Plancherel Formulae associated to Filtrations of Nilpotent Lie Groups

2012/12/09 by Joseph A. Wolf, Wolf, Joseph A
Mathematics · #32L10 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Primary 32L25 #Representation Theory (math.RT) #Secondary 22E46

paper · pdf · doi:10.48550/arxiv.1212.1908

openalex publication_date 2012/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the conditions for a nilpotent Lie group to be foliated into subgroups that have square integrable (relative discrete series) unitary representations, that fit together to form a filtration by normal subgroups. Then we use that filtration to construct a class of "stepwise square integrable" representations on which Plancherel measure is concentrated. Further, we work out the character formulae for those stepwise square integrable representations, and we give an explicit Plancherel formula. Next, we use some structure theory to check that all these constructions and results apply to nilradicals of minimal parabolic subgroups of real reductive Lie groups. Finally, we develop multiplicity formulae for compact quotients N/Γ where Γ respects the filtration.

Related

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• Stepwise Square Integrable Representations: the Concept and Some\n Consequences
• The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula II. Non-tube type groups
• Stepwise Square Integrability for Nilradicals of Parabolic Subgroups and\n Maximal Amenable Subgroups