Affine grassmannian loop group book

Ginzburg, conjugacy classes in loop groups and gbundles on elliptic curves, int. They play a key role in topology and geometry as the universal spaces of vector bundles. 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. Affine insertion and pieri rules for the affine grassmannian 5 pairing between homology h. In mathematics, the affine grassmannian of an algebraic group g over a field k. These manifolds represent the constraints that arise in such areas as the symmetric eig. Differential geometry of grassmannian embeddings of based. The language used is one of vertex algebras, otherwise the approach chosen is that suggested by beilinson and drinfeld. 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. Ill describe the affine grassmannian as a moduli space, and construct it.

One can obtain the fundamental group of all grassmannians in a different way, but that way doesnt help find the homology groups. The first four chapters contain all the geometry needed for the second half of the book which deals with affine groups. Below is a list of publications that cite sagemath andor the sagemath cluster. In this paper we develop new newton and conjugate gradient algorithms on the grassmann and stiefel manifolds. We let the local a ne grassmannian to be the following functor on the category of. They are all cohenmacaulay, some are mildly singular.

We study algebraic loop groups and affine grassmannians in positive. In mathematics, there are two distinct meanings of the term affine grassmannian. Im trying to reconcile the differences between the algebraic based loop group and the affine grassmannian. One reason for study ing such groups is that they are the simplest examples of infinite dimensional lie groups. 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. In this talk a loop group lg will mean the group of smooth maps from the circle s i to. It develops the geometric langlands correspondence for loop groups, a new approach, from a unique.

Ill describe the a ne grassmannian as a moduli space, and construct it concretely. We give a survey on the notion of affine grassmannian, on affine springer fibers. An introduction to algebras of chiral differential operators. 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. I once believed that i understood the relationship, but i just read a paper which has. Geometry of flag varieties meetings, 732 evans, tuesdays 12.

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. This is the ktheoretic analogue of a construction of. 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. 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. 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. Grothendieck resolution, a ne grassmannian, and yangian by alexander shapiro doctor of philosophy in mathematics university of california, berkeley professor nicolai y. Equivariant homology and ktheory of affine grassmannians. The module structure of the equivariant ktheory of the based loop group of s u 2. Lurie and categorifying the fourier transform not even wrong. Langlands correspondence for loop groups edward frenkel. There is a sense in which it describes the local geometry of such moduli spaces. 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. Arithmetic and algebraic geometry a conference in honor of. 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.

I once believed that i understood the relationship. We refer to the not yet constructed affine grassmannian of g aff as the double affine grassmannian. 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 the geometric satake in mixed characteristic pages 403492 from volume 185 2017. See also grassmannian 2 construction and examples 2. Affine springer fibers and affine delignelusztig varieties. This requires finitizing an affine schubert variety to produce a matrix affine schubert variety. Affine flag manifolds and principal bundles ebook, 2010. As a result, we prove the representability of affine grassmannians and establish the geometric satake equivalence in mixed characteristic. Notes on the geometric satake equivalence springerlink.

Mirkovicvilonen cycles and polytopes annals of mathematics. Mar 17, 2016 we introduce various affine grassmannians, study their geometric properties, and give some applications. Hamacher, the dimension of affine delignelusztig varieties in the affine grassmannian, int. They both have a commutative ring structure with respect to convolution. 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. These notes are an informal introduction to algebras of chiral differential operators. Affine springer fibers and affine delignelusztig varieties unidue. Combinatorics of the loop grassmannian, algebra seminar, sydney university 14 june 2016. A new chapter to this grand project is provided in this book. Msri topics in combinatorial representation theory. Affine grassmannians and the geometric satake in mixed. This labelling is a refinement of the labelling of crystal bases by littelmann paths. An introduction to affine grassmannians and the geometric. Matrix schubert varieties for the affine grassmannian.

This will be an ongoing learning seminar devoted to current research directions where infinitesimal symmetries play a distinguished role. Grothendieck resolution, a ne grassmannian, and yangian. Another way of describing this is as the group of formal loops in g. 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. Ktheory schubert calculus of the affine grassmannian. 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. I will explain what the loop grassmannian and the affine flag variety are and how to label their points. 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 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. Tutorialon schubertvarietiesandschubertcalculus icerm. For additional information and updates on this book, visit. Affine grassmannians as moduli spaces of gbundles, and as quotients of loop groups. The module structure of the equivariant ktheory of the based. 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. Local geometric langlands correspondence and affine kac. The goal of this book is to present a systematic and selfcontained introduction. Affine grassmannians and geometric satake equivalences.

There is also a section that constructs a onedimensional extension of the loop group by loop rotation, which is a fairly transparent definition. Affine sets and affine groups london mathematical society. We establish equivalences of derived categories of the following 3 categories. 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. 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. 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. Langlands correspondence for loop groups math berkeley. One reason for study ing such groups is that they are the simplest examples of infinite. The publications listed in each section are sorted in chronological order. 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. Affine flag manifolds are infinite dimensional versions of familiar objects such as gramann varieties. Reshetikhin, chair in this thesis we address several questions on the structure and representation theory of quantum groups. Gortz, affine springer fibers and affine delignelusztig varieties. Part of the lecture notes in mathematics book series lnm.

Reconciling the affine grassmannian and the based loop group. Arithmetic and algebraic geometry a conference in honor. Differential geometry of grassmannian embeddings of based loop groups. Grassmann manifolds are named after hermann grassmann, a german school teacher in stettin who developed basic notions of linear algebra. I plan to write up the lectures as a book in fall 2007. Alternatively the first part provides a sure introduction to the foundations of algebraic geometry. 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. Let g be a connected reductive group, and let lg be the corresponding loop group. 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 introduce various affine grassmannians, study their geometric properties, and give some applications. These are the expanded lecture notes for a minicourse in 2015 pcmi summer school. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Part of the progress in mathematics book series pm, volume 253. Local geometric langlands correspondence and affine kacmoody.