Equivalence of Quasi-regular Representat
Equivalence of Quasi-regular Representations of Two and Three-Step Nilpotent Lie Groups
โœ R. Gornet ๐Ÿ“‚ Article ๐Ÿ“… 1994 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 645 KB

We characterize all pairs of cocompact, discrete subgroups \(\Gamma_{1}\) and \(\Gamma_{2}\) of a twostep nilpotent Lie group \(M\) such that the quasi-regular representations of \(M\) on \(L^{2}\left(\Gamma_{1} \backslash M\right)\) and \(L^{2}\left(\Gamma_{2} \backslash M\right)\) are unitarily eq