Abstract
This article is devoted to the Toeplitz Operators [4] in the context of the geometric quantization [11, 15]. We propose an ansatz for their Schwartz kernel. From this, we deduce the main known properties of the principal symbol of these operators and obtain new results : we define their covariant and contravariant symbols, which are full symbols, and compute the product of these symbols in terms of the Kahler metric. This gives canonical star products on the Kahlerian manifolds. This ansatz is also useful to introduce the notion of microsupport.
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