By Alexander Schmitt

Affine flag manifolds are endless dimensional models of customary items akin to Gra?mann types. The publication positive aspects lecture notes, survey articles, and learn notes - in response to workshops held in Berlin, Essen, and Madrid - explaining the importance of those and similar gadgets (such as double affine Hecke algebras and affine Springer fibers) in illustration thought (e.g., the idea 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 valuable bundles). Novel points of the idea of relevant bundles on algebraic forms also are studied within the booklet.

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**Example text**

Aﬃne Deligne–Lusztig varieties: the Iwahori case Now we come to the Iwahori case. 20. The aﬃne Deligne–Lusztig variety Xx (b) in the aﬃne ﬂag variety associated with b ∈ G(L) and x ∈ W is given by Xx (b)(k) = g ∈ G(L); g −1 bσ(g) ∈ IxI /I. The same remarks as in the case of the aﬃne Grassmannian apply. In fact, in the Iwahori case it is much harder (and not yet completely settled) to give a criterion for which Xx (b) are non-empty, and a closed formula for their dimensions. Note that Xx (b) = ∅ whenever x and b do not lie in the same connected component of G(L).

For practical purposes, the following description is often good enough, however: The restriction NG T (L) → B(G) of the natural map from G(L) to B(G) factors through the extended aﬃne Weyl group W = NG T (L)/T (O). This follows from a variant of Lang’s theorem. 2). Now if w ∈ W , its Newton vector ν can be computed as follows. , the order of the image of w under the projection W = X∗ (A) W → W . Then wn = λ for some translation element λ ∈ X∗ (A), and ν = n1 λ ∈ X∗ (A)Q /W . The resulting map B(G) → X∗ (A)Q /W is called the Newton map.

We sketch the deﬁnition of equivariant cohomology in the -adic setting. Though elegant, it is not easy to digest because it uses -adic cohomology of algebraic stacks. As long as one works over the ﬁeld of complex numbers, one can also use the classical topological version of equivariant cohomology, see [30] and Tymoczko’s introductory paper [76]. The reference we follow in the -adic setting is the paper [15] by Chaudouard and Laumon. Let k be an algebraic closure of the ﬁeld Fq with q elements, let p = char k.