One can obtain the fundamental group of all grassmannians in a different way, but that way doesnt help find the homology groups. The construction is applied to the study of the topology of the affine grassmannian of g and to establishing a langlands type correspondence for automorphic sheaves on the moduli space of g. This is the ktheoretic analogue of a construction of. Affine grassmannians as moduli spaces of gbundles, and as quotients of loop groups. Gortz, affine springer fibers and affine delignelusztig varieties. The book features lecture notes, survey articles, and research notes based on workshops held in berlin, essen, and madrid explaining the significance of these and related objects such as double affine hecke algebras and affine springer fibers in representation theory e. I plan to write up the lectures as a book in fall 2007. In one it is the manifold of all kdimensional affine subspaces of r n described on this page, while in the other the affine grassmannian is a quotient of a group ring based on formal laurent series.
Ginzburg, conjugacy classes in loop groups and gbundles on elliptic curves, int. In mathematics, the affine grassmannian of an algebraic group g over a field k is an indschemea colimit of finitedimensional schemeswhich can be thought of as a flag variety for the loop group gkt and which describes the representation theory of the langlands dual group l g through what is known as the geometric satake correspondence. Affine flag manifolds are infinite dimensional versions of familiar objects such as gramann varieties. Alternatively the first part provides a sure introduction to the foundations of algebraic geometry. Part of the progress in mathematics book series pm, volume 253. The goal of this book is to present a systematic and selfcontained introduction. There is a book by kumar kacmoody groups, their flag varieties, and representation theory that does the construction for the general kacmoody case, but i find the presentation dense. Local geometric langlands correspondence and affine kac. Grassmann manifolds are named after hermann grassmann, a german school teacher in stettin who developed basic notions of linear algebra. An introduction to algebras of chiral differential operators. Lurie and categorifying the fourier transform not even wrong. Tutorialon schubertvarietiesandschubertcalculus icerm.
We study algebraic loop groups and affine grassmannians in positive. The present thesis is the start of a project to recover schubert polynomials for the borelmoore homology of the special linear affine grassmannian by an analogous process. We give a survey on the notion of affine grassmannian, on affine springer fibers. The publications listed in each section are sorted in chronological order. Geometry of moduli spaces and representation theory. Affine flag manifolds and principal bundles ebook, 2010. Perhaps the most famous example of an indscheme is an infinite grassmannian which is a quotient of the loop group of an algebraic group g. I will explain what the loop grassmannian and the affine flag variety are and how to label their points. The a ne grassmannian is an important object that comes up when one studies moduli spaces of the form bun gx, where xis an algebraic curve and gis an algebraic group. We establish equivalences of derived categories of the following 3 categories. Eddie frierson 1,985 words case mismatch in snippet view article find links to article adr loop group the big bounce adr loop group bobby adr loop group commandents adr loop group choke adr loop group crossroads adr loop group dark. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Another way of describing this is as the group of formal loops in g. The key geometric input is a version of the beilinsondrinfeld grassmannian in which the looprotation action of c on the affine grassmannian gr is deformed to the action of c on grp by cyclic permutations.
Ill describe the a ne grassmannian as a moduli space, and construct it concretely. See also grassmannian 2 construction and examples 2. In mathematics, there are two distinct meanings of the term affine grassmannian. Ktheory schubert calculus of the affine grassmannian. Local geometric langlands correspondence and affine kacmoody. Let g be a connected reductive group, and let lg be the corresponding loop group. The module structure of the equivariant ktheory of the based. An introduction to affine grassmannians and the geometric. They both have a commutative ring structure with respect to convolution. Below is a list of publications that cite sagemath andor the sagemath cluster. We introduce various affine grassmannians, study their geometric properties, and give some applications.
Ktheory schubert calculus of the affine grassmannian with anne schilling and mark shimozono we study the ktheory schubert calculus of the affine grassmannian of a simple algebraic group g. A new chapter to this grand project is provided in this book. Mar 17, 2016 we introduce various affine grassmannians, study their geometric properties, and give some applications. Matrix schubert varieties for the affine grassmannian. Arithmetic and algebraic geometry a conference in honor of. The module structure of the equivariant ktheory of the based loop group of s u 2. An intrinsic construction of the tensor category of finite dimensional representations of the langlands dual group of g in terms of a tensor category of perverse sheaves on the loop group, lg, is given. It develops the geometric langlands correspondence for loop groups, a new approach, from a unique. Week 5, february 23, 25 loop groups and affine grassmannians a affine grassmannians as indschemes, the case of gln b affine grassmannians in general references.
I once believed that i understood the relationship. Ktheory schubert calculus of the a ne grassmannian thomas lam, anne schilling and mark shimozono abstract we construct the schubert basis of the torusequivariant khomology of the a ne grassmannian of a simple algebraic group g, using the ktheoretic nilhecke ring of kostant and kumar. Grothendieck resolution, a ne grassmannian, and yangian. Our main goal is to construct a perverse tstructure on the derived category of ad lgequivariant sheaves on lg and to show that the affine grothendieckspringer sheaf belongs to its core. Differential geometry of grassmannian embeddings of based.
Mirkovicvilonen cycles and polytopes annals of mathematics. I once believed that i understood the relationship, but i just read a paper which has. The first four chapters contain all the geometry needed for the second half of the book which deals with affine groups. In this paper we develop new newton and conjugate gradient algorithms on the grassmann and stiefel manifolds. Equivariant homology and ktheory of affine grassmannians. In this talk a loop group lg will mean the group of smooth maps from the circle s i to a compact lie group g. One reason for study ing such groups is that they are the simplest examples of infinite dimensional lie groups. Affine springer fibers and affine delignelusztig varieties unidue. Affine grassmannians and the geometric satake in mixed characteristic pages 403492 from volume 185 2017. Affine grassmannians and the geometric satake in mixed. Algebraic loop groups and moduli spaces of bundles article in journal of the european mathematical society 51 march 2003 with 55 reads how we measure reads. Arithmetic and algebraic geometry a conference in honor.
This will be an ongoing learning seminar devoted to current research directions where infinitesimal symmetries play a distinguished role. We let the local a ne grassmannian to be the following functor on the category of. Affine sets and affine groups london mathematical society. I general remarks in this talk a loop group lg will mean the group of smooth maps from the circle s i to a compact lie group g. This requires finitizing an affine schubert variety to produce a matrix affine schubert variety. One reason for study ing such groups is that they are the simplest examples of infinite. The purpose of this article is to try to generalize the above results to the case when g is replaced by the corresponding affine kacmoody group g aff.
This book is based on lectures given at the graduate summer school of the 2015 park city mathematics institute program geometry of moduli spaces and representation theory, and is devoted to several interrelated topics in algebraic geometry, topology of algebraic varieties, and representation theory. In mathematics, the affine grassmannian of an algebraic group g over a field k. They play a key role in topology and geometry as the universal spaces of vector bundles. For v v a vector space and r r a cardinal number generally taken to be a natural number, the grassmannian gr r, v grr,v is the space of all r rdimensional linear subspaces of v v. Langlands correspondence for loop groups math berkeley. For additional information and updates on this book, visit. Affine insertion and pieri rules for the affine grassmannian 5 pairing between homology h. The language used is one of vertex algebras, otherwise the approach chosen is that suggested by beilinson and drinfeld. These notes are an informal introduction to algebras of chiral differential operators. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Langlands correspondence for loop groups edward frenkel. Reshetikhin, chair in this thesis we address several questions on the structure and representation theory of quantum groups.
There is also a section that constructs a onedimensional extension of the loop group by loop rotation, which is a fairly transparent definition. As a result, we prove the representability of affine grassmannians and establish the geometric satake equivalence in mixed characteristic. We refer to the not yet constructed affine grassmannian of g aff as the double affine grassmannian. Affine springer fibers and affine delignelusztig varieties. Part of the lecture notes in mathematics book series lnm. Once we have this and have determined the attaching maps, we can also get the integral homology, so that is how i like would to proceed if possible. Affine grassmannians and geometric satake equivalences. Combinatorics of the loop grassmannian, algebra seminar, sydney university 14 june 2016. Grothendieck resolution, a ne grassmannian, and yangian by alexander shapiro doctor of philosophy in mathematics university of california, berkeley professor nicolai y. There is a sense in which it describes the local geometry of such moduli spaces.
Ill describe the affine grassmannian as a moduli space, and construct it. In this talk a loop group lg will mean the group of smooth maps from the circle s i to. This labelling is a refinement of the labelling of crystal bases by littelmann paths. Geometry of flag varieties meetings, 732 evans, tuesdays 12. Differential geometry of grassmannian embeddings of based loop groups. Hamacher, the dimension of affine delignelusztig varieties in the affine grassmannian, int. They are all cohenmacaulay, some are mildly singular. This is an application of the general idea that one can approximate an action of the circle by the action of its subgroup c of order p. These manifolds represent the constraints that arise in such areas as the symmetric eig.
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