# The Laguerre calculus on the nilpotent Lie groups of step two

## TL;DR

This paper extends the Laguerre calculus to nilpotent Lie groups of step two, enabling new methods for inverting differential operators and computing fundamental solutions and Szeg"o kernels in these groups.

## Contribution

It introduces a generalized Laguerre calculus for step-two nilpotent groups, expanding its applicability beyond the Heisenberg group.

## Key findings

- Derived fundamental solutions for sub-Laplace operators on these groups.
- Computed Szeg"o kernels for projection operators on quaternion Heisenberg groups.
- Demonstrated the effectiveness of the extended calculus in specific harmonic analysis problems.

## Abstract

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.