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 semisimple. Is $\rho$ decomposable uniquely into irreducible representations? Is a similar statement true for the semisimplification of the weil representation (in the case it is not semisimple)?
