Cuspidal integrals and subseries for $\mathrm{SL}(3)/K_{\epsilon }$

Abstract

We show that for the symmetric spaces $\mathrm{SL}(3,\mathbb{R})/\mathrm{SO}(1,2{)}_e$ and $\mathrm{SL}(3,\mathbb{C})/\mathrm{SU}(1,2)$ the cuspidal integrals are absolutely convergent. We further determine the behavior of the corresponding Radon transforms and relate the kernels of the Radon transforms to the different series of representations occurring in the Plancherel decomposition of these spaces. Finally we show that for the symmetric space $\mathrm{SL}(3,\mathbb{H})/\mathrm{Sp}(1,2)$ the cuspidal integrals are not convergent for all Schwartz functions.