Affine Kac-Moody Groups Research Articles

Affine Kac-Moody groups and Lie algebras in the language of SGA3

In infinite-dimensional Lie theory, the affine Kac-Moody Lie algebras and groups play a distinguished role due to their many applications to various areas of mathematics and physics. Underlying these infinite-dimensional objects there are closely related group schemes and Lie algebras of finite type over Laurent polynomial rings. The language of SGA3 is perfectly suited to describe such objects. The purpose of this short article is to provide a natural description of the affine Kac-Moody groups and Lie algebras using this language.

E 9 exceptional field theory. Part II. The complete dynamics

We construct the first complete exceptional field theory that is based on an infinite-dimensional symmetry group. E9 exceptional field theory provides a unified description of eleven-dimensional and type IIB supergravity covariant under the affine Kac-Moody symmetry of two-dimensional maximal supergravity. We present two equivalent formulations of the dynamics, which both rely on a pseudo-Lagrangian supplemented by a twisted self-duality equation. One formulation involves a minimal set of fields and gauge symmetries, which uniquely determine the entire dynamics. The other formulation extends {mathfrak{e}}_9 by half of the Virasoro algebra and makes direct contact with the integrable structure of two-dimensional supergravity. Our results apply directly to other affine Kac-Moody groups, such as the Geroch group of general relativity.

PRESENTATIONS OF AFFINE KAC–MOODY GROUPS

How many generators and relations does $\text{SL}\,_{n}(\mathbb{F}_{q}[t,t^{-1}])$ need? In this paper we exhibit its explicit presentation with $9$ generators and $44$ relations. We investigate presentations of affine Kac–Moody groups over finite fields. Our goal is to derive finite presentations, independent of the field and with as few generators and relations as we can achieve. It turns out that any simply connected affine Kac–Moody group over a finite field has a presentation with at most 11 generators and 70 relations. We describe these presentations explicitly type by type. As a consequence, we derive explicit presentations of Chevalley groups $G(\mathbb{F}_{q}[t,t^{-1}])$ and explicit profinite presentations of profinite Chevalley groups $G(\mathbb{F}_{q}[[t]])$.

Unramified Whittaker functions on p-adic loop groups

We define unramified Whittaker functions on the $p$-adic points of an affine Kac-Moody group, and establish an analogue of the Casselman-Shalika formula for these functions.

Presentation of affine Kac–Moody groups over rings

Tits has defined Steinberg groups and Kac-Moody groups for any root system and any commutative ring R. We establish a Curtis-Tits-style presentation for the Steinberg group St of any rank > 2 irreducible affine root system, for any R. Namely, St is the direct limit of the Steinberg groups coming from the 1- and 2-node subdiagrams of the Dynkin diagram. This leads to a completely explicit presentation. Using this we show that St is finitely presented if the rank is > 3 and R is finitely generated as a ring, or if the rank is 3 and R is finitely generated as a module over a subring generated by finitely many units. Similar results hold for the corresponding Kac-Moody groups when R is a Dedekind domain of arithmetic type.

Iwahori–Hecke algebras for p-adic loop groups

This paper is a continuation of Braverman and Kazhdan (Ann Math (2) 174(3):1603–1642, 2011) in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac–Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori–Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik’s double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald’s formula for the spherical function in the finite-dimensional case. The results of this paper have been previously announced in Braverman and Kazhdan (European Congress of Mathematics. European Mathematical Society, Zurich, 2014).

Orthogonal symmetric affine Kac-Moody algebras

Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues, known as affine Kac-Moody groups. We solve this problem and construct affine Kac-Moody symmetric spaces in a series of several papers. This paper focuses on the algebraic side; more precisely, we introduce OSAKAs, the algebraic structures used to describe the connection between affine Kac-Moody symmetric spaces and affine Kac-Moody algebras and describe their classification.

Tame Fréchet structures for affine Kac-Moody groups

We construct holomorphic loop groups and their associated affine Kac-Moody groups and prove that they are tame Frechet manifolds; furthermore we study the adjoint action of these groups. These results form the functional analytic core for a theory of affine Kac-Moody symmetric spaces, that will be developed in forthcoming papers. Our construction also solves the problem of complexification of completed Kac-Moody groups: we obtain a description of complex completed Kac-Moody groups and, using this description, deduce constructions of their non-compact real forms.

Eisenstein series on affine Kac-Moody groups over function fields

In his pioneering work, H. Garland constructed Eisenstein series on affine Kac-Moody groups over the field of real numbers. He established the almost everywhere convergence of these series, obtained a formula for their constant terms, and proved a functional equation for the constant terms. In his subsequent paper, the convergence of the Eisenstein series was obtained. In this paper, we define Eisenstein series on affine Kac-Moody groups over global function fields using an adelic approach. 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.

Bounded presentations of Kac–Moody groups

Abstract. In this article we extend the result of Guralnick, Kantor, Kassabov and Lubotzky to the affine Kac–Moody groups: we show that there exists a constant such that every affine Kac–Moody group defined over a finite field , (with the exception of and ), has a presentation σ with . We then derive the consequences of this result for the 2-spherical Kac–Moody groups.

Double MV cycles and the Naito–Sagaki–Saito crystal

The theory of MV cycles associated to a complex reductive group G has proven to be a rich source of structures related to representation theory. We investigate double MV cycles, which are analogues of MV cycles in the case of an affine Kac–Moody group. We prove an explicit formula for the Braverman–Finkelberg–Gaitsgory (2006) [7] crystal structure on double MV cycles, generalizing a finite-dimensional result of Baumann and Gaussent (2008) [2]. As an application, we give a geometric construction of the Naito–Sagaki–Saito [23] crystal via the action of SL̂n on Fermionic Fock space. In particular, this construction gives rise to an isomorphism of crystals between the set of double MV cycles and the Naito–Sagaki–Saito crystal. As a result, we can independently prove that the Naito–Sagaki–Saito crystal is the B(∞) crystal. In particular, our geometric proof works in the previously unknown case of sl̂2.

Kac-Moody groups, infinite dimensional differential geometry and cities

Minimal affine Kac-Moody groups act on affine twin buildings by isometries. However there is no way to extend this action to any completion of the Kac-Moody groups. To remedy this, we introduce in this paper affine twin cities, a new class of objects, whose elements behave like completions of twin buildings. Twin cities are defined as special arrays of affine buildings connected among themselves by twinnings. Corresponding to completed affine Kac-Moody groups they are characterized by the type of the affine buildings and by some kind of regularity conditions describing the completion. The isometry groups of affine twin cities are (completions of) affine Kac-Moody groups. We study applications of cities in infinite dimensional differential geometry by proving infinite dimensional versions of classical differential geometric results: For example, we show that points in an isoparametric submanifold in a Hilbert space correspond to all chambers in a city. In two sequels we will describe the theory of twin cities for formal completions.

Pursuing the double affine Grassmannian II: Convolution

This is the second paper of a series (started by Braverman and Finkelberg, 2010 [2]) which describes a conjectural analog of the affine Grassmannian for affine Kac–Moody groups (also known as the double affine Grassmannian). The current paper is dedicated to describing a conjectural analog of the convolution diagram for the double affine Grassmannian. In the case when G=SL(n) our conjectures can be derived from Nakajima (2009) [12].

The spherical Hecke algebra for affine Kac-Moody groups I

We dene the spherical Hecke algebra for an (untwisted) ane KacMoody group over a local non-archimedian eld. We prove a generalization of the Satake isomorphism for this algebra, relating it to integrable representations of the Langlands dual ane Kac-Moody group. In the next publication we shall use these results to dene and study the notion of Hecke eigenfunction for the group Ga .

A Brylinski filtration for affine Kac–Moody algebras

Braverman and Finkelberg have recently proposed a conjectural analogue of the geometric Satake isomorphism for untwisted affine Kac–Moody groups. As part of their model, they conjecture that (at dominant weights) Lusztig's q-analog of weight multiplicity is equal to the Poincare series of the principal nilpotent filtration of the weight space, as occurs in the finite-dimensional case. We show that the conjectured equality holds for all affine Kac–Moody algebras if the principal nilpotent filtration is replaced by the principal Heisenberg filtration. The main body of the proof is a Lie algebra cohomology vanishing result. We also give an example to show that the Poincare series of the principal nilpotent filtration is not always equal to the q-analog of weight multiplicity. Finally, we give some partial results for indefinite Kac–Moody algebras.

Gerbal Representations of Double Loop Groups

A crucial role in representation theory of loop groups of reductive Lie groups and their Lie algebras is played by their nontrivial second cohomology classes, which give rise to their central extensions (the affine Kac–Moody groups and Lie algebras). Loop groups embed into the group GL∞ of continuous automorphisms of C((t)), and these classes come from a second cohomology class of GL∞. In a similar way, double loop groups embed into a group of automorphisms of C((t))((s)), denoted by GL∞,∞, which has a nontrivial third cohomology. In this paper, we explain how to realize a third cohomology class in representation theory of a group: it naturally arises when we consider representations on categories rather than vector spaces. We call them “gerbal representations”. We then construct a gerbal representation of GL∞,∞ (and hence of double loop groups), realizing its nontrivial third cohomology class, on a category of modules over an infinite-dimensional Clifford algebra. This is a two-dimensional analog of the fermionic Fock representations of the ordinary loop groups.

Representation theory of p-adic groups and canonical bases

In this paper, we interpret the Gindikin–Karpelevich formula and the Casselman–Shalika formula as sums over Kashiwara–Lusztigʼs canonical bases, generalizing the results of Bump and Nakasuji (2010) [7] to arbitrary split reductive groups. We also rewrite formulas for spherical vectors and zonal spherical functions in terms of canonical bases. In a subsequent paper Kim and Lee (preprint) [14], we will generalize these formulas to p-adic affine Kac–Moody groups.

Pursuing the double affine Grassmannian, I: Transversal slices via instantons on Ak-singularities

This article is the first in a series that describes a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. (For simplicity, we only consider the untwisted and simply connected case here.) The usual geometric Satake isomorphism for a reductive group G identifies the tensor category Rep(G∨) of finite-dimensional representations of the Langlands dual group G∨ with the tensor category PervG(O)(GrG) of G(O)-equivariant perverse sheaves on the affine Grassmannian GrG=G(K)/G(O) of G. (Here K=C((t)) and O=C[[t]].) As a by-product one gets a description of the irreducible G(O)-equivariant intersection cohomology (IC) sheaves of the closures of G(O)-orbits in GrG in terms of q-analogs of the weight multiplicity for finite-dimensional representations of G∨. The purpose of this article is to try to generalize the above results to the case when G is replaced by the corresponding affine Kac-Moody group Gaff. (We refer to the (not yet constructed) affine Grassmannian of Gaff as the double affine Grassmannian.) More precisely, in this article we construct certain varieties that should be thought of as transversal slices to various Gaff(O)-orbits inside the closure of another Gaff(O)-orbit in GrGaff. We present a conjecture that computes the intersection cohomology sheaf of these varieties in terms of the corresponding q-analog of the weight multiplicity for the Langlands dual affine group Gaff∨, and we check this conjecture in a number of cases. Some further constructions (such as convolution of the corresponding perverse sheaves, analog of the Beilinson-Drinfeld Grassmannian, and so forth) will be addressed in another publication

Five-Branes in M-Theory and a Two-Dimensional Geometric Langlands Duality

A recent attempt to extend the geometric Langlands duality to affine Kac-Moody groups, has led Braverman and Finkelberg [1] to conjecture a mathematical relation between the intersection cohomology of the moduli space of G-bundles on certain singular complex surfaces, and the integrable representations of the Langlands dual of an associated affine G-algebra, where G is any simply-connected semisimple group. For the AN−1 groups, where the conjecture has been mathematically verified to a large extent, we show that the relation has a natural physical interpretation in terms of six-dimensional compactifications of M-theory with coincident five-branes wrapping certain hyperkahler four-manifolds; in particular, it can be understood as an expected invariance in the resulting spacetime BPS spectrum under string dualities. By replacing the singular complex surface with a smooth multi-Taub-NUT manifold, we find agreement with a closely related result demonstrated earlier via purely field-theoretic considerations by Witten [2]. By adding OM five-planes to the original analysis, we argue that an analogous relation involving the non-simply-connected DN groups, ought to hold as well. This is the first example of a string-theoretic interpretation of such a two-dimensional extension to complex surfaces of the geometric Langlands duality for the A−D groups. On leave of absence from the National University of Singapore.

Conjugacy classes in affine Kac–Moody groups and principal G-bundles over elliptic curves

For a simple complex Lie group G the connected components of the moduli space of semistable G-bundles over an elliptic curve are weighted projective spaces or quotients of weighted projective spaces by a finite group action. In this note we will provide a new proof of this result using the invariant theory of affine Kac–Moody groups, in particular the action of the (twisted) Coxeter element on the root system of G.