On p-adic lattices and Grassmannians

TL;DR

This paper explores the geometric structure of p-adic affine Grassmannians for SL_n, linking lattice descriptions over Witt vectors to projective subvarieties and Schubert varieties, extending concepts from the function field case.

Contribution

It provides a description of the p-adic affine Grassmannian for SL_n using lattices over Witt vectors and constructs Schubert-like varieties within a multigraded Hilbert scheme.

Findings

Describes R-valued points of the p-adic affine Grassmannian in terms of lattices over Witt vectors.

Constructs projective subvarieties that map to the p-adic affine Grassmannian, analogous to Schubert varieties.

Establishes bijections between R-valued points of these subvarieties and Schubert cells for SL_n.

Abstract

It is well-known that the coset spaces G(k((z)))/G(k[[z]]), for a reductive group G over a field k, carry the geometric structure of an inductive limit of projective k-schemes. This k-ind-scheme is known as the affine Grassmannian for G. From the point of view of number theory it would be interesting to obtain an analogous geometric interpretation of quotients of the form G(W(k)[1/p])/G(W(k)), where p is a rational prime, W denotes the ring scheme of p-typical Witt vectors, k is a perfect field of characteristic p and G is a reductive group scheme over W(k). The present paper is an attempt to describe which constructions carry over from the function field case to the p-adic case, more precisely to the situation of the p-adic affine Grassmannian for the special linear group G=SL_n. We start with a description of the R-valued points of the p-adic affine Grassmannian for SL_n in terms of lattices over W(R), where R is a perfect k-algebra. In order to obtain a link with geometry we further construct projective k-subvarieties of the multigraded Hilbert scheme which map equivariantly to the p-adic affine Grassmannian. The images of these morphisms play the role of Schubert varieties in the p-adic setting. Further, for any reduced k-algebra R these morphisms induce bijective maps between the sets of R-valued points of the respective open orbits in the multigraded Hilbert scheme and the corresponding Schubert cells of the p-adic affine Grassmannian for SL_n.