Local models of Shimura varieties and a conjecture of Kottwitz

TL;DR

This paper defines local models for Shimura varieties using group theory, studies their singularities, and proves Kottwitz's conjecture relating Frobenius traces to central functions in the Hecke algebra.

Contribution

It introduces a group theoretic framework for local models in mixed characteristic and proves a conjecture connecting Frobenius traces with Hecke algebra centers.

Findings

Defined local models as degenerations of Grassmannian varieties in mixed characteristic.

Analyzed singularities and monodromy actions of local models.

Proved Kottwitz's conjecture on the centrality of Frobenius traces in the Hecke algebra.

Abstract

We give a group theoretic definition of "local models" as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a $p$-adic local field that are expected to model the singularities of integral models of Shimura varieties with parahoric level structure. Our local models are certain mixed characteristic degenerations of Grassmannian varieties; they are obtained by extending constructions of Beilinson, Drinfeld, Gaitsgory and the second-named author to mixed characteristics and to the case of general (tamely ramified) reductive groups. We study the singularities of local models and hence also of the corresponding integral models of Shimura varieties. In particular, we study the monodromy (inertia) action and show a commutativity property for the sheaves of nearby cycles. As a result, we prove a conjecture of Kottwitz which asserts that the semi-simple trace of Frobenius on the nearby cycles gives a function which is central in the parahoric Hecke algebra.