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

TL;DR

This paper proves Kottwitz's nearby cycles conjecture for certain unramified unitary Shimura varieties, showing that the nearby cycles complexes are central in the convolution algebra, with implications for the structure of related Hecke algebras.

Contribution

It establishes the centrality of nearby cycles complexes for a class of PEL local models of unramified unitary groups, confirming Kottwitz's conjecture.

Findings

Nearby cycles complexes are central in the convolution algebra.

Semisimple trace functions lie in the centers of Iwahori-Hecke algebras.

The proof follows the method of Haines-Ngo 2002.

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 to the PEL local models under consideration are unramified unitary similitude groups with even dimension. The proof follows the method of Haines-Ngo 2002. Upon completion of the first version of this paper, Pappas and Zhu released a preprint, now published, which contained within its scope the main theorem of this paper. However, the methods of Pappas-Zhu are very different and some of the proofs from this paper have been useful in forthcoming work of Haines-Stroh.