On the $l$-adic cohomology of some $p$-adically uniformized Shimura varieties

We determine the Galois representations inside the $l$-adic cohomology of some unitary Shimura varieties at split places where they admit uniformization by finite products of Drinfeld upper half spaces. Our main results confirm Langlands-Kottwitz’s description of the cohomology of Shimura varieties in new cases.