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On p-adic loop groups and Grassmannians

Martin Kreidl

Number 11
Author Martin Kreidl
Year 2010

It is well-known that the coset spaces G(k ( (z) ) )=G(k [ [z] ]), for a reductive group G over a field k, carry a geometric structure, notably the structure of an ind-projective k-ind-scheme. 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 gain an analogous geometric understanding of the quotients of the form G(W(k)[1/p])=G(W(k)), where W denotes the ring of Witt vectors. The present paper is an attempt to describe which constructions carry over from the `function field case' to the `p-adic case' and in particular to describe a construction of a p-adic affine Grassmannian for Sl_n as an fpqc-sheaf on the category of k-algebras for a perfect field k. In order to obtain a link with geometry we construct, inside a multigraded Hilbert scheme, projective k-schemes which map equivariantly to the p-adic affine Grassmannian inducing an isomorphism of Schubert cells and describe these morphisms on the level of k-valued points. Finally, we describe the R-valued points, where R is a perfect k-algebra, of the p-adic affine Grassmannian in terms of `lattices over W(R)', analogously to the function field case.

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