---
tag: F2E3317P78
kind: concept
name: Harish-Chandra character
msc: ["22E46", "22E45", "22E50"]
specializes: [RMWMYBY8TC]
sources:
  - url: https://en.wikipedia.org/wiki/Harish-Chandra_character
same_as:
  wikidata: Q5657842
  wikipedia: Harish-Chandra character
---

## Summary

The Harish-Chandra character extends the [[character of a representation|RMWMYBY8TC]] 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.

## Explanations

### L1 human

For a finite group, the character of a representation records the trace of each group element and tells the representation apart from all others.
For a group like $\mathrm{SL}_2(\R)$ acting on an infinite-dimensional space, the trace of a single element is usually infinite.
The trick is to smear the group element out with a smooth bump function before taking the trace.
The result is no longer a function on the group but a "generalised function", and it still remembers the representation.