Benson-Ratcliff conjecture on solvable Lie groups
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The paper deals with the invariant for unitary irreducible representation introduced by Benson and Ratcliff in \cite{BR1}. We first focus on the study of this invariant for the class of reduced semi-direct product of nilpotent Lie algebras, which leads to a way of constructing families of counter-examples to the B-R conjecture, as well as identifying another class of $r$-step nilpotent Lie algebras for which the conjecture holds. Finally we extend the study of the B-R invariant of a coadjoint orbit of a nilpotent Lie group to the class of almost abelian solvable (non-nilpotent) Lie group $\mathbb R^n\rtimes\mathbb R$.
Abstract
The paper deals with the invariant for unitary irreducible representation introduced by Benson and Ratcliff in \cite{BR1}. We first focus on the study of this invariant for the class of reduced semi-direct product of nilpotent Lie algebras, which leads to a way of constructing families of counter-examples to the B-R conjecture, as well as identifying another class of $r$-step nilpotent Lie algebras for which the conjecture holds. Finally we extend the study of the B-R invariant of a coadjoint orbit of a nilpotent Lie group to the class of almost abelian solvable (non-nilpotent) Lie group $\mathbb R^n\rtimes\mathbb R$.