Speaker & Affiliation 
Title and Abstract

G. Benkart
(Wisconsin)

Affine Dynkin Diagrams Reviewed
The simplylaced 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.

S. Cautis
(Southern California) 
Quantum affine algebra actions on categories
A quantum affine algebra has two presentations: the KacMoody
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 finitedimensional 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 finitedimensional representation V is prime iff the space
of selfextensions Ext(V,V) is onedimensional. We shall also see
that this feature is quite unusual and has no counterpart in the case
of the affine Lie algebra for instance.

I. Dimitrov (Queens) 
Lagrangian subalgebras of classical Lie superalgebras
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.

B. Elias (MIT) 
The Soergel conjecture: a proof and a counterexample
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 KazhdanLusztig 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 BottSamelson
bimodules, which are the equivariant cohomology of BottSamelson 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 BottSamelson bimodules should split into summands as expected.
Inspired by de Cataldo and Migliorini's Hodgetheoretic 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 Hodgetheoretic properties expected
of an equivariant intersection cohomology space. This is joint work
with Geordie Williamson.
If time permits, we will advertise a counterexample to the original
version of Soergel's conjecture: a quantized version of the geometric
Satake equivalence.

P. Etingof
(MIT) 
Symplectic Rreflection Algebras and Affine
Lie Algebras
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 ShanVasserot and GordonLosev.

G. Fourier (Cologne)
Lecture Notes

Weyl modules and subalgebras
We fix a simple finitedimensional 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 finitedimensional simple
\gmodule to \a, then the component of the highest weight vector is
a simple \amodule. 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.

T. Gannon (Alberta) 
Equivariant Ktheory and affine algebras
In a recent series of papers, FreedHopkinsTeleman have established
a deep connection between twisted equivariant Ktheory 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 Kgroups
and KKgroups naturally associated to affine algebras and their corresponding
conformal field theories. My talk will be an overview of this work.

Y. Gao (York) 
Representation for a class of multiloop
Lie algebras
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.

P. Gille
(Paris) 
The group of points of loop group schemes
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.

N. Guay (Alberta) 
Yangians for affine KacMoody algebras
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 KacMoody 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.

J. Kamnitzer
(Toronto) 
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.

M. Lau (Laval) 
LiePoisson structures for gl(infinity)
Let g be a finitedimensional Lie algebra. The dual g^* of g has
a wellknown (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 GelfandTsetlin
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 LiePoisson structure, symplectic foliation,
and a polarisation given by GelfandTsetlin systems. I will discuss
some of this geometry, based on joint work with Mark Colarusso.

KH. Lee
(Connecticut) 
Rank 2 symmetric hyperbolic KacMoody algebras
and Hilbert modular forms
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 KacMoody 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 KacMoody superalgebra whose denominator function
is a Hilbert modular form given by a Borcherds product.
This is a joint work with Henry Kim.

O. Mathieu (Lyon) 
On the Classification of ${\bf Z}^N$graded Lie Algebras
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 GelfandKirillov 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 note on the simplicity and the universal covering of
some KacMoody group
A recent topic including a joint work with Bertrand Rémy will
be shown. We will deal with not only the simplicity of some KacMoody
group, but also its universal covering. Recall that a universal KacMoody
group is obtained by the socalled Tits group functor associated with
a certain realization of a given generalized Cartan matrix. Furthermore,
we would like to discuss the corresponding Schur multiplier.

KarlH. Neeb (Erlangen) 
Bounded Unitary Representations of Lie Algebras of Smooth
Sections
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 LFtopology . A bounded unitary representation is a continuous
homomorphism into the Lie algebra u(H) of bounded skewhermitian 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.

M. Patnaik
(Alberta) 
Entirety of Cuspidal Eisenstein Series on
Loop Groups
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 finitedimensional 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 rapiddecay statements for cusp forms on finitedimensional
groups.
This is joint work with H. Garland and S.D. Miller

H. Salmasian
(Ottawa) 
A rigidity property of the discrete spectrum
of adele groups
Let G be a classical Qisotropic algebraic group and G(A) be the
group of adelepoints 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 padic) Lie groups. In the
real case, one needs functional calculus on Lie groups, and in the
paid case one needs to analyze representations of certain Hecke algebras
of biinvariant 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

Hecke Algebras and Formal Group Laws
Motivated by geometric realizations of (degenerate) affine Hecke
algebras via convolution products on the equivariant Ktheory (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 Ktheory 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 MalagonLopez, and Kirill Zainoulline.

V. Serganova
(Berkeley) 
Tensor representations
of classical Lie superalgebras at infinity
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 finitedimensional 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 LittlewoodRichardon coefficients. As an example,
we interpret Howe duality between orthogonal and symplectic groups
at infinity in terms of the Lie superalgebra osp.

Y. Yoshii (Akita National
College of Technology)
Lecture Notes

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.
