0
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ You should probably refer to the similar question that you mention. $\endgroup$ – Alex Degtyarev Feb 27 '15 at 15:04
1
$\begingroup$

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).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.