Kottwitz's nearby cycles conjecture for a class of unitary Shimura varieties

Abstract

This paper proves that the nearby cycles complex on a certain family of PEL local models is central with respect to the convolution product of sheaves on the corresponding affine flag variety. As a corollary, the semisimple trace function defined using the action of Frobenius on that nearby cycles complex is, via the sheaf-function dictionary, in the center of the corresponding Iwahori-Hecke algebra. This is commonly referred to as Kottwitz's conjecture. The reductive groups associated to the PEL local models under consideration are unramified unitary similitude groups with even dimension. The proof follows the method of [Haines-Ngo 2002].