# What is meant by an analytic function on a Lie group?

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:

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?

• A Lie group has a canonical real analytic structure: locally around 1 one chart is given by the exponential map and elsewhere first left-translate to 1; then the product and inverse are both analytic.
– YCor
Jun 2, 2019 at 6:17
• I am sorry but I don't see what is your problem. Since you start with an algebraic object(semisimple Lie group), so as you mentioned, Harish-Chandra tried to find a well-defined 'trace' function(note that the fixed irreducible representation may not be finite dimensional). Indeed they are functions on conjugacy classes of (regular, strongly-.) semisimple elements of rational points of G, and these are widely studied objects nowadays. On the other hand, on geometric side, you have GAGA for underlying field being complex or p-adic. So you can really find 'analytic' functions, eg rigid geomrtry. Jun 2, 2019 at 14:39