Weyl calculus on graded groups

We present a pseudo-differential Weyl calculus on graded nilpotent Lie groups, especially the Heisenberg group, which extends the celebrated Weyl calculus on R^n. This Weyl calculus is a particular instance of a general symbolic calculus we develop for a large class of quantization schemes that are defined via the (operator-valued) group Fourier transform. The symbol classes we consider are the Hörmander-type classes introduced by Fischer and Ruzhansky, which on R^n coincide with the classical ones. As a by-product, we also recover the classical Kohn-Nirenberg calculus on R^n and Fischer and Ruzhansky's KN-calculus on general graded groups. A few immediate applications of our theory are the expected mapping properties on Sobolev spaces, the existence of one-sided parametrices and the G\aa rding inequality for elliptic operators, and a generalized Poisson bracket on stratified groups. We also discuss two simple algebraic criteria which determine the Weyl quantization uniquely at least on R^n and the Heisenberg group. The talk is based on joint-work with Serena Federico and Michael Ruzhansky.