Let $G$ denote a connected semisimple Lie group and $\pi$ an irreducible representation on $G$. Define $\pi':C^{\infty}_c(G) \to \mathbb{C}$ by:
$$\pi'(f) = \int_G f(x) \pi(x) dx $$
where $dx$ is the Haar measure on $G$. Then $\pi'$ is of trace class and the map $T_{\pi}$ defined by $f \mapsto tr(\pi (f))$ is a distribution on $G$ (the character of $\pi$).
Denote the regular set of $G$ by $G'$. Harish Chandra showed that $T_{\pi}$ coincides on $G'$ (an open dense subset of $G$) with an analytic function $F_{\pi}$.
How does one define an analytic function on a Lie group? As is shown at this link on stackexchange:
https://math.stackexchange.com/questions/2532282/analytic-functions-on-smooth-manifolds
the notion of an analytic function on a smooth manifold doesn't make sense a priori. I had read somewhere though that Lie groups have a unique analytic atlas making them into an analytic manifold, so perhaps is an analytic function on a Lie group defined to be analytic in analytic charts?