Harish-Chandra character

Remarks

Example: if $\dim H < \infty$, then $\operatorname{tr} \pi(f) = \int_G f(g) \operatorname{tr} \pi(g) \, dg$ by linearity of the trace. So $\Theta_\pi$ is the distribution given by the ordinary character $g \mapsto \operatorname{tr} \pi(g)$.

Non-example: for $\dim H = \infty$, the operator $\pi(e) = \id_H$ is not trace class. So the naive rule $g \mapsto \operatorname{tr} \pi(g)$ does not define a function, and the smearing by $f$ is necessary.

The character $\Theta_\pi$ is invariant under conjugation: $\Theta_\pi(f^x) = \Theta_\pi(f)$ for $f^x(g) = f(x g x^{-1})$. Two irreducible admissible representations have the same character if and only if they are infinitesimally equivalent. By Harish-Chandra's regularity theorem, $\Theta_\pi$ is given by a locally integrable function. The same definition works for $G(F)$ with $F$ a non-archimedean local field, with $f$ locally constant of compact support.