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

中文翻译：

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基里洛夫性格公式的正性

我们直接证明了 Kirillov 的特征在连接的、简单连接的幂零李群 G 上的平滑、紧支持函数的卷积代数上的正性。然后我们使用这个正性结果来构造 G × G 的表示，并在我们的表示和 G 的底层表示上的 Hilbert-Schmidt 算子之间建立 G × G -等变等距同构。事实上，我们提供了一个框架，在该框架中，我们在额外的假设下建立了基里洛夫特征对 SL ( 2 , ℝ ) $\text {SL}(2, \mathbb {R})$ 等群的共伴随轨道的正性在幂零情况下自动满足。这些假设包括真实极化的存在和普坎茨基条件。