Coadjoint orbits of stepwise square integrable representations

TL;DR

This paper offers a new perspective on stepwise square integrable representations of nilpotent Lie groups by linking them to stratifications of dual Lie algebra spaces, revealing their topological and measure-theoretic properties.

Contribution

It introduces an alternative approach to these representations, demonstrating their correspondence with specific stratifications and analyzing their topological and measure characteristics.

Findings

Representations correspond to stratifications of dual Lie algebra spaces

They form a subset with relative Hausdorff topology and dense interior

They have total Plancherel measure in the unitary dual

Abstract

Nilpotent Lie groups with stepwise square integrable representations were recently investigated by J.A.~Wolf. We give an alternative approach to these representations by relating them to the stratifications of the duals of nilpotent Lie algebras, thus proving that they correspond to a subset with relative Hausdorff topology, dense interior, and total Plancherel measure in the unitary dual of the Lie group under consideration.