THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th ANNIVERSARY YEAR

Affine Dynkin Diagrams Re-viewed

The simply-laced affine Dynkin diagrams have remarkable connections with the finite subgroups of SU(2) and their representations via the McKay Correspondence. This talk will discuss some new connections between the diagrams and certain combinatorial objects.

Quantum affine algebra actions on categories

A quantum affine algebra has two presentations: the Kac-Moody and the loop. What does it mean to have an action of these algebras on categories? The answer should be motivated by examples. A notable phenomenon is that in the loop presentation the categories seem to require a triangulated structure.

V. Chari (Riveside)

Lecture Notes

Prime representations and extensions

We are interested in the category of finite--dimensional representations of quantum affine algebras. This is a tensor category and a prime representation is one which is not isomorphic to the tensor product of two non trivial representations in the category. Understanding the prime simple representations is an important problem and several important families of examples are known of such representations. But no unifying feature is known to connect these families. In joint work with Charles Young and Adriano Moura, we show that the notion of prime is closely connected with the homological properties of these representations. In this talk we shall give evidence for our conjecture: a simple finite--dimensional representation V is prime iff the space of self--extensions Ext(V,V) is one--dimensional. We shall also see that this feature is quite unusual and has no counterpart in the case of the affine Lie algebra for instance.

We classify the Lagrangian subalgebras of all classical simple Lie superalgebras and discuss the corresponding super varieties.
The talk is based on a joint work with Milen Yakimov.

For any Coxeter group, Soergel gave a straightforward construction of a collection of bimodules, now called Soergel bimodules, over the coordinate ring of the reflection representation. Soergel bimodules form a monoidal category, whose Grothendieck ring is isomorphic to the Hecke algebra of the Coxeter group. Soergel conjectured that (when defined over a field of characteristic zero) the indecomposable bimodules would descend to the Kazhdan-Lusztig basis, which would give an algebraic proof of the various positivity conjectures put forth by Kazhdan and Lusztig.

For Weyl groups, Soergel bimodules are constructed to agree precisely with the equivariant intersection cohomology of Schubert varieties. That is, Soergel bimodules are (by definition) summands of Bott-Samelson bimodules, which are the equivariant cohomology of Bott-Samelson resolutions of Schubert varieties. Therefore, the Decomposition theorem implies that the indecomposable bimodules agree with the intersection cohomology of the simple perverse sheaves on the flag variety. However, without the use of the Decomposition theorem, there is no a priori reason why the Bott-Samelson bimodules should split into summands as expected.

Inspired by de Cataldo and Migliorini's Hodge-theoretic proof of the Decomposition Theorem, we provide an algebraic proof of the Soergel conjecture for a general Coxeter group. Moreover, we show algebraically that Soergel bimodules have the Hodge-theoretic properties expected of an equivariant intersection cohomology space. This is joint work with Geordie Williamson.

If time permits, we will advertise a counter-example to the original version of Soergel's conjecture: a quantized version of the geometric Satake equivalence.

The speaker will present some results and conjectures suggesting that the representation theory of symplectic reflection algebras for wreath products (in particular, cyclotomic rational Cherednik algebras) categorifies certain structures in the representation theory of affine Lie algebras (namely, decompositions of the restriction of the basic representation to finite dimensional and affine subalgebras). These conjectures arose from the insight due to R. Bezrukavnikov and A. Okounkov on the link between quantum connections for Hilbert schemes of resolutions of Kleinian singularities and representations of symplectic reflection algebras. Some of these conjectures were recently proved in the works of Shan-Vasserot and Gordon-Losev.

G. Fourier (Cologne)

Lecture Notes

Weyl modules and subalgebras

We fix a simple finite-dimensional complex Lie algebra \g and a simple Lie subalgebra \a of \g induced by a closed subset of positive roots. If one considers the restriction of a given finite-dimensional simple \g-module to \a, then the component of the highest weight vector is a simple \a-module. If we generalize this to (generalized) current algebras, we obtain an analog picture. What happens to the restriction of local and global Weyl modules to the highest weight vector component?

Here the answer depends on \g, \a and the highest weight of the Weyl module, but we will give necessary and sufficient conditions such that this component is again a local or global Weyl module.

Equivariant K-theory and affine algebras

In a recent series of papers, Freed-Hopkins-Teleman have established a deep connection between twisted equivariant K-theory and integrable modules of affine algebras. In work with David Evans, we're showing that this is just one (central) spot of an interconnected web of K-groups and KK-groups naturally associated to affine algebras and their corresponding conformal field theories. My talk will be an overview of this work.

Various representations for a class of multiloop Lie algebras coordinated by a quantum torus $C_Q$ where $Q (q_{ij})$ is an n by n matrix have been studied with a limitation that $q_{ij}=1$ for $2\leq i, j\leq n$. In this talk, we will construct a representation by using Wakimoto's idea without the limitation.

It is a report on our running joint work with V. Chernousov and A. Pianzola. Let R be the Laurent polynomial ring in n variables over a field k of characteristic 0. We are interested in loop semisimple simply connected groups G/R, that is a certain kind of reductive group schemes related to (multi)-loop Lie algebras. When such a group is isotropic, we shall investigate the question of its generation by one parameter additive subgroups, with special attention to the nullity one case.

The first part of my talk will be devoted to an overview of recent developments about Yangians and their connections to other interesting mathematical objects. Afterwards, it will be explained how to construct a coproduct on a certain completion of the Yangian attached to an affine Kac-Moody algebra. Vertex operator representations of Yangians will also be introduced: they are the analogs of the vertex operator representations of quantum affine algebras due to I. Frenkel and N. Jing. One application of these representations to the structure of Yangians will be presented.

Affine MV polytopes and preprojective algebras

MV polytopes give a combinatorial model for representation theory of semisimple Lie algebras. They were originally defined in the finite type case using MV cycles in the affine Grassmannian. I will explain how they can be generalized to affine type using generic modules for preprojective algebras. This is joint work with Pierre Baumann and Peter Tingley.

Let g be a finite-dimensional Lie algebra. The dual g^* of g has a well-known (linear) Poisson structure and a symplectic foliation into coadjoint orbits. In the case of g=gl(n,C), Kostant and Wallach have constructed a completely integrable system on each regular coadjoint orbit. This is a geometric analogue of the classical Gelfand-Tsetlin bases for irreducible representations.

In the context of the (infinite dimensional) direct limit algebra gl(infinity), the situation is somewhat more delicate. Nonetheless, there is still a beautiful Lie-Poisson structure, symplectic foliation, and a polarisation given by Gelfand-Tsetlin systems. I will discuss some of this geometry, based on joint work with Mark Colarusso.

The notion of automorphic correction of a Lie algebra was originated from Borcherds's work on Monster Lie algebras. In this talk we consider rank two symmetric hyperbolic Kac-Moody algebras H(a) and their automorphic correction in terms of Hilbert modular forms. We associate a family of H(a)'s to the quadratic field Q(\sqrt p) for each odd prime p and show that there exists a chain of embeddings in each family. When p=5, 13, 17, we show that the first H(a) in each family is contained in a generalized Kac-Moody superalgebra whose denominator function is a Hilbert modular form given by a Borcherds product.

This is a joint work with Henry Kim.

Set $\Lambda={\bf Z}^N$.

We consider $\Lambda={\bf Z}^N$-graded Lie algebras ${\cal L}=\oplus_{\lambda\in\Lambda}\,{\cal L}_{\lambda}$ such that each homogenous component ${\cal L}_{\lambda}$ is one dimensional. Moreover, we assume that ${\cal L}$ is graded simple.

Around 1978, V.G. Kac conjectured the classification of such Lie algebras in the case $N=1$ (proved by the author in 1983). In the 90's, I.M. Gelfand and A. Kirillov raise the question of the classification of such Lie algebras in the case $N\geq 2$, but without providing an explicit list.

In this talk, we will explain how to solve Gelfand-Kirillov question. Although the setting of the question is very abstract, it turns out that the Lie algebras occuring in the classification are very concrete. Most of them are connected with the Lie algebras of symbols of twisted PDO on the circle. The remaining Lie algebras are classified by using Jordan algebras theory.

The proofs appear in two long papers, in Proc. London Math. Soc. and Math. Z.

J. Morita (Tsukuba)

Lecture Notes

A recent topic including a joint work with Bertrand Rémy will be shown. We will deal with not only the simplicity of some Kac-Moody group, but also its universal covering. Recall that a universal Kac-Moody group is obtained by the so-called Tits group functor associated with a certain realization of a given generalized Cartan matrix. Furthermore, we would like to discuss the corresponding Schur multiplier.

Let L be the Lie algebra of smooth sections of a Lie algebra bundle over the manifold X whose typical fibre is a compact semisimple Lie algebra. We endow L with its natural Frechet topology and the ideal L_c of compactly supported smooth sections with its natural direct limit LF-topology . A bounded unitary representation is a continuous homomorphism into the Lie algebra u(H) of bounded skew-hermitian operators on a Hilbert space H.

In this talk we describe a classification of all bounded irreducible unitary representations of L and L_c. Due to the rather coarse topology, for L, the result is rather simple: the irreducible bounded representations are the finite tensor products of evaluation representations (compositions of a representation of a fiber with an evaluation in a point x). For the Lie algebra L_c also infinite tensor products occur, and this lead to a bounded representation theory that is ``wild'' in the sense that factor representations of type II and III occur.

If $\Gamma$ is a finite group acting on the compact Lie algebra K and freely on the smooth manifold Y and accordingly on the algebra A of smooth functions on Y, then the corresponding equivariant map algebra (K\otimes A)^$\Gamma$ coincides with the Lie algebra of smooth sections of a Lie algebra bundle over the orbit space.

This is joint work with Bas Janssens.

Starting from a special type of function on a finite dimensional group called a cusp form, we define an object on the corresponding loop group which depends on one complex variable. This object, the cuspidal loop Eisenstein series, can then be shown to be entire on the complex plane, which is a phenomenon quite unusual from the point of view of finite-dimensional automorphic forms. We explain how to deduce this result from two ingredients: (a) inequalities between the classical and central "directions" of elements in a certain discrete family in a loop symmetric space; and (b) a strengthening of the usual rapid-decay statements for cusp forms on finite-dimensional groups.

This is joint work with H. Garland and S.D. Miller

Let G be a classical Q-isotropic algebraic group and G(A) be the group of adele-points of G. In the 1980's Roger Howe defined a notion of rank for irreducible unitary representations of G(A) and its local components G(R) and G(Q_p). Among many other results, he proved that for an automorphic representation of G(A) all of these ranks are equal. The latter technique has found a number of applications, namely in the study of multiplicities of automorphic forms, Howe duality, etc. In this talk we extend the rigidity result of Howe in a uniform and conceptual way to include exceptional G. Our approach is based on the orbit method for nilpotent (real and p-adic) Lie groups. In the real case, one needs functional calculus on Lie groups, and in the p-aid case one needs to analyze representations of certain Hecke algebras of bi-invariant functions. As a special case of our result, we obtain a new proof of the following theorem due to Kazhdan (and Gan and Savin): if one local component of a unitary representation of G(A) is "minimal", then all of its local components are "minimal".

A. Savage (Ottawa)
Lecture Notes

Motivated by geometric realizations of (degenerate) affine Hecke algebras via convolution products on the equivariant K-theory (or homology) of the Steinberg variety, we define a "formal (affine) Hecke algebra" associated to any formal group law. Formal group laws are associated to algebraic oriented cohomology theories. When specialized to the formal group laws corresponding to K-theory and (co)homology, our definition recovers the usual affine and degenerate affine Hecke algebras. However, other formal group laws (such as those corresponding to elliptic and cobordism cohomology theories) give rise to apparently new algebras with interesting properties. This is joint work with Alex Hoffnung, Jose Malagon-Lopez, and Kirill Zainoulline.

There are four series of classical Lie superalgebras: sl, osp, P and Q. In this talk I consider the direct limits of these superalgebras and study their representations in the tensor algebra generated by the standard and costandard representation.

We will see that complications related to the lack of complete reducibility for finite-dimensional superalgebras disappear at infinity. I define an abelian category of tensor modules for those superalgebras and discuss its properties. In particular, we will see that all categories in question are Koszul and extensions between simple modules can be described in terms of Littlewood--Richardon coefficients. As an example, we interpret Howe duality between orthogonal and symplectic groups at infinity in terms of the Lie superalgebra osp.

Locally Affine Lie Algebras

We introduce a local version of affine Lie algebras, called a locally affine Lie algebra. A certain ideal, called the core, of such a Lie algebra is a directed union of loop algebras. We explain the classification and some isomorphisms of minimal locally affine Lie algebras.