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

Abstract

This paper proves that the nearby cycles complexes on a certain family of PEL local models are central with respect to the convolution product of sheaves on the corresponding affine flag varieties. As a corollary, the semisimple trace functions defined using the action of Frobenius on those nearby cycles complexes are, via the sheaf-function dictionary, in the centers of the corresponding Iwahori–Hecke algebras. This is commonly referred to as Kottwitz’s conjecture. The reductive groups associated with the PEL local models under consideration are unramified unitary similitude groups with even dimension. The proof follows the method of Haines and Ngô (Compos Math 133:117–150, 2002). Upon completion of the first version of this paper, Pappas and Zhu released a preprint (now published as Pappas and Zhu in Invent Math 194(1):147–254, 2013) which contained within its scope the main theorem of this paper. However, the methods of Pappas and Zhu (2013) are very different, and some of the proofs from this paper have been useful in forthcoming work of Haines–Stroh.