Harish–Chandra character of a p-adic representation

Statement

Let $G$ be a connected reductive group over a nonarchimedean local field $F$ , and let $\pi$ be an admissible representation of $G(F)$ . For a test function $f\in C_c^\infty(G(F))$ , the operator

\pi(f)=\int_{G(F)} f(g)\pi(g)\,dg

has finite rank. The Harish–Chandra distribution character is

\Theta_\pi(f)=\operatorname{tr}\pi(f).

It is a conjugation-invariant distribution—that is, a continuous linear functional on the nonarchimedean test-function space—and is independent of the chosen Haar measure once the test-function measure convention is fixed.

Regularity theorem

Harish–Chandra's regularity theorem says that $\Theta_\pi$ is represented by a locally integrable conjugation-invariant function on $G(F)$ . This function is locally constant on the regular semisimple locus. The same symbol $\Theta_\pi(g)$ usually denotes that representing function there.

Near singular elements the character has a local expansion in Fourier transforms of nilpotent orbital integrals. Thus the distribution is the primary object; pointwise values are available on the regular set, not on every element by definition.

Character identities

Endoscopic transfer and local Langlands often produce identities between sums of these characters over an L-packet and stable distributions on an endoscopic group. These are distributional identities and depend on the normalization of transfer factors.

References

Harish-Chandra, “Admissible invariant distributions on reductive $p$ -adic groups,” in Lie Theories and Their Applications, Queen's Papers in Pure and Applied Mathematics 48, 1978.
Stephen DeBacker, “Homogeneity results for invariant distributions of a reductive $p$ -adic group,” Annales scientifiques de l'École Normale Supérieure 35 (2002), 391–422. Numdam.