Characteristically nilpotent Lie groups with flat coadjoint orbits
Dietrich Burde, Jordy Timo van Velthoven
TL;DR
The paper addresses the existence of characteristically nilpotent Lie groups with flat coadjoint orbits that still admit square‑integrable representations modulo the center but do not admit a family of dilations on the quotient by the center. It constructs an explicit two‑parameter family of 11‑dimensional filiform Lie algebras $\mathfrak{g}(\alpha,\beta)$, with one‑dimensional center, such that $\mathfrak{g}(\alpha,\beta)/\mathfrak{z}$ is characteristically nilpotent for all $(\alpha,\beta)\neq(0,0)$ and there exists $\ell\in\mathfrak{g}(\alpha,\beta)^*$ with $\mathfrak{g}(\alpha,\beta)(\ell)=\mathfrak{z}$, yielding square‑integrable representations modulo the center. Consequently, the associated groups $G(\alpha,\beta)/Z$ admit square‑integrable projective representations but do not admit a family of dilations, illustrating a separation between square‑integrability and dilation‑structure in nilpotent Lie groups. The results rely on a careful analysis of automorphisms, gradings, and derivations, and provide explicit CN examples with flat orbits relevant to the theory of orthonormal bases in square‑integrable representations.
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 $G(α,β)$ in dimension $11$, admitting square-integrable representations modulo the center $Z$, such that $G(α,β)/Z$ does not admit a family of dilations.