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

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 appeared 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.