Affine Flag Manifolds and Principal Bundles (Trends in Mathematics)

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The claim you want is Proposition 2. You can find this in the paper "Conformal blocks and generalized theta functions" by Beauville and Laszlo Section 1 here , together with the paper "Un lemme de descente" by the same authors here.

Beauville, Arnaud; Laszlo, Yves. Conformal blocks and generalized theta functions. Un lemme de descente. This gluing isomorphism can be given by an element of the loop group, but is not uniquely determined. If we pass from vector bundles to pairs of vector bundles and a trivialization as in the original post, then the non-uniqueness corresponds exactly to taking the quotient of the loop group by the positive loop group if one stayed with vector bundles, one would instead arrive at a double quotient of the loop group.

See the above-mentioned papers. If you're familiar with the "structure" group in terms of gluing charts perspective on vector bundles and torsors, then you can see this because they have the same structure group.

Bibliography of Georg Hein

It turns out to be an equivalence; the inverse is given by the "frame bundle" construction. There are some details on topologizing this which I'm not sure of offhand.

That is, a torsor has to satisfy a cocycle condition and two torsors are conditions if they are related by a coboundary. On the other hand, you can do the same for vector bundles: This construction is also functorial. There are two "variables" on the left hand side of this functor.

Homepage of Jochen Heinloth

We can also fix instead of the representation, fix the torsor; this is the "Tannakian" point of view where I think. This is exactly the definition of a lattice. By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service , privacy policy and cookie policy , and that your continued use of the website is subject to these policies. Home Questions Tags Users Unanswered.

Definition of Affine Grassmannian Ask Question. I appreciate any comments or answers. Bezrukavnikov , Cohomology of tilting modules over quantum groups and t -structures on derived categories of coherent sheaves , Invent. Bezrukavnikov , Noncommutative counterparts of the Springer resolution , in International Congress of Mathematicians.

Bezrukavnikov , On two realizations of an affine Hecke algebra , preprint arXiv: MR Zbl pre Rumynin , Singular localization and intertwining functors for reductive Lie algebras in prime characteristic , Nagoya Math. Rumynin , Localization of modules for a semisimple Lie algebra in prime characteristic , Ann. Riche , Presentation of B aff ' , appendix to [54].

SFB 647: Search results for "affine"

Thomas , Braid groups and Kleinian singularities , Math. Bridgeland , Stability conditions and Kleinian singularities , Int. Kumar , Frobenius splitting methods in geometry and representation theory , Progress in Math. Gordon , The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras , Math. Kamnitzer , Braiding via geometric Lie algebra actions , Compos.

Kapranov , Derived Quot schemes , Ann.

Dodd , Equivariant coherent sheaves , Soergel bimodules, and categorification of affine Hecke algebras, preprint arXiv: Eisenbud , Commutative algebra. With a view towards algebraic geometry, Graduate Texts in Math. Ginzburg , Variations on themes of Kostant , Transform. Groups 13 , Ginzburg , Harish-Chandra bimodules for quantized Slodowy slices , Represent.

Research Interests

Theory 13 , Grinberg , A generalization of Springer theory using nearby cycles , Represent. Theory 2 , Hartshorne , Residues and duality , Lecture Notes in Math. Hartshorne , Algebraic geometry , Graduate Texts in Math. Humphreys , Conjugacy classes in semisimple algebraic groups , Mathematical Surveys and Monographs 43, Amer. Uehara , Stability conditions on A n -singularities , J. Jantzen , Subregular nilpotent representations of Lie algebras in prime characteristic , Represent. Theory 3 , Jantzen , Nilpotent orbits in representation theory , in Lie theory, Progr. Weisfeiler , Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic p , Indag.

Saito , Geometric construction of crystal bases , Duke Math. Tanisaki , The characteristic cycles of holonomic systems on a flag manifold related to the Weyl group algebra , Invent.

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