Proper $r$-harmonic functions from Riemannian manifolds

TL;DR

This paper presents a novel method for constructing complex-valued $r$-harmonic functions on Riemannian manifolds and applies it to various important semisimple Lie groups, expanding the toolkit for harmonic analysis.

Contribution

It introduces a new construction method for $r$-harmonic functions and demonstrates its application to key semisimple Lie groups, advancing the field of harmonic analysis on manifolds.

Findings

New method for constructing $r$-harmonic functions

Application to classical semisimple Lie groups

Enhanced understanding of harmonic functions on manifolds

Abstract

We introduce a new method for constructing complex-valued $r$-harmonic functions on Riemannian manifolds. We then apply this method for the important semisimple Lie groups $SO(n)$, $SU(n)$, $Sp(n)$, $SL_n(R)$, $Sp(R,n)$, $SU(p,q)$, $SO(p,q)$, $Sp(p,q)$, $SO^*(2n)$ and $SU^*(2n)$.