Characteristically nilpotent Lie groups with flat coadjoint orbits

TL;DR

This paper constructs specific 11-dimensional characteristically nilpotent Lie groups with flat coadjoint orbits that admit square-integrable representations, even though their quotients by the center lack dilation structures.

Contribution

It introduces a new two-parameter family of nilpotent Lie groups with flat coadjoint orbits that defy previous assumptions about dilation structures.

Findings

Constructed a two-parameter family of Lie groups in dimension 11.

These groups admit square-integrable representations modulo the center.

Their quotients by the center do not admit a family of dilations.

Abstract

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 $G(\alpha,\beta)$ in dimension $11$, admitting square-integrable representations modulo the center $Z$, such that $G(\alpha,\beta)/Z$ does not admit a family of dilations.