On a conjecture of Pappas and Rapoport about the standard local model for $GL_d$

TL;DR

This paper proves a conjecture by Pappas and Rapoport regarding the reducedness of a subscheme of matrices related to local models of Shimura varieties, using techniques from affine Grassmannians and Frobenius splitting.

Contribution

It provides a complete proof of the conjecture on reducedness, extending previous work on affine Grassmannians and Schubert varieties with an almost elementary approach.

Findings

Confirmed the reducedness of the subscheme in question.

Connected the conjecture to affine Grassmannian slices and Frobenius splitting.

Extended the understanding of local models for Shimura varieties.

Abstract

In their study of local models of Shimura varieties for totally ramified extensions, Pappas and Rapoport posed a conjecture about the reducedness of a certain subscheme of $n \times n$ matrices. We give a positive answer to their conjecture in full generality. Our main ideas follow naturally from two of our previous works. The first is our proof of a conjecture of Kreiman, Lakshmibai, Magyar, and Weyman on the equations defining type A affine Grassmannians. The second is the work of the first two authors and Kamnitzer on affine Grassmannian slices and their reduced scheme structure. We also present a version of our argument that is almost completely elementary: the only non-elementary ingredient is the Frobenius splitting of Schubert varieties.