An estimate for spherical functions on $\mathrm{SL}(3,\mathbb{R})$

TL;DR

This paper establishes uniform decay estimates for spherical functions on SL(3,R), removing previous regularity restrictions by analyzing oscillatory integrals and classifying singularities via stationary phase methods.

Contribution

It improves existing bounds on spherical functions by extending the analysis to non-regular group parameters, using a detailed classification of singularities.

Findings

Uniform decay estimates for spherical functions on SL(3,R)

Extension of previous results to non-regular parameters

Analysis of singularities via stationary phase and normal forms

Abstract

We prove an estimate for spherical functions $\varphi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.