Harish-Chandra's regularity theorem

Remarks

The character of an irreducible admissible representation satisfies the hypotheses, because $\mathfrak Z$ acts on such a representation by scalars, its infinitesimal character. The proof first treats invariant eigendistributions on the Lie algebra, using the radial parts of invariant differential operators along Cartan subalgebras. It then transfers the result to the group through the exponential map near each semisimple element. On $G'$ the eigen-equations become an elliptic system, which gives analyticity there; the hard part is the local integrability across the singular set $G \setminus G'$. For a finite-dimensional representation, $F$ is the ordinary character $g \mapsto \operatorname{tr} \pi(g)$, which is analytic on all of $G$. For infinite-dimensional representations $F$ is usually singular on $G \setminus G'$, so analyticity on all of $G$ cannot be expected.

Going deeper

For a student

The theorem is the starting point of Harish-Chandra's harmonic analysis on real reductive groups. Explicit formulas for the function $F$ on each Cartan subgroup lead to the construction of the discrete series and to the Plancherel formula. An analogue holds for reductive groups over $p$-adic fields, also due to Harish-Chandra, where the proof is different.