Equivalence of Quasi-regular Representations of Two and Three-Step Nilpotent Lie Groups

✍ Scribed by R. Gornet

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 645 KB
• Volume: 119
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1994.1005

✦ Synopsis

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 equivalent. This characterization is used to construct new examples. We also describe a pair of nonisomorphic, cocompact, discrete subgroups (\Gamma_{1}) and (\Gamma_{2}) of a three-step 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(r_{2} \backslash M\right)) are unitarily equivalent. |C 1994 Academic Press, Inc.