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(, ) in dimension 11 , admitting square-integrable representations modulo the center Z , such that G(, )/Z does not admit a family of dilations.
Burde et al. (Wed,) studied this question.