Journal of Lie Theory 35 (2025), No. 2, 227--237
Copyright Heldermann Verlag 2025
Characteristically Nilpotent Lie Groups with Flat Coadjoint Orbits
Dietrich Burde
Fakultät für Mathematik, Universität Wien, Wien, Austria
dietrich.burde@univie.ac.at
Jordy Timo van Velthoven
Fakultät für Mathematik, Universität Wien, Wien, Austria
jordy.timo.van.velthoven@univie.ac.at
We study the existence of certain characteristically nilpotent Lie algebras with flat coadjoint orbits. Their connected, simply connected Lie groups admit square-integrable representations modulo the center. There are many examples of nilpotent Lie groups admitting families of dilations and square-integrable representations. Much less is known about examples admitting square-integrable representations for which the quotient by the center does not admit a family of dilations. In this paper we construct a two-parameter family of characteristically nilpotent Lie groups G(α, β) in dimension 11, admitting square-integrable representations modulo the center Z, such that G(α, β)/Z does not admit a family of dilations.
Keywords: Characteristically nilpotent Lie algebras, dilations, flat coadjoint orbits.
MSC: 17A36, 17B30, 22E27.
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