In the context of studying some representation theoretic aspects of Fourier transforms (coming from physics), I would like to know the significance of some spaces of functions on a Lie group, as a module for the group.
The space of complex functions on a complex semisimple Lie Group $G$, let's call it $F(G)$, carries an action of $G$ induced from the left action of $G$ on itself, for any $f \in F(G)$ the action can be defined as: $$G \ni g: F(G) \to F(G)\,, \qquad g: f \mapsto f'\,, \qquad f'(gh) := f(h)\,.$$ I don't have a specific characterization of $F(G)$ in mind, for example, $F(G)$ can be $L^1(G)$, $L^2(G)$ etc.
In most cases, $F(G)$ seems to be a rather large module containing many irreducible modules, my question is this: is there a precise statement that can be made about which representations are contained in $F(G)$ for different choices of $F(G)$ (different $L^p$ spaces or something else interesting)? By "which representations" I mean whether all the finite dimensional/highest weight/irreducible/verma etc etc modules are contained in these function paces. Does assuming $G$ to be abelian make any significant simplification?
If the answer to this question is in some available literature I will be happy with a reference. Thanks in advance for any pointers.