## Schedule

 Mon Tue Wed Thu Fri 10:00 - 11:00 x Losev He Bezrukavnikov Fiebig 11:15 - 12:15 Nadler Finkelberg Tachikawa Achar Cherednik 13:30 - 14:30 Varshavsky Ip Nevins Ginzburg Abe 14:45 - 15:45 Negut Kashiwara Braden Kapranov Williamson 16:00 - 17:00 Braverman Kuwabara x Feigin x

• ## 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.

• ## 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.