The motivic Hecke algebra for PEL Shimura varieties

We construct a motivic lift of the action of the Hecke algebra on the cohomology of PEL Shimura varieties $S_K$. To do so, when $S_K$ is associated with a reductive algebraic group $G$ and $V$ is a local system on $S_K$ coming from a $G$-representation, we define a motivic Hecke algebra ${\mathcal{H}}^M(G,K)$ as a natural sub-algebra of the endomorphism algebra, in the triangulated category of motives, of the constructible motive associated with $S_K$ and $V$. The algebra ${\mathcal{H}}^M(G,K)$ is such that realizations induce an epimorphism from it onto the classical Hecke algebra. We then consider Wildeshaus’ theory of interior motives, along with the necessary hypotheses for it to be employed. Whenever those assumptions hold, one gets a Chow motive realizing to interior $V$-valued cohomology of $S_K$, equipped with an action of ${\mathcal{H}}^M(G,K)$ as an algebra of correspondences modulo rational equivalence. We give a list of known cases where this applies.