On the flatness of spin local models for split even orthogonal groups

TL;DR

This paper proves the flatness and reduced special fiber of spin local models for split orthogonal groups over discretely valued fields, confirming a conjecture and constructing a flat moduli space of PEL-type D.

Contribution

It establishes the flatness and reducedness of spin local models for split orthogonal groups, confirming a conjecture of Pappas and Rapoport in this case.

Findings

Spin local models are flat schemes with reduced special fibers.

The results confirm the conjecture of Pappas and Rapoport for split orthogonal groups.

A flat moduli space of PEL-type D is constructed.

Abstract

Let $F$ be a complete discretely valued field with ring of integers $\mathcal{O}$ and residue field of characteristic $p>2$. Let $G=\operatorname{GO}_{2n}$ denote the split orthogonal similitude group over $F$. For any parahoric level structure, we prove that the associated spin local model for $G$ is a flat $\mathcal{O}$-scheme with reduced special fiber. This confirms a conjecture of Pappas and Rapoport in the split case. As a corollary, we construct a flat (integral) moduli space of PEL-type D.