Generalized Hua-operators and parabolic subgroups. The cases of $\textrm {SL}(n, \textbf {C})$ and $\textrm {SL}(n, \textbf {R})$

Abstract:

Suppose $G = {\text {SL}}(n,{\mathbf {C}})$ or ${\text {SL}}(n,{\mathbf {R}})$ and $K$ is a maximal compact subgroup of $G$. If $P$ is any parabolic subgroup of $G$, we determine a system of differential equations on $G/K$ with the property that any function on $G/K$ satisfies these differential equations if and only if it is the Poisson integral of a hyperfunction on $G/P$.