On the Positivity of Kirillov's Character Formula

TL;DR

This paper proves the positivity of Kirillov's character for certain Lie groups, constructs related representations, and extends the positivity framework to more general group settings under specific conditions.

Contribution

It provides a direct proof of Kirillov's character positivity for nilpotent Lie groups and extends the framework to groups like SL(2,R) with additional hypotheses.

Findings

Established positivity of Kirillov's character for nilpotent Lie groups.

Constructed a G×G-equivariant isometric isomorphism with Hilbert--Schmidt operators.

Extended the positivity framework to groups like SL(2,R) under certain conditions.

Abstract

We give a direct proof for the positivity of Kirillov's character on the convolution algebra of smooth, compactly supported functions on a connected, simply connected nilpotent Lie group $G$. Then we use this positivity result to construct a representation of $G\times G$ and establish a $G\times G$-equivariant isometric isomorphism between our representation and the Hilbert--Schmidt operators on the underlying representation of $G$. In fact, we provide a more general framework in which we establish the positivity of Kirillov's character for coadjoint orbits of groups such as $\operatorname{SL}(2, \mathbb{R})$ under additional hypotheses that are automatically satisfied in the nilpotent case. These hypotheses include the existence of a real polarization and the Pukanzsky condition.