Twisted loop groups and their affine flag varieties. The emphasis is on the study of the kacmoody groups 9 and their flag varieties xy, including their detailed construction, and their applications to the representation theory of g. Chapter 1 gives a general overview of the theory of kac moody groups and is a condensed version of 6. In the course of proving the convergence of these eisenstein series, we also calculate a formula for the constant terms and prove their convergence and functional equations. In this paper, we define eisenstein series on affine kac moody groups over global function fields using an adelic approach. Special aspects of infinite or finite groups cohomology theory. Here n is the prounipotent progroup with lie algebra the. A group corresponding to a kacmoody algebra is a kacmoody group. Abstract simplicity of locally compact kacmoody groups. Kac, infinitedimensional lie algebras kac moody groups, their flag varieties and representation theory topological.
In spite of hard work by many people, we dont have a uniform construction of the overextended hyperbolic kacmoody groups starting. We prove a generalization of the satake isomorphism for this algebra, relating it to integrable representations of the langlands dual affine kac moody group. The classifying space bg of every kacmoody group is a homotopy colimit over classifying. This can be viewed as a generalization of the theory of affine flag varieties for loop groups to a twisted case. Chapter 2 contains some results about the hopfalgebra structure of the. Twisted loop groups and their affine flag varieties arxiv. Kac moody groups and cosheaves on davis building, arxiv. Generalized flag varieties of kac moody groups viii. There are other versions of groups associated to ga, e. Kacmoody lie algebras 9 were introduced in the mid1960s independently by v. Most of the material presented here is not available anywhere in the book literature. This subsumes the case of affine lie algebras in which case the.
First, we show that one can recover finite hecke categories realized by d modules on flag varieties from affine hecke categories realized by coherent sheaves on steinberg varieties via. We develop a theory of affine flag varieties and of their schubert varieties for reductive groups over a laurent power series local field k t with k a perfect field. Kacmoody groups and cosheaves on davis building, arxiv. This subsumes the case of affine lie algebras in which case the corresponding kacmoody group is a centrally extended loop group see there for more and often the words kacmoody group are used synonomously with centrally extended loop group. Combinatorics in affine flag varieties sydney mathematics and. Suitable for those interested in flag varieties and representation theory in the semisimple or kac moody case, this book includes different topics. Infinite dimensional lie algebras notes for part i notes for part ii. On topological split kacmoody groups and their twin buildings. The book of kumar shrawan kumar, kacmoody groups, their flag varieties and representation theory, progress in mathematics, vol. The cohomology rings of kacmoody groups,their flag manifolds.
This book is an excellent way of introducing todays students to representation. Table of contents frobenius splitting methods in geometry and representation theory with m. Kac moody lie algebras 9 were introduced in the mid1960s independently by v. Nitu kitchloo, on the topology of kac moody groups. The emphasis is on the study of the kac moody groups 9 and their flag varieties xy, including their detailed construction, and their applications to the representation theory of g.
Kac moody groups, their flag varieties and representation theory, progress in mathematics vol. Kacmoody groups, their flag varieties and representation. Spherical subgroups of kacmoody groups and transitive. Ams transactions of the american mathematical society. We first construct certain groupsb,nand pi1 kacmoody groups, their flag varieties and representation theory, 1st edition, birkhauser 2002. Special issue on kacmoody algebras and applications. Shrawan kumar, kacmoody groups, their flag varieties and representation theory, progress in mathematics, vol. From the data of the kacmoody lie algebra and a choice of a commutative. Using twin bnpairs g,b,n and g,b,n for g we show that if k is sufficiently large, then the subgroup b is a nonuniform lattice in g. This section gives a brief treatment of the theory of chevalley groups. Moody, generalizing the finitedimensional semisimple lie alge bras which we refer to as the finite case.
Definition of kacmoody groups and parabolic subgroups 174 2. Quiver hecke algebras and 2lie algebras algebra colloquium. Shrawan kumar, kacmoody groups, their flag varieties and representation theory, birkhauser 2002. Representation theory of symmetric groups is the most uptodate abstract algebra book on the subject of symmetric groups and representation theory. Twisted loop groups and their affine flag varieties core. Nitu kitchloo, kac moody groups over the last decade. We provide an introduction to the 2 representation theory of kac moody algebras, starting with basic properties of nil hecke algebras and quiver hecke algebras, and continuing with the resulting monoidal categories, which have a geometric description via quiver varieties, in certain cases. Loop groups section 2 of nitu kitchloos paper dominant k theory arxiv. Klappentext zu kac moody groups, their flag varieties and representation theory kac moody lie algebras 9 were introduced in the mid1960s independently by v. Kac moody groups, their flag varieties and representation theory shrawan kumar auth. The cohomology rings of kac moody groups, their flag manifolds and classifying spacesj. The equivariant category and its parabolic version 14 3.
Shrawan kumar, kacmoody groups, their flag varieties and representation theory, 1st edition, birkhauser 2002. Hklr of representation spaces of a quiver on a graph,whicharefinitedimensional. Generalized flag varieties of kacmoody groups 199 1. Brion, lectures on the geometry of flag varieties, pdf. Unc mathematics department professor shrawan kumar.
Finally in appendix c, we recall the basic theory of alne kac moody groups and their flag varieties. We define the spherical hecke algebra for an untwisted affine kac moody group over a local nonarchimedian field. The quiver varieties will be described as hyperkihler quotients see hitchin et al. The spherical hecke algebra for affine kacmoody groups i. This research project is motivated by various recent developments in the literature on infinitedimensional groups, in particular on topics that, in the classical case, are closely related to spherical varieties. Download group and representation theory pdf books pdfbooks. Twisted loop groups and their affine flag varieties request pdf. Parker distinguished professor of mathematics at the university of north carolina at chapel hill.
Reductive group actions with onedimensional quotient. In particular, we apply our previously developed theory to flag varieties, and we obtain new insights into fundamental categories in representation theory. Cohomology algebra of schubert varieties associated to kacmoody groups. Shrawan kumar, kac moody groups, their flag varieties and representation theory, progress in mathematics, 204. Kumar, kacmoody groups, their flag varieties and representation theory.
The theory has undergone tremendous developments in various directions and connections with diverse areas. We develop a theory of affine flag varieties and of their schubert varieties for reductive groups over a laurent power series local field kt with k a perfect field. Invariant bilinear form and the casimir operator 29 ii. Appendix b is an introduction to ind varieties and ind groups. Kacmoody groups, flag varieties, equivariant ktheory, positivity. Mixed perverse sheaves on flag varieties for coxeter groups. Kac moody groups, their flag varieties, and representation theory and frobenius splitting methods in geometry and representation theory jointly with michel brion. Abstract simplicity of locally compact kacmoody groups volume 150 issue 4 timothee marquis. Kacmoody groups, their flag varieties and representation theory. Kumar, positivity in tequivariant ktheory of flag varieties associated to kacmoody groups. In this thesis, we explore the topology of kac moody groups. Is there a geometric construction of hyperbolic kacmoody groups. In the finite case, 9 is nothing but a semisimple y simplyconnected algebraic group and x is the flag variety 9 py for a parabolic subgroup p y c g.
Nitu kitchloo, kacmoody groups over the last decade pdf. Kumarkacmoody groups, their flag varieties and representation theory. First of all the representation theory of kacmoody groups and the geometry of their flag manifolds are wellestablished central. Kacmoody groups, their flag varieties, and representation. Kacmoody groups and their representations request pdf. Kac, infinitedimensional lie algebras kacmoody groups, their flag varieties and representation theory topological. Infinite dimensional lie algebras an introduction progress in. Groups, their flag varieties and representation theory. Kacmoody groups, their flag varieties and representation theory, 1992. Kacmoody groups, their flag varieties and representation theory shrawan kumar auth.
We give an interpretation of tits associated group functor using representation theory of and we construct a locally compact kacmoody group g over a finite field k. Kacmoody groups and cluster algebras sciencedirect. Download citation kacmoody groups, their flag varieties and representation theory kacmoody lie algebras 9 were introduced in the mid1960s. Shrawan kumar, kacmoody groups, their flag varieties and representation theory. This chapter is devoted to the construction of the maximal kacmoody group c associated to a kacmoody lie algebra g ga, for a any. Utilizing new research and results, this book can be studied from a combinatorial, algorithmic or algebraic viewpoint. Here n is the prounipotent progroup with lie algebra the completion n. Ams representation theory of the american mathematical society.