Harish-Chandra character

Summary

The Harish-Chandra character extends the character of a representation to infinite-dimensional representations of reductive groups. The operators $\pi(g)$ have no trace, but their averages $\int f(g) \pi(g) \, dg$ against smooth compactly supported functions $f$ do. The map $f \mapsto \operatorname{tr} \pi(f)$ is a distribution on the group, the character of $\pi$. It determines an irreducible admissible representation up to infinitesimal equivalence, and it is the basic tool of harmonic analysis on reductive groups.