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 × G and establish a G × G-equivariant isometric isomorphism between our representation and the Hilbert–Schmidt operators on the underlying representation of G. In fact, we provide a framework in which we establish the positivity of Kirillov's character for coadjoint orbits of groups such as SL (2 , 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. [ABSTRACT FROM AUTHOR]