Explicit fundamental solutions of some second order differential operators on Heisenberg groups

TL;DR

This paper derives explicit fundamental solutions for certain second order differential operators on Heisenberg groups associated with the action of unitary groups over complex or quaternionic fields, advancing understanding of these operators.

Contribution

It provides explicit formulas for fundamental solutions of specific second order differential operators on Heisenberg groups linked to $U(p,q,f)$ actions, a novel contribution.

Findings

Explicit fundamental solutions for differential operators on Heisenberg groups.

Connections between group invariances and differential operator solutions.

Enhanced tools for analysis on Heisenberg groups with complex or quaternionic structures.

Abstract

Let $p,q,n$ be natural numbers such that $p+q=n$. Let $\FF$ be either $\CC$, the complex numbers field, or $\HH$, the quaternionic division algebra. We consider the Heisenberg group $N(p,q,\FF)$ defined as $N(p,q,\FF)=\FF^{n}\times \mathfrak{Im}\FF$, with group law given by $$(v,\zeta)(v',\zeta')=(v+v', \zeta+\zeta'-{1/2} \mathfrak{Im} B(v,v')),$$ where $B(v,w)=\sum_{j=1}^{p} v_{j}\bar{w_{j}} - \sum_{j=p+1}^{n} v_{j}\bar{w_{j}}$. Let $U(p,q,\FF)$ be the group of $n\times n$ matrices with coefficients in $\FF$ that leave invariant the form $B$. In this work we compute explicit fundamental solutions of some second order differential operators on $N(p,q,\FF)$ which are canonically associated to the action of $U(p,q,\FF)$.