  \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 $M$.
