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.