Construction of torsion cohomology classes for KHT Shimura varieties

Let $Sh_K(G,\mu)$ be a Shimura variety of KHT type, as introduced in Harris-Taylor book, associated to some similitude group $G/\mathbb Q$ and a open compact subgroup $K$ of $G(\mathbb A)$. For any irreducible algebraic $\overline{\mathbb Q}_l$-representation $\xi$ of $G$, let $V_\xi$ be the $\mathbb Z_l$-local system on $Sh_K(G,\mu)$. From my paper about p-stabilization, we know that if we allow the local component $K_l$ of $K$ to be small enough, then there must exists some non trivial cohomology classes with coefficient in $V_\xi$. The aim of this paper is then to construct explicitly such torsion classes with the control of $K_l$. As an application we obtain the construction of some new automorphic congruences between tempered and non tempered automorphic representations of the same weight and same level at $l$.