Let $Y (N) $ be the moduli scheme of dimension two principally polarized Abelian schemes with level $N$. It is claimed in "[G.Laumon - Fonctions zeta des variétés de Siegel](http://www.numdam.org/article/AST_2005__302__1_0.pdf)" (Lemma 4.1) that to an algebraic representation $W$ of $\mathrm{GSp}\_{4}(\mathbb{Q})$ we can associate an $l$-adic smooth sheaf on $Y (N) [1/l] $.


Where can I find a proof of this please?



$l$-adic sheaf associated to an algebraic representation of $\mathrm{GSp}_{4}(\mathbb{Q})$
This is a special case of Pink's "canonical construction" functor, which associates various kinds of coefficient sheaves on a Shimura variety (etale $\ell$-adic sheaves, vector bundles with connection, variations of Hodge structures, etc) to algebraic representations of the underlying group.


For more information see e.g. this paper by Torzewski: <http://link.springer.com/article/10.1007/s00229-019-01150-9#Sec10>.


