Towards a conjecture of Pappas and Rapoport on a scheme attached to the symplectic group

TL;DR

This paper proves a special case of a conjecture by Pappas and Rapoport by analyzing a scheme related to symplectic groups, showing it has a basis of pfaffians and is an integral domain under certain conditions.

Contribution

It establishes the structure of the coordinate ring of a scheme linked to symplectic groups, confirming a case of the Pappas-Rapoport conjecture under specific assumptions.

Findings

Coordinate ring has a basis of pfaffians labeled by symplectic tableaux.

Proves the scheme's coordinate ring is an integral domain when n is divisible by 4.

Confirms a special case of the Pappas-Rapoport conjecture for certain characteristics.

Abstract

Let n = 2r be an even integer. We consider a closed subscheme V of the scheme of n-by-n skew-symmetric matrices, on which there is a natural action of the symplectic group Sp(n). Over a field F of characteristic not equal to 2, the scheme V is isomorphic to the scheme appearing in a conjecture by Pappas and Rapoport on local models of unitary Shimura varieties. With the additional assumption char F = 0 or char F > r, we prove the coordinate ring of V has a basis consisting of products of pfaffians labelled by King's symplectic standard tableaux with no odd-sized rows. When n is a multiple of 4, the basis can be used to show that the coordinate ring of V is an integral domain, and this proves a special case of the conjecture by Pappas and Rapoport.