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

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 $U(p,q,\FF)$.