• DOI:10.1007/s00209-013-1225-y
• Corpus ID: 119313862

On $$p$$p-adic lattices and Grassmannians

@article{Kreidl2014OnL,
  title={On \$\$p\$\$p-adic lattices and Grassmannians},
  author={Martin Kreidl},
  journal={Mathematische Zeitschrift},
  year={2014},
  volume={276},
  pages={859-888}
}

• M. Kreidl
• Published 2 October 2010
• Mathematics
• Mathematische Zeitschrift

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