Explicit fundamental solutions of some second order differential operators on Heisenberg groups
Volume 129 / 2012
Abstract
Let $p,q,n$ be natural numbers such that $p+q=n$. Let $\mathbb F$ be either $\mathbb C$, the complex numbers field, or $\mathbb H$, the quaternionic division algebra. We consider the Heisenberg group $N(p,q,\mathbb F)$ defined $\mathbb F^{n}\times \mathop{\mathfrak{Im}}\nolimits \mathbb F$, with group law given by $$ (v,\zeta)(v',\zeta')=\biggl( v+v', \zeta+\zeta'-{\frac{1}{2}} \mathop{\mathfrak{Im}}\nolimits B(v,v') \biggr), $$ where $B(v,w)=\sum_{j=1}^{p} v_{j}\overline{w_{j}} - \sum_{j=p+1}^{n} v_{j}\overline{w_{j}}$. Let $U(p,q,\mathbb F)$ be the group of $n\times n$ matrices with coefficients in $\mathbb F$ that leave the form $B$ invariant. We compute explicit fundamental solutions of some second order differential operators on $N(p,q,\mathbb F)$ which are canonically associated to the action of $U(p,q,\mathbb F)$.
