The mod p cohomology of locally symmetric spaces for definite unitary groups at infinite level is expected to realize the mod p local Langlands correspondence for $GL_n$. In particular, one expects the (component at p) of the associated Galois representation to be determined by cohomology as a smooth representation. I will describe how one can establish this expectation in many cases when the local Galois representation is Fontaine-Laffaille. \\ \\ This is joint work with D. Le, S. Morra, C. Park and Z. Qian.