204 research outputs found
The 2-group of linear auto-equivalences of an abelian category and its Lie 2-algebra
For any abelian category \calC satsifying (AB5) over a separated,
quasi-compact scheme S, we construct a stack of 2-groups \GL(\calC) over the
flat site of S. We will give a concrete description of \GL(\calC) when \calC is
the category of quasi-coherent sheaves on a separated, quasi-compact scheme X
over S. We will show that the tangent space \gl(\calC) of \GL(\calC) at the
origin has a structure as a Lie 2-algebra.Comment: 47 pages, preliminary version. Comments welcome at any tim
Affine Demazure modules and -fixed point subschemes in the affine Grassmannian
Let be a simple algebraic group of type or defined over \C and
be a maximal torus of . For a dominant coweight of , the
-fixed point subscheme of the Schubert variety
in the affine Grassmannian is a finite scheme. We
prove that there is a natural isomorphism between the dual of the level one
affine Demazure module corresponding to and the ring of functions
(twisted by certain line bundle on ) of . We use
this fact to give a geometric proof of the Frenkel-Kac-Segal isomorphism
between basic representations of affine algebras of type and lattice
vertex algebras.Comment: 25 pages
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
An example of the derived geometrical Satake correspondence over integers
Let G^\vee be a complex simple algebraic group. We describe certain morphisms
of G^\vee(\calO)-equivariant complexes of sheaves on the affine Grassmannian
\Gr of G^\vee in terms of certain morphisms of G-equivariant coherent sheaves
on \frakg, where G is the Langlands dual group of G^\vee and \frakg is its Lie
algebra. This can be regarded as an example of the derived Satake
correspondence.Comment: 16 page
Any flat bundle on a punctured disc has an oper structure
We prove that any flat G-bundle, where G is a complex connected reductive
algebraic group, on the punctured disc admits the structure of an oper. This
result is important in the local geometric Langlands correspondence proposed in
arXiv:math/0508382. Our proof uses certain deformations of the affine Springer
fibers which could be of independent interest. As a byproduct, we construct
representations of affine Weyl groups on the homology of these deformations
generalizing representations constructed by Lusztig.Comment: 12 page
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