In this paper, we compute the cohomology sheaves of the -adic nearby cycles on the local model of the PEL GU (n-1, 1) Shimura variety over a ramified prime. The local model is known to have isolated singularities. If n=2 it has semi-stable reduction, and if n 3 the blow-up at the singular point has semi-stable reduction. Thus, in principle one may compute the nearby cycles at least on the blow-up, then use proper base change to describe them on the original local model. As a result, we prove that the nearby cycles are trivial when n is odd, and that only a single higher cohomology sheaf does not vanish when n is even. In this case, we also describe the Galois action by computing the associated eigenvalue of the Frobenius.
Joseph Müller (Mon,) studied this question.