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Roman Bezrukavnikov
Coherent sheaves on the Hilbert scheme and perverse sheaves on spaces of stable pairs
I will report on joint works in progress with Andrei Okounkov and Galyna Dobrovolska. The former leads
to t-structures on the derived category of coherent sheaves on the Hilbert scheme of points on the plane,
related to modular representations of the rational Cherednik algebra, quantum cohomology etc. The latter us devoted to
realizing the hearts of these t-structures as full subcategories in the category of perverse sheaves on the space of stable
pairs on an elliptic curve.
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Philip Boalch
Non-perturbative symplectic geometry and non-commutative algebras
Just as a Lie group is a multiplicative version of a Lie algebra,
there is a class of holomorphic symplectic varieties which are precise
multiplicative versions of simpler ("additive") symplectic varieties.
In this theory the role of the exponential map is played by the
Riemann-Hilbert map, or its extension to the irregular case.
I'll describe some examples, and show how a new theory of
multiplicative quiver varieties emerges.
Then I will recall how certain (multiplicative) symplectic quotients
lead to noncommutative algebras and thereby construct the fission
algebras (which are new multiplicative analogues of deformed
preprojective algebras).
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Alexander Braverman
Symplectic duality, the Coulomb branch and the affine Grassmannian
(joint work in progress with M.Finkelberg and H.Nakajima)
To a (reductive) group G and a symplectic representation M of G, physicists
associate an (N=4) super-symmetric gauge theory in 3 dimensions. Any
such theory
is supposed to give rise to two (affine) "singular symplectic" moduli
spaces - the Higgs branch and the Coulomb branch. These spaces are
related by a certain mysterious "symplectic duality" (also called Gale
duality) studied in mathematical literature by many authors.
While the Higgs branch is easy to define (usually, it is just the
symplectic reduction of M by G), the Coulomb branch has no
mathematically rigorous definition.The purpose of the talk will be to
suggest such a definition in the case when M is of the form V\( \oplus\) V*
for some representation V of G, and to show that in many examples it
gives the right answer. The suggestion is based on the geometry of the
affine Grassmannian of G.
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Adrien Brochier
Topological field theories and quantum D-modules
The goal of this talk is to describe a topological construction of a
certain quantum version of the category of D-modules on a reductive
algebraic group G, and of its equivariant version. The latter is, at
least conjecturally, closely related to the category of modules over the
double affine Hecke algebra. More precisely, these categories are
obtained as the value of a certain (partially defined) 4-dimensional
topological field theory (TFT) on the punctured and the closed torus
respectively. This TFT is constructed from the braided tensor category
of modules over the quantum group of G. More generalluy, our main result
is an explicit construction, in the framework of factorization homology,
of the 2-dimensional part of a 4-dimensional TFT constructed from any
ribbon tensor category. Time permitting I will discuss some applications
of this formalism to quantization of character varieties, Witten-Reshetikhin-Turaev theory,
and the so-called AJ conjecture relating the coloured Jones polynomial and the A-polynomial of a link.
This is a joint work with David Ben-Zvi and David jordan.
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Olivier Dudas
Projective modules in the cohomology of Deligne-Lusztig varieties
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Pavel Etingof
Non-integral tensor powers
Let \(t\) be a complex number, and \(V\) a complex vector space. I will explain how to define
the tensor product \(V^{\otimes t}\). This can be done canonically if we fix a nonzero vector in \(V\).
However, the result is not a vector space but rather an (ind-)object in the tensor category \({\rm Rep}(S_t)\), defined
by P. Deligne as an interpolation of the representation category of the symmetric group \(S_n\) to complex values of \(n\).
This category is semisimple abelian for \(t\notin\mathbf{Z}_+\), but only Karoubian (=idempotent complete) for \(t\in\mathbf{Z}_+\),
in which case it projects onto the usual representation category of \(S_n\). I will recall the definition of the category
\({\rm Rep}(S_t)\), and explain how Schur-Weyl duality works in this category when \(t\notin\mathbf{Z}_+\). Then I will explain
what happens at nonnegative integer \(t\), which is more subtle and is due to Inna Entova-Aizenbud, and indicate possible
applications.
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Stephen Griffeth
Eigenvalues of monodromy for parabolic restrictions of Cherednik algebra modules
This is a report on joint work with Armin Gusenbauer, Daniel Juteau, and Martina Lanini. Bezrukavnikov-Etingof discovered that
modules in category O for the rational Cherednik algebra of a complex reflection group restrict to D-modules on each stratum of
the reflection representation of the group. Category O is a quasi-hereditary cover of the category of representations of the finite
Hecke algebra, with maps between standard modules delimited by a well-known ordering usually referred to as the "c-ordering".
In certain cases, such as for the symmetric group, a coarser order combining dominance order and cores of partitions is known to
wotk, giving much more detailed information about the possible maps between standard objects. We explain how the calculation
of the residues of the connections studied by Bezrukavnikov-Etingof may be used to coarsen the c-ordering for arbitrary groups in
such a way that one obtains versions of dominance order and cores for all reflection groups.
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Bogdan Ion
BGG reciprocity for current algebras
Current algebras are special maximal parabolic subalgebras of
affine Lie algebras. In the case of untwisted affine Lie algebras
they are isomorphic to the tensor product of a finite dimensional
simple Lie algebra and the ring pf polynomials in one variable. The
category of finite dimensional representations of current algebras
is not semisimple. In 2011 Chari et al. have conjectured a version
of the BGG reciprocity in this context, which connects simple finite
dimensional representations, their projective covers, and standard
modules. I will present a proof of this conjecture based on a
connection with Macdonald theory (joint work with V. Chari).
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Joel Kamnitzer
K-theoretic quantum geometric Satake
The geometric Satake corresponds relates morphism spaces between
representations to homology of ``Steinberg-like'' varieties
constructed using the affine Grassmannian. We formulate a quantum
group version of geometric Satake, where homology is replaced with
equivariant K-theory. We prove this conjecture in type A.
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Volodymyr Mazorchuk
Simple fiat 2-categories and the center
The aim of this talk is to explain classification
of simple fiat 2-categories and describe some
parts of the 2-representation theory for such
categories. In particular, we will try to
explain why, from this point of view, it is
difficult to derive information on surjectivity
of the action of the center.
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Ivan Mirkovic
Local spaces and loop Grassmannians
The notion of a "local space" over a curve is a version of the
fundamental structure of a factorization space introduced and
studied by Beilinson and Drinfeld. The change of emphases leads to
new constructions. The main example will be generalizations of loop
Grassmannians.
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Alexei Oblomkov
Cohomology of the homogeneous Hitchin fibers and representation theory of Cherednik algebras
The talk is based on the joint work with Z. Yun.
I will explain a geometric construction of an action of the rational Cherednik algebra on the cohomology of the homogeneous
Hitchin fiber for the Hitchin system on the weighted projective line. The perverse and cohomological fibrations on the cohomology
have a natural representation theoretic interpretation which will be explained. In type \(A\) case, the ring structure of the cohomology
will be presented.
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Gerhard Röhrle
Equivariant K-theory of generalized Steinberg varieties
We discuss the equivariant K-groups of a family of generalized
Steinberg varieties that interpolates between the Steinberg variety of a
reductive, complex algebraic group and its nilpotent cone in terms of the extended
affine Hecke algebra and double cosets in the extended affine Weyl group.
As an application, we use this description to define Kazhdan-Lusztig "bar" involutions
and Kazhdan-Lusztig bases for these equivariant K-groups.
This is a report on joint work with J.M. Douglass.
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Olivier Schiffmann
Cohomology of the moduli spaces of stable Higgs bundles
We give a closed formula for the number of absolutely indecomposable vector bundles of fixed rank and degree, over a smooth projective curve
defined over a finite field. We then show that this same number computes the Poincare polynomial of the moduli space of stable Higgs bundles
(of same rank and degree) over the same curve. We propose analogues, in this context, of the conjectures of Kac in the setting of quivers and
Kac-Moody algebras.
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Mark Shimozono
Conjectures on the Quantum K-theory of flag varieties and K-theory of affine Grassmannians
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Catherina Stroppel
Coideal subalgebras, categorification and finite dimensional representations
Given a reductive Lie algebra and an involution I, the fixed point Lie algebra is a reductive subalgebra imbedded into the original Lie algebra.
Unfortunately this does not lift to the quantum level. However, there is a coideal subalgebra which can be viewed as a quantization of the fixed point Lie algebra. In this talk we will study a particular example of such a construction, address the problem of finite dimensional representations and finally describe a categorification.
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Lewis Topley
Small modules for reduced enveloping algebras
The dimensions of modules for a reduced enveloping
algebras of a Lie algebra of a reductive group over a field of
positive characteristic are known to be bounded below by a number
depending upon the dimension of the coadjoint orbit of the chosen
p-character. In the past year this lower bound has been shown to be
best possible, provided the characteristic of the underlying field
is large enough. In this talk I shall describe a work in progress
which attempts to construct the modules which achieve this lower
bound as induced modules.
