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

TL;DR

This paper investigates the convergence of cuspidal integrals and the behavior of Radon transforms on certain symmetric spaces related to SL(3), revealing convergence in some cases and divergence in others, with implications for representation theory.

Contribution

It establishes convergence of cuspidal integrals for specific SL(3) symmetric spaces and analyzes the Radon transforms and their kernels in relation to the Plancherel decomposition.

Findings

Cuspidal integrals are absolutely convergent for SL(3,R)/SO(1,2) and SL(3,C)/SU(1,2).

Radon transforms' behavior is characterized and linked to representation series.

Cuspidal integrals are not convergent for SL(3,H)/Sp(1,2) for all Schwartz functions.

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.