Lusztig asterisque1011021983, 208209 is a tensor category for convolution without truncation, closely related to the tensor category of representations of the group dual, in the sense of langlands. One novel feature is the use of geometric techniques to study walgebras. Michael finkelbergs research works national research university. Ill show the picture which summarises the connection to the affine hecke algebra and the spherical affine hecke algebra. Dmodules on the affine grassmannian and representations of affine.
Equivariant homology and ktheory of affine grassmannians and. Geometric and modular representation theory seminar. Casselman notes on macdonalds book on padic spherical functions. Affine springer fibers and affine delignelusztig varieties unidue. Let g be a connected, reductive algebraic group over the field k c. Let grr, r n denote the grassmannian of rdimensional subspaces of r n.
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. In this paper, we present a version of the matsuki correspondence for the affine grassmannian gr gkgo of a connected reductive complex algebraic group g. Indeed, let us trivialize the tangent bundle of and the symplectic form in a neighbourhood of. I once believed that i understood the relationship. They both have a commutative ring structure with respect to convolution. 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. The loop group is representable by a strict indaffine indgroup scheme over k cf. Let k ct be the field of formal laurent series, and let o ct be the ring of formal power series. We present a version of the matsuki correspondence for the affine grassmannian gr gctgct of a connected reductive complex algebraic group g. We begin with a reductive g and consider the affine grassmannian qx, the grassmannian for the loop group of g. For affine grassmannians, geometric satake, and related, see zhus notes. Let kbe a eld and let k kpp tqq be the eld of formal laurent series over k.
Affine springer fibers and affine delignelusztig varieties. Our main statement is an antiisomorphism between the orbit posets of two subgroups of gk acting on gr. The main results are normality of schubertvarieties, the construction of linebundles on the affine grassmannian, and the proof that they induce linebundles on the modulistack of torsors. Ktheory schubert calculus of the affine grassmannian. The equivalence 23 is a step towards resolving an old mystery surrounding the existense of two completely different realizations of the affine hecke algebra which have played a key role in. 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. The book bk, chapter 1 provides a detailed account of the general theory.
Im trying to reconcile the differences between the algebraic based loop group and the affine grassmannian. While, there is a more general definition for any smooth affine k group. Segal, loop groups and their representations book i. Introduction to heegaard floer homology iopscience. Integral homology of loop groups via langlands dual groups. As a functor, it maps an a ne cscheme s specrto a gtorsor f g on p1 s together with a trivialization. Apr 23, 2019 a disk determines a loop in the lagrangian grassmannian. Part of the progress in mathematics book series pm, volume 253 abstract. The goal of these lectures will be to start from scratch and learn what the affine grassmannian of a reductive group is, and how its geometry is related to the representation theory of reductive groups. In the case of affine grassmannian slices, i will describe how this deformation theory is related to the beilinsondrinfeld grassmannian. If we add the loop rotation equivariance, we obtain a noncommutative deformation of the khomology ring, and thus a poisson structure on its spectrum. Nov 19, 2018 it is also well known that there is an algebraic analog of loop grassmannians, known as the affine grassmannian of an algebraic groop this is an indvariety.
Differential geometry of grassmannian embeddings of based. Jonathan weitsman, northeastern university enhanced symmetry in the semiclassical category and characters of loop groups. Bezrukavnikov, perverse sheaves on affine flags and nilpotent cone of the langlands dual group, arxiv. Gaitsgory, dmodules on the affine grassmannian and representations of affine kacmoody algebras, duke math. On the action of the dual group on the cohomology of. Seminars and colloquia by series school of mathematics. Jul 15, 1997 this contains as a special case the category of equivariant perverse sheaves on an affine grassmannian, which, as a consequence of g. This gluing isomorphism can be given by an element of the loop group, but is not uniquely determined. Let gbe a simply connected semisimple and simple group over kwhich is split over k. In this talk a loop group lg will mean the group of smooth maps from the circle s i to. Therefore, we have some affine kacmoody loop algebra of meromorphic functions with values in lie algebra use the construction of fock space reprsentations to have representations of such. Ill describe the a ne grassmannian as a moduli space, and construct it concretely. Convex polytopes for the central degeneration of the. Exercises on the a ne grassmannian northeastern university.
The associated twisted loop group is the group of maps r gsuch that. Contents of individual talks, prerequisites, and logical. Another way of describing this is as the group of formal loops in g. Ill describe the affine grassmannian as a moduli space, and construct it. Ktheory schubert calculus of the affine grassmannian volume 146 issue 4. On the action of the dual group on the cohomology of perverse. You are asking several questions here, so it may be useful to separate out what is going on first in the setting of affine reflection groups.
Localization of gmodules on the affine grassmannian 41 0. One reason for study ing such groups is that they are the simplest examples of infinite dimensional lie groups. Bfs roman bezrukavnikov, michael finkelberg, and vadim schechtman, factorizable sheaves and quantum groups, lecture notes in mathematics, vol. Affine grassmannian, affine flag variety, and the central degeneration 2. Higgs bundles, kostant section, and local triviality of gbundles dennis gaitsgory.
This book is an exposition of the subject spanning kschur functions and affine. Ar, n if and only if the three conditions are satisfied. Equivariant homology and ktheory of affine grassmannians. Citeseerx matsuki correspondence for the affine grassmannian. In mathematics, the affine grassmannian of an algebraic group g over a field k is an indscheme a colimit of finitedimensional schemes which can be thought of as a flag variety for the loop group g k t and which describes the representation theory of the langlands dual group lg through what is known as the geometric satake correspondence. This kindscheme is known as the affine grassmannian for g. This is work in progress with gwyn bellamy, dinakar muthiah and oded yacobi. This is independent of the application to algebraic groups or loop groups. Pages 403492 from volume 185 2017, issue 2 by xinwen zhu. Such representations can be realized in spaces of global sections of twisted dmodules on the quotient of the loop group gt. Oct 02, 2010 it is wellknown that the coset spaces gkzgkz, for a reductive group g over a field k, carry the geometric structure of an inductive limit of projective kschemes. This series of noaudience performances took place for 2 days a week, 1 hour 50 minutes a day, at the affine grassmannian class under the instruction of ngo bao chau, professor of mathematics at the university of chicago, from january 1st to march th, 2014. As a result, the functor carries an important additional structure, an endomorphism coming from monodromy of nearby cycles.
Quadrics and grassmann varieties, cambridge, at the university. Perverse sheaves on a loop group and langlands duality. Ktheory schubert calculus of the affine grassmannian compositio. Quantum groups, the loop grassmannian, and the springer resolution. Show that fis invertible if and only if there is an integer msuch that a k. Using the gauss equation of the embedding of a based loop group into the restricted grassmannian we show that the conditional ricci curvature of a based loop group is proportional to its metric. If we pass from vector bundles to pairs of vector bundles and a trivialization as in the original post, then the nonuniqueness corresponds exactly to taking the quotient of the loop group by the positive loop group if one stayed with vector. The amplituhedron and the oneloop grassmannian measure with yuntao bai. It was proved by ginzburg, mirkovic and vilonen that the goequivariant perverse sheaves on the affine grassmannian of a connected reductive group g form a tensor category equivalent to the tensor category of finite dimensional representations of the dual group g in this paper we construct explicitly the action of g. For each finite bps chamber of an model which is uv superconformal, one gets a periodic ysystem, while for each finite bps chamber of an asymptoticallyfree qft one gets a qsystem i. The shape of a random affine weyl group element, and random core partitions. Cells in affine weyl groups and tensor categories sciencedirect.
This labelling is a refinement of the labelling of crystal bases by littelmann paths. The affine grassmannian g r g l n for g l n is the presheaf which takes every k algebra r to the set of r families of lattices. According to wikipedia, the affine grassmannian of an algebraic group ggover a field kkis an indschemewhich can be thought of as a flag variety for the loop group gktgkt the affine grassmannian is indrepresentable. We show that principal quiver grassmannians for the one vertex loop quiver provide. The class of the resulting loop in is called the maslov index. Pdf crystals via the affine grassmannian semantic scholar. Let a be a ring and let g be an affine agroup scheme equipped with a closed. The functor is given as nearby cycles for a family over a curve whose general fiber is the affine grassmannian times the finite flag variety and whose special fiber is the affine flag variety. Another way of describing this is as the group of fo. The twisted loop grouplg is the functor on the category of k algebras lg. The grassmannian as a real affine algebraic variety. Affine grassmannians and the geometric satake in mixed characteristic. Ginzburg notes on perverse sheaves on a loop group and langlands duali. Geometric and modular representation theory seminar special.
The tangent spaces to and on the boundary of define two paths in, which can be connected by standard paths at the points and. Quantum groups, the loop grassmannian, and the springer. As a set we want the a ne grassmannian gr parametrize the set of lattices in v, i. Affine grassmannians and affine flag varieties 425 2. Ginzburg, conjugacy classes in loop groups and gbundles on elliptic curves, int. As a result, the functor carries an important additional structure, an endomorphism coming from monodromy of.
The main statement is an antiisomorphism between the orbit posets of two subgroups of gct acting on gr. Affine insertion and pieri rules for the affine grassmannian. Eudml algebraic loop groups and moduli spaces of bundles. Local geometric langlands correspondence and affine kacmoody. Affine grassmannians and geometric satake equivalences. This is independent of the application to algebraic groups or loop. Convex polytopes for the central degeneration of the affine. There is a sense in which it describes the local geometry of such moduli spaces. Introduction to affine grassmannians and the geometric satake equivalence.
Schubert polynomials for the affine grassmannian of the symplectic group. Global nilpotent cone and flatness of the hitchin map xinwen zhu oct. These are used to give an infinite family of affine analogues of the bezrukavnikovbravermanmirkovic theorem, providing a geometric version of rodiers compact approximation to the whittaker model from the arithmetic setting. Schubert classes of a loop group michigan state university. I will explain what the loop grassmannian and the affine flag variety are and how to label their points. In these notes, we survey the homology of the loop group. As a result, we prove the representability of affine grassmannians and establish the geometric satake equivalence in mixed characteristic. Perverse sheaves on affine grassmannians and langlands. For technical reasons we work with formal algebraic loops. Residues on affine grassmannians archive ouverte hal.
Reconciling the affine grassmannian and the based loop group. We use intensively the notions of the book 8, for example the different radicals of a smooth algebraic kgroup h. Local geometric langlands correspondence and affine kac. Affine grassmannian and the loop group dennis gaitsgory, notes by d. Degenerate affine grassmannians and loop quivers articles.
We consider a certain category of sheaves, the spherical perverse sheaves, on. Conjectured langlands dualities typically relate automorphic objects on a reductive group g to galois objects on a dual reductive group g. F gj d1, where d 1is the formal disk specrt 1 at in nity. Affine grassmannians and the geometric satake in mixed. The affine grassmannian is an infinite dimen sional complex space. The associated loop group is then the group of c0, or c1, or c. Citeseerx document details isaac councill, lee giles, pradeep teregowda. We give a survey on the notion of affine grassmannian, on affine springer fibers. We use intensively the notions of the book cgp, for example the different rad icals of a.
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