Decomposition of semi simple local systems

Question

I found A question similar to this, but the answer wasn't clear to me and I'm not supposed to ask for further clarification in the answer section.

Let $L$ a semi simple local system defined over an algebraic variety. I have seen the following decomposition

$$L \cong \bigoplus_\rho \rho\otimes L_\rho$$

where $\rho$ runs over a set of representatives of the set of irreducible representations of the endomorphism algebra, and $L_\rho=Hom(\rho,L)$.

I have several doubts concerning this decomposition. For example, I think $\rho$ should be consider in this case as a local system, (because $L$ is one) but then the sum should be done over the set of irreducible representations of the fundamental group of the variety. I'm not quite sure what $Hom(\rho,L)$ means either (Is it the functor $U\mapsto Homp(\rho_{|U},L_{|U})$ where again we see $\rho$ as a local system). I don't understand either how to build the isomorphism. I hope someone can give a hand. A good reference would be great. Thank you very much.

Answer 1

I think you're getting confused about duality (in the sense of Schur duality here). The endomorphism algebra $E$ of $L$ is just an algebra, and its irreducible representations are just vector spaces. However, if $\rho$ is such a representation, then $\mathrm{Hom}_E(\rho,L)$ is a local system, defined by $\mathrm{Hom}_E(\rho,L)(U)=\mathrm{Hom}_E(\rho,L(U))$.

The isomorphism above holds when the local system $L$ is semi-simple (for example if the underlying field is characteristic 0, and the local system is unitary). What you're trying to describe is instead the dual decomposition where you decompose a fiber of $L$ with respect to the action of the fundamental group. The "duality" I mentioned above is that the isotypic components of this decomposition are exactly one of the terms $\rho\otimes L_\rho$ in the decomposition above, and this defines a bijection between representations of $\pi_1$ appearing in $L$ and representations of $E$ (again, I'm assuming semi-simplicity here).