Analysis of invariant PDO's on the Heisenberg group

TL;DR

This paper reviews the representation theory of the Heisenberg group and Howe's construction of the metaplectic group, applying these tools to analyze local solvability of second order invariant differential operators.

Contribution

It introduces a novel application of representation theory and twisted convolution operators to study local solvability on the Heisenberg group.

Findings

Demonstrates how representation theory aids in understanding differential operators.

Shows the effectiveness of generalized complex Gaussians in analysis.

Provides insights into the structure of invariant PDOs on the Heisenberg group.

Abstract

In these lecture notes, the representation theory of the Heisenberg group as well as Howe's construction of the metaplectic group by means of twisted convolution operators with generalized, complex Gaussians are reviewed, and it is shown how these tools can be successfully applied to the study of local solvability for second order left-invariant differential operators on the Heisenberg group.