Spectral theory for commutative algebras of differential operators on Lie groups

TL;DR

This paper develops a spectral theory for commuting differential operators on Lie groups, characterizing their joint spectrum via eigenfunctions and linking it to Gelfand pair theory under certain algebraic conditions.

Contribution

It introduces a framework for analyzing the joint spectrum of commuting invariant differential operators on Lie groups, extending spectral theory with new algebraic and representation-theoretic insights.

Findings

Joint spectrum characterized by positive-type eigenfunctions

Connections established with Gelfand pair theory

Spectral properties linked to algebraic structures of differential operators

Abstract

The joint spectral theory of a system of pairwise commuting self-adjoint left-invariant differential operators L_1,...,L_n on a connected Lie group G is studied, under the hypothesis that the algebra generated by them contains a "weighted subcoercive operator" of ter Elst and Robinson (J. Funct. Anal. 157 (1998) 88-163). The joint spectrum of L_1,...,L_n in every unitary representation of G is characterized as the set of the eigenvalues corresponding to a particular class of (generalized) joint eigenfunctions of positive type of L_1,...,L_n. Connections with the theory of Gelfand pairs are established in the case L_1,...,L_n generate the algebra of K-invariant left-invariant differential operators on G for some compact subgroup K of Aut(G).