In this talk, we shall discuss the existence of a Weyl pseudo-differential calculus on a graded Lie G. To begin with, we will introduce a family of admissible quantizations on G and develop the corresponding pseudo-differential calculus. Afterwards, following classical criteria, we shall identify the suitable candidate Weyl quantization on G among all the admissible symmetric ones. Finally, we will show that in the case of the Heisenberg group, the Weyl quantization is uniquely determined.

This talk is based on a joint work with D. Rottensteiner and M. Ruzhansky.