July 28 - Aug.1 Conference
Schedule
Noriyuki Abe (Hokkaido university)
A classification of irreducible admissible modulo $p$ representations of
reductive $p$-adic groups
- Abstract:
We describe a classification of irreducible admissible modulo $p$
representations (representations over a field of characteristic $p$) of
a reductive $p$-adic group in terms of supercuspidal representations.
This generalizes the result of Barthel-Livne for $\mathrm{GL}(2)$,
Herzig for $\mathrm{GL}(n)$ and my previous work for split groups. This
is a joint work with G. Henniart, F. Herzig and M.-F. Vigneras.
Pramod N Achar (Louisiana State University)
The affine Grassmannian and the Springer resolution in positive
characteristic
- Abstract:
An important 2004 theorem of Arkhipov-Bezrukavnikov-Ginzburg asserts that
the derived category of constructible sheaves on the affine Grassmannian
(for the Schubert stratification) is equivalent to the derived category of
equivariant coherent sheaves on the Springer resolution for the Langlands
dual group over C. This equivalence is compatible with (and contains a copy
of) the geometric Satake equivalence. Recent advances have now made it
possible to prove an ABG equivalence with coefficients in a field of
positive characteristic. Key ingredients in the proof include the new
theory of "mixed modular derived categories" (joint work with S. Riche) and
the Mirkovic-Vilonen conjecture. This is joint work with L. Rider.
Roman Bezrukavnikov (MIT, HSE International Laboratory of Representation Theory and Mathematical Physics)
Representations as complexes of coherent sheaves and dualities
- Abstract:
In several examples a derived category of coherent sheaves is
known
out to be equivalent to a derived category of an abelian category of
representations.
The corresponding abelian subcategory turns out to have favorable properties
in relation to dualities, such as geometric Langlands and homological mirror
duality. I will describe some results and conjectures illustrating this
phenomenon.
Tom Braden (University of Massachusetts, Amherst)
Ringel duality and perverse sheaves on hypertoric varieties
(Joint work with Carl Mautner)
- Abstract:
Affine hypertoric varieties have many properties similar to
singularities of nilpotent cones and affine Grassmannian Schubert
varieties. In particular, they can be stratified with even-dimensional
strata, and the category of perverse sheaves constructible with respect
to this stratification is semisimple in characteristic zero. However,
with positive characteristic coefficients this category can be quite
complicated; we show that it is highest weight, and that it is Ringel dual
to perverse sheaves on another hypertoric variety, related to the
first by Gale duality. This means that the endomorphisms of a
projective generator on one variety is isomorphic to the endomorphisms
of a tilting generator on the other. This is analogous to results for
the nilcone of $GL(n)$ proved by Mautner and Achar-Mautner. Our result
is obtained by means of explicit combinatorial descriptions of the
category of perverse sheaves and its tilting objects.
Alexander Braverman
Kazhdan-Lusztig conjecture via quasi-maps and Uhlenbeck spaces
Ivan Cherednik (University of North Carolina at Chapel Hill, RIMS)
A surprising application of Nil-DAHA to the PBW-filtration
- Abstract:
A fundamental but difficult question in the representation
theory is counting the minimal number of $f$-operators for
all positive roots (not only for simple ones) needed to reach
any vector from the highest vector. E. Feigin, G. Fourier
and P. Littlemann constructed the corresponding abstract
PBW-basis for the Lie algebras of types A,C. A surprising
conjecture due to the speaker, D. Orr and E. Feigin connects
the count of PBW-degrees in Demazure level-one modules
with the degeneration of the nonsymmetric(!) Macdonald
polynomials at $t=\infty$. This resembles the connection of
the BK-filtrartion with the Hall-Littlewood polynomials ($q=0$);
both filtrations are related to the Kostant $q$-partition functions,
but in very different ways. The conjecture was justified by the
speaker and E. Feigin for extremal vectors in finite-dimensional
irreducible representations (the top part of the corresponding
Demazure module) for classical Lie algebras and G2. These
and some latest developments will be discussed in the talk.
Peter Fiebig (Friedrich-Alexander-Universität Erlangen-Nürnberg)
Periodic Patterns in the representation theory of affine Kac-Moody algebras
- Abstract:
We will study a certain subcategory of the category $O$
for an affine Kac-Moody algebras (in positive level) that is governed
by periodic polynomials. It can be interpreted as a characteristic
zero analogue of the category of $G_1T$-modules (for big enough
characteristics) and hopefully it helps us to understand the critical
level category $O$, which should, by a conjecture of Feigin, Frenkel and
Lusztig, be governed by periodic polynomials as well. This is joint
work with Martina Lanini.
Michael Finkelberg (State University Higher School of Economics)
Twisted Whittaker sheaves on zastava (after Gaitsgory, Gaiotto and Witten)
- Abstract:
D.Gaitsgory discovered a localization of $U_q(\check{\mathfrak g})$-modules as twisted Whittaker D-modules on the Drinfeld compactification of $Bun_B(C)$. We (together with A.Braverman and G.Dobrovolska) describe these D-modules explicitly on the zastava space $Z_G(C)$ and compare with the Gaiotto-Witten superpotential on the space of $G$-monopoles.
Victor Ginzburg (Univ. of Chicago)
Counting indecomposables over a finite field
- Abstract:
We develop an alternative approach to the theorem
of Hausel, Letellier, and Rodriguez-Villages
on counting indecomposable quiver representations.
Our approach is based on character sheaves and it
does not involve combinatorial tools.
A similar approach applies for counting indecomposable
vector bundles on an algebraic curve in terms of the
geometry of Higgs bundles. There seems to be a connection
with a recent work of Deligne on counting irreducible
local systems.
Xuhua He (The Hong Kong Univ. of Science and Technology)
Cocenters and representations of affine Hecke algebras
- Abstract:
It is known that the number of conjugacy classes of a finite
group equals the number of irreducible representations (over complex
numbers). The conjugacy classes of a finite group give a natural basis
of the cocenter of its group algebra. Thus the above equality can be
refomulated as a duality between the cocenter of the group algebra and
the Grothendieck group of its finite dimensional representations.
For affine Hecke algebras, the situtation is much more complicated.
First, the cocenter of affine Hecke algebras is harder to understand
than the cocenter of group algebras. Second, for an affine Hecke
algebra, the dimension of its cocenter is countablly infinite and the
number of irreducible representations is uncountablly infinite. However,
the ``cocenter-representation duality'' is still valid. This is what I
am going to explain in this talk. It is based joint works with S. Nie,
and joint work with D. Ciubotaru.
If time allows, I will also mention affine Hecke algebras at roots of
unity and modular representations of affine Hecke algebras.
Ivan Ip (Kavli IPMU)
Positive representations and quantum higher Teichmüller theory
- Abstract:
We review the notion of positive representations of split real quantum
groups introduced in a joint work with Igor Frenkel, and describe the
tensor product decomposition when restricted to the Borel part, through the
use of multiplier Hopf algbera. This generalized the essential step of the
construction of the quantum Teichmüller theory from the representation of
the quantum plane studied by Frenkel-Kim, and provide a candidate for the
quantum higher Teichmüller theory.
On going joint work with Hyun-Kyu Kim.
Mikhail Kapranov (Kavli IPMU)
Perverse sheaves on real hyperplane arrangements
- Abstract:
An arrangement of hyperplanes
in $\C^n$ gives a natural stratificaion of $\C^n$ and the
corresponding category of perverse sheaves. In the case
when the hyperplanes have real equations,
I will present an explicit description of this category
on terms of a quiver with relations built from the
face complex of the corresponding real arrangement.
The relations are such that they suggest the possibility
of categorifying the very concept of a perverse sheaf
in this and possibly in other situations.
Joint work with V. Schechtman.
Masaki Kashiwara (RIMS)
Simplicity of tensor products (joint work with
Seok-Jin Kang, Myungho Kim and Ser-jin Oh)
- Abstract:
In this talk, I explain
that, for simple modules $M$ and $N$ over a quantum affine algebra,
their tensor product $M\otimes N$
has a simple head and a simple socle if $M\otimes M$ is simple.
A similar result is proved for the convolution product of simple modules over quiver Hecke algebras.
This affirmatively answer some conjecture of Bernard Leclerc
on real upper global basis.
The use of $R$-matrix is essential for the proof.
Toshiro Kuwabara (Higher School of Economics)
BRST cohomologies for rational Cherednik algebras
- Abstract:
Quantization of Kleinian singularities can be realized as two different quantum
Hamiltonian reductions. They are known as rational Cherednik algebras
(symplectic reflection algebras) and finite W-algebras. Losev showed that these
two quantizations are isomorphic by using realization of these algebras in terms
of deformation-quantization.
One can define a cohomology theory associated with Hamiltonian reduction,
which is known as BRST cohomologies. In this talk, we see that higher BRST
cohomologies corresponding to the rational Cherednik algebras do not vanish,
while ones corresponding to the finite W-algebras vanish. Moreover, we see that
the higher BRST cohomologies can be determined explicitly. To
determine the higher
cohomologies, we use the realization as deformation-quantization algebras and
affinity properties of these sheaves of deformation-quantization algebras.
Ivan Losev (Northeastern University)
Representation theory of quantized Gieseker moduli spaces
- Abstract:
Gieseker moduli spaces are an especially nice and interesting quiver varieties.
I will discuss the representation theory of their quantizations and, in particular, the
categories $O$.
David Nadler (UC Berkeley)
Combinatorics of microlocal sheaves
- Abstract:
To the germ of a Legendrian singularity, there is assigned a
derived category of microlocal sheaves. It is traditionally constructed as
a subquotient of sheaves specified by singular support. We will explain how
to non-characteristically deform the germ of a Legendrian singularity to a
Legendrian with arboreal singularities (as appear in the preprint
1309.4122). One can then calculate microlocal sheaves in an elementary way
in terms of quiver representations.
Andrei Negut (Columbia Univ., RIMS)
Pieri rules and Hilbert schemes
- Abstract:
I will discuss work recently done at RIMS on certain combinatorial
consequences of two rich structures arising on Hilbert schemes of surfaces:
one geometric (via the K-theoretic stable basis) and one representation
theoretic (via toroidal algebras).
Thomas Nevins (Univ. of Illinois at Urbana-Champaign)
Morse Decomposition of $D$-Module Categories
- Abstract:
Algebraic varieties with actions of reductive algebraic groups possess special Morse-theoretic properties that yield decompositions of their equivariant cohomology. I will explain parallel structures ``one categorical level higher," for equivariant sheaves (more precisely, D-modules) on varieties, that are closely related to vanishing properties of equivariant D-modules. I will discuss representation-theoretic and cohomological consequences as well.
Yuji Tachikawa (Univ. of Tokyo)
Nilpotent orbits and supersymmetric gauge theories
- Abstract:
Various properties of nilpotent orbits are now known to govern many aspects of four-dimensional supersymmetric gauge theories. Combined with information on the gauge theory side already known to physicists, this relation can sometimes lead to new mathematical conjectures concerning nilpotent orbits. Two such conjectures will be discussed: one is about a graded polynomial ring assigned to each nilpotent orbit whose property is controlled by the duality of Spaltenstein-Lusztig-Sommers-Achar, and another is about a nice class of holomorphic symplectic varieties from which the instanton moduli space of type E groups can be obtained.
Yakov Varshavsky (Hebrew University of Jerusalem)
Geometric local Langlands correspondence and the (stable) Bernstein center
- Abstract:
This is part of the joint project in progress with Roman
Bezrukavnikov and David Kazhdan.
The goal of the project is to develop a geometric local Langlands
correspondence in the $\ell$-adic setting and apply it to the study
of the (stable) Bernstein center.
To illustrate the general picture, I will explain how to provide a
geometric construction of the Bernstein projector to the depth
zero spectrum. As an application, I will give its explicit formula
and show its stability.
Geordie Williamson (Max-Planck-Institut für Mathematik)
Lusztig's conjecture and torsion explosion
- Abstract:
I will explain how recent results in number theory (the affine
sieve of Bourgain, Gamburd and Sarnak) allow one to show that the torsion
in the intersection cohomology of Schubert varieties in $GL_n$ grows
exponentially in $n$. Using results of Soergel one may deduce that any bound
for Lusztig's conjecture has to be at least exponential in $n$. The
applicability of the BGS sieve to our setting was pointed out to me by
Peter McNamara.
