Local models for Weil-restricted groups

TL;DR

This paper extends the construction of local models for reductive groups to Weil restrictions, proving normality, reduced special fibers, and confirming a conjecture relating Frobenius traces to Hecke algebra functions.

Contribution

It generalizes local model constructions to Weil-restricted groups, completes the case for all reductive groups when p ≥ 5, and verifies a key conjecture of Kottwitz.

Findings

Local models are normal with reduced special fibers.

Proved Kottwitz's conjecture on Frobenius traces.

Studied monodromy and inertial actions in the context of Weil restrictions.

Abstract

We extend the group theoretic construction of local models of Pappas and Zhu to the case of groups obtained by Weil restriction along a possibly wildly ramified extension. This completes the construction of local models for all reductive groups when $p \geq 5$. We show that the local models are normal with special fiber reduced and study the monodromy action on the sheaves of nearby cycles. As a consequence, we prove a conjecture of Kottwitz that the semi-simple trace of Frobenius gives a central function in the parahoric Hecke algebra. We also introduce a notion of splitting model and use this to study the inertial action in the case of an unramified group.