Dixmier traces, Wodzicki residues, and determinants on compact Lie groups: the paradigm of the global quantisation

TL;DR

This paper develops a global analysis framework for computing Dixmier traces, Wodzicki residues, and determinants of pseudo-differential operators on compact Lie groups, linking these to representation theory and extending to arbitrary closed manifolds.

Contribution

It introduces a global symbol-based method for explicit computation of traces and residues, connecting operator analysis with group representation theory and broadening applicability to general manifolds.

Findings

Explicit formulas for Dixmier trace and Wodzicki residue on compact Lie groups.

Connection established between traces, residues, and representation theory.

Extension of formulas to invariant pseudo-differential operators on arbitrary closed manifolds.

Abstract

\begin{abstract} By following the paradigm of the global quantisation, instead of the analysis under changes of coordinates, in this work we establish a global analysis for the explicit computation of the Dixmier trace and the Wodzicki residue of (elliptic and subelliptic) pseudo-differential operators on compact Lie groups. The regularised determinant for the Dixmier trace is also computed. We obtain these formulae in terms of the global symbol of the corresponding operators. In particular, our approach links the Dixmier trace and Wodzicki residue to the representation theory of the group. Although we start by analysing the case of compact Lie groups, we also compute the Dixmier trace and its regularised determinant on arbitrary closed manifolds $M$, for the class of invariant pseudo-differential operators in terms of their matrix-valued symbols. This analysis includes e.g. the family of positive and elliptic pseudo-differential operators on $M$.