Seminars will be held at the Fields Institute
Webs and quantum skew Howe duality
We give a diagrammatic presentation of the representation
category of SLn (and its quantum version). Our main tool
is an application of quantum skew Howe duality.
The mirabolic Hecke algebra
The Iwahori-Hecke algebra of the symmetric group is the
convolution algebra arising from the variety of pairs of
complete flags over a finite field. Considering convolution
on the space of triples of two flags and a vector we obtain
the mirabolic Hecke algebra, which had originally been described
by Solomon. We will see a new presentation of this algebra
which shows that it is a quotient of a cyclotomic Hecke
algebra. This lets us recover Siegel's results about its
representations, as well as proving new 'mirabolic' analogues
of classical results about the Iwahori-Hecke algebra.
Generalizing Kashiwara's equivalence to conic quantized symplectic
Kashiwara's equivalence, saying that the category of D-modules
on a variety X supported on a smooth, closed subvariety
Y is equivalent to the category of D-modules on Y, is a
key result in the theory of D-modules. In this talk I will
explain how one can generalize Kashiwara's result to modules
for deformation-quantization algebras on a conic symplectic
manifold. As an illustrative application, one can use this
result to calculate the additive invariants such as the
K-theory and Hochschild homology of these module categories.
This is based on joint work C. Dodd, K. McGerty and T. Nevins.
Quantization in positive characteristic and Langlands duality
I will first explain a version of geometric Langlands correspondence
in positive characteristic for a reductive group. Then I
will explain how to use the method of quantization in positive
characteristic to construct generic parts of the correspondence.
(This is a joint work with Xinwen Zhu.)
Double affine Hecke algebras and Jones polynomials
For a (reductive) group G, let O(K; G) be the ring of functions
on the variety of G-representations of the fundamental group
of the complement of a knot K in S^3. There is an algebra
map from the spherical double affine Hecke algebra H^+(G;
q=1,t=1) to O(K; G), which leads to the question "does
O(K) deform to a module over H^+(G; q,t)?" We give
a conjectural positive answer for G=SL_2(C) and discuss
some corollaries of this conjecture involving the SL_2(C)
Jones polynomials of K. (This is joint work with Yuri Berest.)
Representation homology and strong Macdonald conjectures
In the early 1980s, I. Macdonald discovered a number of
highly non-trivial combinatorial identities related to a
semisimplecomplex Lie algebra g. These identities were under
intensive study for a decade until they were proved by I.Cherednik
using representation theory of his double affine Hecke algebras.One
of the key identities in the Macdonald list - the so-called
constant term identity - has a natural homological interpretation:
it formally follows from the fact that the Lie algebra cohomology
of a truncated current Lie algebra over g is a free exterior
algebra with generators of prescribed degree (depending
on g). This last fact (called the strong Macdonal conjecture)
was proposed by P.Hanlon and B.Feigin in the 80s and proved
only recently by S. Fishel, I.Grojnowski and C. Teleman
In this talk, I will discuss analogues (in fact, generalizations)
of strong Macdonald conjectures arising from homology of
derived representations schemes.
Double Mirkovi\'c-Vilonen Cycles and the Naito-Sagaki-Saito
The theory of Mirkov\'c-Vilonen (MV) cycles associated
to a complex reductive group $G$ has proven to be a rich
source of structures related to representation theory. I
investigate double MV cycles, which are analogues of MV
cycles in the case of an affine Kac-Moody group.
I will shortly review some aspects of the theory of MV
cycles for finite-dimensional groups. The story gives rise
to MV polytopes and a surprising connection with Lusztig's
canonical basis. Then I will discuss double MV cycles. Here
the finite-dimensional story does not naively generalize.
Nonetheless, in type A, I will present a method to parameterize
double MV cycles. This method gives rise to exactly the
combinatorics of the Naito-Sagaki-Saito crystal. If I have
time, I will discuss some related work and some open problems.
Omar Ortiz Branco
On the torus-equivariant cohomology of p-compact flag varieties
p-compact groups are the homotopy analogues of compact
Lie groups. The torus-equivariant cohomology of p-compact
flag varieties can be described as a quotient ring of polynomials.
I will give another description of this cohomology via moment
graph theory and its relation with the polynomial description,
generalizing results of Goresky-Kottwitz-MacPherson from
classical Schubert calculus.
Cactus group and monodromy of Bethe vectors
The cactus group is the fundamental group of the real locus
of the Deligne-Mumford moduli space of stable rational curves.
This group appears naturally as an analog of braid group
in coboundary monoidal categories; the main example of this
is the category of crystals where the cactus group acts
on tensor product of crystals by crystal commutors. We define
an action of the cactus group on the set of Bethe vectors
of the Gaudin magnet chain (for Lie algebra sl_2) and prove
that this action is isomorphic to the action of cactus group
on the tensor product of sl_2-crystals. We also relate this
to the Berenstein-Kirillov group of piecewise linear transformations
of the Gelfand-Tsetlin polytope. Some conjectures generalizing
our construction will be discussed.
K-theoretic AGT conjecture
I will explain the AGT conjecture and its K-theoretic analogue.
The conjecture states (deformed) W-algebras should act on
the equivariant cohomology groups (or K-theory) of instanton
moduli spaces over CP^2. If have enough time, I will also
explain combinatorical conjectures on the Whittaker vectors
of deformed W-algebras.
Quantum loop algebras and the conjectural monoidal categorification
of cluster algebras
Beyond the case of sl_2, the tensor structure of the category
of finite-dimensional representations of a quantum loop
algebra is still not very well-understood. For instance,
we do not have a general factorization theorem (in terms
of prime objects) for simple objects. We will explain a
conjecture by Hernandez and Leclerc relating this tensor
structure to the combinatorics in some cluster algebra of
geometric type. Special cases of this conjecture are proved
by Hernandez-Leclerc and Nakajima.