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Let $X$ be a smooth, separated scheme of finite type over $\mathbb{F}_q$ where $q=p^r$ for some $r>0$. Let $gcd(l,p)=1, \rho:W(X) \to GL_r(\mathbb{Q}_l)$ be a Weil representation which is semi-simple. Is $\rho$ decomposable uniquely into irreducible representations? Is a similar statement true for the semi-simplification of the weil representation (in the case it is not semi-simple)?

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It would help to have a background reference for Weil representations in your setting. Aside from that, keep in mind that classical representation theory of finite groups (when the characteristic of the field doesn't divide the group order) involves completely reducible representations (as well as complications about splitting fields which I'll ignore). There one can't expect irreducible summands to be unique unless they occur with multiplicity one. The same principle would apply to semi-simplifications in the classical theory. – Jim Humphreys Feb 11 '13 at 15:37
@Humphreys: Weil representations are representations where $W(X)$ is a weil group, a subgroup of the etale fundamental group of $X$. It is not a finite group. A good reference is Deligne's paper on the proof of the weil conjectures. – Naga Venkata Feb 11 '13 at 16:24
What is preventing a completely abstract argument of the form "by dimension considerations it's a sum of irreps" (assuming your "semisimple" is the same as mine) and then "any non-zero map between irreps is an isomorphism"? Why doesn't such an argument do the job for you? – user30035 Feb 11 '13 at 21:08
Pleas do not omit the prfix 'nt.' in the tag (I changed it). – quid Feb 12 '13 at 12:39

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