I'm a beginner on representation theory and group theory, and maybe my language is not too technically correct.
But I just want to know why the action of a group elements $\lambda \in \Lambda$ on functions $f \in M $ is: $$(\lambda \cdot f)(x) = f(\lambda^{-1}x)$$
In the basic texts I read it appears like a definition, or they don't talk too much about it, just works. But it seems to me a non-trivial expression, specially the $\lambda^{-1}$ on the argument. What's the intuition to understand the expression? Is there a prove for this relation?
It's to satisfy the group action axioms, namely the composition one, that $$(\mu \cdot (\lambda \cdot f)) = ((\mu\lambda) \cdot f).$$
If we write $g(x) = f(\lambda^{-1}x)$, we have $$ (\mu \cdot (\lambda \cdot f))(x) = (\mu \cdot g)(x) = g(\mu^{-1}x) = f(\lambda^{-1}\mu^{-1}x) = f((\mu\lambda)^{-1}x) = ((\mu\lambda) \cdot f)(x), $$ as it should be. If you don't use the inverse, you get the wrong order in the product in the composition.