On the Positivity of Kirillov's Character Formula

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.