The Test Function Conjecture for Local Models of Weil-restricted groups

TL;DR

This paper proves the test function conjecture for local models of Shimura varieties associated with Weil-restricted groups, extending the result to all connected reductive groups over p-adic fields with p≥5.

Contribution

It establishes the test function conjecture for a broad class of local models, including all connected reductive groups over p-adic fields with p≥5, and provides a detailed study of related affine Grassmannians and loop groups.

Findings

Proved the test function conjecture for local models of Weil-restricted groups.

Extended results to all connected reductive groups over p-adic fields with p≥5.

Developed a self-contained framework for relative affine Grassmannians and loop groups.

Abstract

We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached to all connected reductive groups over $p$-adic local fields with $p\geq 5$. In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme.