Seminars will be held at the Fields Institute
How to integrate the quantum group over a surface
I'll explain joint work with D. Ben-Zvi and A. Brochier, in which
we construct a partially defined 4D ``Quantum Geometric Langlands"
TFT from the quantum group U_q(g) associated to a reductive group
G, and develop completely explicit computations of the QGL theory
of arbitrary surfaces, using factorization homology of Francis. For
once-punctured tori we recover algebras of quantum differential operators
on G. For closed tori and G=GL_N, we expect to recover the double
affine Hecke algebra associated to G.
The QGL theory of 3-manifolds yields knot invariants valued in modules
for the DAHA, and hence is closely related to several recently conjectured
knot invariants of Cherednik, Gukov, Oblomkov-Rasmussen-Shende, and
Casselman-Shalika Formula for Loop Groups
The usual Casselman-Shalika formula relates the Weyl character to
unramified Whittaker functions on the p-adic points of a reductive
group. We will explain how to define Whittaker functions on a p-adic
loop group (i.e, the group of p-adic points of an affine Kac-Moody
group), present the analogue of the Casselman-Shalika formula in this
context, and then sketch its proof.
Based on joint work with A. Braverman, H. Garland, and D. Kazhdan.
|March 20, 3pm Stewart Library
Tensor isomorphism between Yangians and quantum loop algebras
The Yangian and the quantum loop algebra of a simple Lie algebra
g arise naturally in the study of the rational and trigonometric solutions
of the Yang--Baxter equation, respectively. These algebras are deformations
of the current algebra g[s] and the loop algebra g[z,z^(-1)] respectively.
The aim of this talk is to establish an explicit relation between
the finite--dimensional representation categories of these algebras,
as meromorphic braided tensor
categories. The notion of meromorphic tensor categories was introduced
by Y. Soibelman and finite--dimensional representations of Yangians
and quantum loop algebras are among the first non--trivial examples
The isomorphism between these two categories is governed by the monodromy
of an abelian difference equation. Moreover, the twist relating the
tensor products is a solution of an abelian version of the qKZ equations
of Frenkel and Reshetikhin.
These results are part of an ongoing project, joint with V. Toledano
Derived equivalences of hypertoric varieties
Given a representation of a complex torus, there is a natural hyperplane
arrangement A whose faces correspond to the stability conditions defining
the hypertoric varieties associated to this representation. We describe
ongoing work on constructing a representation of the Deligne groupoid
of A on the derived categories of the hypertoric varieties corresponding
to the chambers of A.
Cohomology rings of quiver moduli from the Harder-Narasimhan stratification
Let Q be a quiver with dimension vector d. The Harder-Narasimhan
filtration on the category Rep(Q) gives an algebraic stratification
of the vector space Rep(Q,d) of representations of fixed dimension.
When the ground field is C, this stratification agrees with the Morse
stratification coming from the norm-square of a moment map. I will
give an explicit description of this stratification, and show that
it leads to an inductive procedure for computing the cohomology ring
of the moduli space of stable representations of Q. I will give a
few worked examples, and time-permitting, I will describe some applications
to Nakajima quiver varieties.
Type A quiver loci and Schubert varieties
I'll describe a closed immersion from each representation space of
a type A quiver with bipartite (i.e., alternating) orientation to
a certain opposite Schubert cell of a partial flag variety. This "bipartite
Zelevinsky map" restricts to an isomorphism from each orbit closure
to a Schubert variety intersected with the above-mentioned opposite
Schubert cell. For type A quivers of arbitrary orientation, I'll discuss
a similar result up to some factors of general linear groups.
These identifications allow us to recover results of Bobinski and
Zwara; namely we see that orbit closures of type A quivers are normal,
Cohen-Macaulay, and have rational singularities. We also see that
each representation space of a type A quiver admits a Frobenius splitting
for which all of its orbit closures are compatibly Frobenius split.
This work is joint with Ryan Kinser.
Yangians and the affine Grassmanian
We will discuss ongoing research linking Yangians and some subvarieties
of the affine Grassmannian, namely transverse slices to Schubert varieties.
The geometry of these spaces hints that the representation theory
of the quantizations should be very interesting, and tied to the representation
theory of simple Lie algebras. We will present a summary of our work
so far on these links, highlighting the case of SL(2, C).
Cycles of Algebraic D-modules in positive characteristic
I will explain some ongoing work on understanding algebraic D-moldules
via their reduction to positive characteristic. I will define the
p-cycle of an algebraic D-module, explain the general results of Bitoun
and Van Den Bergh; and then discuss a new construction of a class
of algebraic D-modules with prescribed p-cycle.
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 (2008).
In this talk, I will discuss analogues (in fact, generalizations)
of strong Macdonald conjectures arising from homology of derived representations
Double Mirkovi\'c-Vilonen Cycles and the Naito-Sagaki-Saito Crystal
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.