Affine Matsuki correspondence for sheaves

We lift the affine Matsuki correspondence between real and symmetric loop group orbits in affine Grassmannians to an equivalence of derived categories of sheaves. In analogy with the finite-dimensional setting, our arguments depend upon the Morse theory of energy functions obtained from symmetrizations of coadjoint orbits. The additional fusion structures of the affine setting lead to further equivalences with Schubert constructible derived categories of sheaves on real affine Grassmannians

On quiver varieties and affine Grassmanians of type A

We construct Nakajima\u27s quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framework for skew (GL(m),GL(n)) duality and identifies the natural basis of weight spaces in Nakajima\u27s construction with the natural basis of multiplicity spaces in tensor products which arises from affine Grassmannians. To cite this article: I. Mirkovi , M. Vybornov, C. R. Acad. Sci. Paris, Ser. I 336 (2003)

Affine Grassmannians and the geometric Satake in mixed characteristic

We endow the set of lattices in Q_p^n with a reasonable algebro-geometric structure. As a result, we prove the representability of affine Grassmannians and establish the geometric Satake correspondence in mixed characteristic. We also give an application of our theory to the study of Rapoport-Zink spaces.Comment: 63 pages. Fix a gap in the proof of Theorem A.29. A few more details added and exposition improve

Affine Grassmannians of group schemes and exotic principal bundles over A¹

Let G be a simple simply-connected group scheme over a regular local scheme U. Let E be a principal G-bundle over A^1_U trivial away from a subscheme finite over U. We show that E is not necessarily trivial and give some criteria of triviality. To this end we define affine Grassmannians for group schemes and study their Bruhat decompositions for semi-simple group schemes. We also give examples of principal G-bundles over A^1_U with split G such that the bundles are not isomorphic to pull-backs from U.Comment: Introduction re-written. Other minor improvements. Final versio