Affine Flag Manifolds and Principal Bundles by Alexander Schmitt

By Alexander Schmitt

Affine flag manifolds are limitless dimensional types of widely used gadgets corresponding to Gra?mann types. The booklet positive factors lecture notes, survey articles, and study notes - in accordance with workshops held in Berlin, Essen, and Madrid - explaining the importance of those and comparable gadgets (such as double affine Hecke algebras and affine Springer fibers) in illustration conception (e.g., the speculation of symmetric polynomials), mathematics geometry (e.g., the basic lemma within the Langlands program), and algebraic geometry (e.g., affine flag manifolds as parameter areas for crucial bundles). Novel elements of the speculation of crucial bundles on algebraic types also are studied within the booklet.

Show description

Read or Download Affine Flag Manifolds and Principal Bundles PDF

Similar mathematics books

Love and Math: The Heart of Hidden Reality

What when you needed to take an paintings classification during which you have been in basic terms taught find out how to paint a fence? What when you have been by no means proven the work of van Gogh and Picasso, weren’t even informed they existed? sadly, this can be how math is taught, and so for many people it turns into the highbrow an identical of gazing paint dry.

Group Testing Theory in Network Security: An Advanced Solution

"Group checking out conception in community Security" explores a brand new department of team trying out thought with an program which reinforces examine leads to community protection. This short offers new options on a number of complicated community protection difficulties and mathematical frameworks in accordance with the gang trying out idea, particularly denial-of-service and jamming assaults.

How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics

Arithmetic is an engaging topic. i'm really not a mathematician, yet deal adequate with it in my selected occupation to be regularly surprised via how logical the applying of arithmetic to proving a theorem or studying an set of rules seems to be. yet wait . .. is it relatively logical? Or does it purely turns out so and what's really occurring is that the writer of the stated facts is utilizing inventive tips and strategies from the mathematical software field to by some means tie every thing up with a pleasant pink bow-tie?

Additional resources for Affine Flag Manifolds and Principal Bundles

Example text

To reduce the proof of the dimension formula to the superbasic case, one has to compare the affine Deligne–Lusztig varieties XμM (b) and XμG (b) for a Levi subgroup A ⊆ M ⊆ G and b ∈ M (L). If P = M N is a parabolic subgroup, then there is a bijection P (L)/P (o) ∼ = G(L)/K, and this defines a map α : G(L)/K ∼ = P (L)/P (o) −→ M (L)/M (o) dim Xμ (b) = ρ, μ − νb − from the affine Grassmannian for G to the affine Grassmannian for M . This map is not a morphism of ind-schemes, but for any connected component Y of the affine Grassmannian for M , the restriction of α to α−1 (Y ) is a morphism of ind-schemes.

If one works over the field of complex numbers, one can replace -adic cohomology by singular cohomology. Shimomura [71] has generalized the theorem to Springer fibers for GLn in partial flag varieties, see also the paper [39] by Hotta and Shimomura. On the other hand, the Springer fibers have severe singularities, and in particular Poincar´e duality fails for these varieties, even on the level of Betti numbers. 2. Affine Springer fibers Now let k be an algebraically closed field, let O = k[[ ]], and let L = k(( )) be the field of Laurent series.

2. σ-conjugacy classes Now and for the following sections we fix a finite field Fq , and let k be an algebraic closure of Fq . The Frobenius σ : x → xq acts on k, and also (on the coefficients) on L = k(( )): σ( ai i ) = aqi i . We write F = Fq (( )), the fixed field of σ in L. As usual, we fix an algebraic group G over Fq . We assume, since that is the case we will consider below, that G is a split connected reductive group (see Kottwitz’ paper for the classification of σ-conjugacy classes in the general case).

Download PDF sample

Rated 5.00 of 5 – based on 33 votes