Topological flatness of orthogonal local models in the split, even case. I

TL;DR

This paper proves that the spin local model for even orthogonal groups is topologically flat in the split, Iwahori case, addressing a conjecture related to the flatness of integral models of PEL Shimura varieties.

Contribution

It establishes topological flatness of the spin local model for split, even orthogonal groups, confirming a conjecture by Pappas and Rapoport in a specific case.

Findings

Proves topological flatness of the spin local model in the split, Iwahori case.

Shows equivalence of mu-admissibility and mu-permissibility for certain root systems.

Addresses flatness issues in local models for orthogonal groups.

Abstract

Local models are schemes, defined in terms of linear algebra, that were introduced by Rapoport and Zink to study the \'etale-local structure of integral models of certain PEL Shimura varieties over p-adic fields. A basic requirement for the integral models, or equivalently for the local models, is that they be flat. In the case of local models for even orthogonal groups, Genestier observed that the original definition of the local model does not yield a flat scheme. In a recent article, Pappas and Rapoport introduced a new condition to the moduli problem defining the local model, the so-called spin condition, and conjectured that the resulting "spin" local model is flat. We prove a weak form of their conjecture in the split, Iwahori case, namely that the spin local model is topologically flat. An essential combinatorial ingredient is the equivalence of mu-admissibility and mu-permissibility for two minuscule cocharacters mu in root systems of type D.