The Laguerre calculus on the nilpotent Lie groups of step two

The Laguerre calculus is widely used for the inversion of differential operators on the Heisenberg group. We extend the Laguerre calculus for nilpotent groups of step two, and test it in the determining of the fundamental solution of the sub-Laplace operator. We also apply it to find the Szeg\"o kernels of the projection operators to a kind of regular functions on the quaternion Heisenberg group.