I'm reading a paper that uses representation theory, and I'm stuck on something simple. Say $\Delta$ is an abelian group, and $\hat{\Delta}$ its group of irreducible characters. Say $M$ is a $\mathbb{C}[\Delta]$-module. The paper uses the decomposition $M=\oplus_{\chi \in \hat{\Delta}} M^{\chi}$, where $M^{\chi}:=e_{\chi}M$ and $e_{\chi}:=\frac{1}{|\Delta|} \sum_{\delta \in \Delta} \chi(\delta)\delta^{-1}$.
The question, roughly speaking, is: how should I think of the $M^{\chi}$'s? What is their meaning, and function? How else are they characterized?
I've taken a course in representation theory a few years back, so it's all a little vague, but if I remember correctly, $\mathbb{C}[\Delta]$ is the direct sum of $|\Delta|$ many simple submodules, in bijection with the $\chi$'s. Then the $e_{\chi}$'s can be interpreted as being the decomposition of $1$ ($1=\sum_{\chi} e_{\chi}$, where $e_{\chi}$ is in the simple submodule of $\mathbb{C}[\Delta]$ corresponding to $\chi$).
But I don't see how this translates to a decomposition of some random module $M$. $M$ is a direct sum of simple submodules. But there's reason to think that there are $|\Delta|$ many (for example: what if $M$ is simple?). So what is the interpretation of the $M^{\chi}$'s?