  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.
