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On the Positivity of Kirillov’s Character Formula

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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 × 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 \(\text {SL}(2, \mathbb {R})\) under additional hypotheses that are automatically satisfied in the nilpotent case. These hypotheses include the existence of a real polarization and the Pukanzsky condition.

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Acknowledgements

The author wishes to thank Nigel Higson for posing the central question in this project and also for numerous conversations about it.

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    Ehssan Khanmohammadi

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  1. Ehssan Khanmohammadi
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Correspondence to Ehssan Khanmohammadi.

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Khanmohammadi, E. On the Positivity of Kirillov’s Character Formula. Math Phys Anal Geom 23, 13 (2020). https://doi.org/10.1007/s11040-020-09337-3

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