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.

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**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).