Fundamental solutions of invariant differential operators on a semisimple Lie group

Guillermo Ames

Vol. 196 (2000), No. 1, 17–44

DOI: 10.2140/pjm.2000.196.17

Abstract

Let G be a connected semisimple Lie group of real rank one. We denote by 𝒰(g)^K the algebra of left invariant differential operators on G right invariant by K, and let 𝒵(𝒰(g)^K) be its center.

In this paper we give a sufficient condition for a differential operator P ∈𝒵(𝒰(g)^K) to have a fundamental solution on G. We verify that this condition implies P C^∞(G) = C^∞(G). If G has a compact Cartan subgroup, we also give a sufficient condition for a differential operator P ∈𝒵(𝒰(g)^K) to have a parametrix on G. Finally we prove a necessary condition for the existence of parametrix of P ∈𝒵(𝒰(g)^K) for a connected semisimple Lie group.

Milestones

Received: 20 April 1999

Revised: 14 September 1999

Published: 1 November 2000