Fall 2013-- Summer 2014

Title: On differential central extensions of Chevalley groups.

Abstract. I will explain what differential algebraic groups are and how they arise in differential Galois theory. Every such a group is built up from so-called almost simple differential algebraic groups, non-abelian of which are certain linear central extensions of Chevalley groups over differential fields. My recent result is the classification of such extensions: they are actually Chevalley groups too. I will describe the main steps of the proof and will give some interesting examples. The talk will be quite elementary.

Title: On differential central extensions of Chevalley groups (Continuation)

Title: Introduction to Lie superalgebras.

Abstract. We will see some examples and define Dynkin diagrams and odd reflections. The lecture will be accessible to students who have taken a course on Lie algebras.

Title: Introduction to Lie superalgebras (continuation).

Title: Vertex algebras and integrable systems (I)

Abstract. Vertex operators and vertex (operator) algebras have arisen in a number of exciting areas of mathematics and physics, from the Monstrous Moonshine conjectures to two-dimensional conformal field theory. I will motivate the notion of a vertex (operator) algebra, giving relevant examples and their relationship to the representation theory of certain infinite dimensional Lie algebras. The talk will be as self-contained as possible, assuming only a basic knowledge of Lie algebras.

Title: Vertex algebras and integrable systems (II)

Abstract. This talk will be a continuation of the talk on November 13th. I will describe the relationship between vertex algebras (in the guise of representations of affine Kac-Moody algebras) and the theory of integrable systems of partial differential equations. I will present some classical results and also a novel result arising from Wakimoto representations.

Title: Abelian varieties, endomorphisms and monodromy.

Abstract. Using Galois theory, we construct explicit examples of abelian varieties with small endomorphism rings.

Title: Distributions on p-adic groups, finite under the action of the Bernstein center.

Title: A Gross-Kohnen-Zagier Type Theorem for Higher-Codimensional Heegner Cycles.

Title: Representations of affine Lie superalgebras and mock theta functions.

Title: On bizarre geometric properties of a counterexample to the two-dimensional Jacobian Conjecture.

Abstract. A lot can be said about a potential counterexample to the Jacobian conjecture, save that it does not exist. In my talk after giving a necessary background I'll discuss some strange properties of a map between affine planes given by a pair of polynomials constituting a counterexample.

Title: Cherednik algebras and torus knots.

Abstract: The Cherednik algebra B(c,n), generated by symmetric polynomials and the quantum Calogero-Moser Hamiltonian, appears in many areas of mathematics. It depends on two parameters - the coupling constant c and number of variables n. I will talk about representations of this algebra, and in particular about a mysterious isomorphism between the representations of B(m/n,n) and B(n/m,m) of minimal functional dimension. This symmetry between m and n is made manifest by the fact that the characters of these representations can be expressed in terms of the colored HOMFLY polynomial of the torus knot T(m/d,n/d), where d=GCD(m,n). The talk is based on my joint work with E. Gorsky and I. Losev.

Title: Categories of tensor modules of Lie algebras of infinite matrices (joint with V.Serganova)

Title: A Coxeter-type presentation for Double affine Hecke algebras.

Title: Residues and Duality for Schemes and Stacks.

Title: The Hochschild category of commutative algebras and schemes via twisting.

Title: Associated varieties and associated cycles of local theta lifts.

Title: The dual Kashiwara functions in type A and wiring diagrams (joint work with Lamprou and Zelikson).

Title:Existence of Klyachko models for GL(n,R).

Abstract : Klyachko models are certain models for all unitary representations of general linear groups over finite and local fields. We will start from the definition and general overview of these models. Then I will concentrate on complex and real numbers, and explain the construction of these models for all unitary representations of GL(n,R) and GL(n,C).

Title: Demazure structure of defining inequalities of Kashiwara's crystal B(infinity).

Abstract : Kashiwara's crystal graph B(infinity) is the combinatorial skeleton of the canonical basis of the positive part Uq(n+) of the quantized envelopping algebra Uq(g). It admits a purely combinatorial construction in terms of tuples of integers, known as Kashiwara's embedding. This embedding depends on the choice of a reduced expression for the longest element w0 of the Weyl group. Its image is the set of integer points of a polyhedral cone, the Kashiwara (or string) cone. Nakashima and Zelevinsky introduced an inductive procedure for generating a system of inequalities for this cone, which fails in general. We shall discuss how this construction can be modified in An case in order to apply for reduced expressions of w0 adapted to quivers. As a result, we obtain that the set of inequalities itself carries a structure of a union of Demazure crystals, as conjectured by Nakashima.

Title: Symmetric Self Adjoint Hopf Categories .

Abstract : We define what we call a {symmetric self-adjoint Hopf} structure on an abelian category, which is a categorical analog of the "Positive self-adjoint Hopf algebras" which were introduced by Zelevinsky. We consider the examples of this structure for the categories of polynomial functors, and equivariant polynomial functors, and obtain a categorical manifestation of Zelevinsky's structure theorem in these examples. We extend our construction to propose a categorification of the Fock space representation for the Heisenberg double. We explain why it is reasonable to expect that the symmetric selfadjoint Hopf category structure implies the existence of such categorification in general, and prove it in the case of (equivariant) polynomial functors.

Title: Coderived categories, contraderived categories, and the comodule-contramodule correspondence.

Abstract: The simple familiar story of the classical homological algebra, where the derived category can be equivalently described either as the category of complexes up to quasi-isomorphism, or as the category of complexes of projective/injective objects up to chain homotopy, becomes more complicated when one passes to DG-modules or to unbounded complexes of modules. Either the definition of the class of resolutions, or otherwise the equivalence relation on complexes has to be changed; hence the philosophy of two kinds of derived categories. The derived category of the second kind comes in two dual versions, called the coderived and the contraderived categories. Exploring the relation between these two triangulated category constructions forces one to recall the classical notions of comodules and contramodules, and leads to the theory of derived comodule-contramodule correspondence. The latter turns out to be a fundamental homological phenomenon on par with, e.g., the Koszul duality, occuring classically under various names both in representation theory and in algebraic geometry.

Title: A Study of the Saturated Tensor Cone for Symmetrizable Kac-Moody Algebras.

Abstract: Let g be a symmetrizable Kac-Moody Lie Algebra and let G be the 'minimal' Kac-Moody group with Lie algebra g. The saturated tensor cone describes which components occur in the tensor product of integrable highest weight g-modules. We give a set of necessary inequalities satisfied by the saturated tensor semigroup indexed by products in H^*(G/B; Z) for B the standard Borel subgroup. The proof relies on the Kac-Moody analogue of the Borel-Weil theorem and Geometric Invariant Theory (specifically the Hilbert-Mumford index). In the case that g is affine of rank 2, we show that these inequalities are necessary and sufficient.

Title: On Kac-Wakimoto character formula for finite-dimensional modules.

Title: Stability and Gelfand property of symmetric pairs.

Abstract. We call a symmetric pair (G,H,theta) of reductive groups stable if every closed H double coset in G is stabilized by the anti-involution associated to theta. Aizenbud and Gourevitch have shown, using the Gelfand-Kazhdan distributional criterion, that for many pairs stability implies the Gelfand property. We present a method, based on non-abelian group cohomology, to check if a pair is stable or not. As a result we get some new examples of symmetric pairs which are Gelfand pairs. We also use the results of Blanc and Delorme to falsify the Gelfand property for many pairs which are not stable, in the case of a non-archimedean ground field.

Title: Procesi bundles on Hilb^n(C^2).

Abstract: A Procesi bundle is a vector bundle on the Hilbert scheme of n points on the plane. It was first constructed by Haiman who used it to prove the Schur positivity for Macdonald polynomials. This bundle also provides a derived McKay equivalence for the Hilbert scheme. I will basically take the latter for an axiomatic description of a Procesi bundle. I will show that there are exactly two bundles with these properties: Haiman's and its dual. The proof is based on the study of Rational Cherednik algebras. The talk is based on arXiv:1303.4617.

Title:A geometric Robinson-Schensted correspondence for the double flag variety of a symmetric pair.

Abstract: The classical Robinson-Schensted correspondence is a combinatorial algorithm that establishes a bijection between permutations and pairs of Young tableaux. It admits interpretations in terms of representations of the symmetric group or in terms of Kazhdan-Lusztig cells. There is also a geometric interpretation of the Robinson-Schensted correspondence, due to Steinberg, which results in a relation between Schubert varieties and nilpotent orbits. The purpose of the talk is to present an analogue of Steinberg's construction in the context of orbits on certain double flag varieties and certain nilpotent varieties, which can be associated to symmetric pairs.

Title:Tame representations of the general linear Lie superalgebra.

Abstract: We shall discuss the character theory of Lie superalgebras and describe a new class of modules that admits the elegant Kac-Wakimoto character formula, namely the piecewise disconnected modules.

Title:Finite multiplicity theorem for spherical pairs.

Abstract: Let (G, H) be a spherical pair over a local field k and let $\pi$ be an admissible representation of G. Kobayashi and Oshima, and independently Kroetz and Schlichtkrull have recently shown that, if k is the field of real or complex numbers, then the space of H-invariant functionals on $\pi$ is finite-dimensional. Both approaches use Casselman's theorem which says that $\pi$ can be presented as a quotient of a principal series representation. We will consider another approach that does not use this theorem. An important step in our proof is to show that the singular support of any H-spherical character is a Lagrangian in the cotangent bundle of G. In the future we hope to generalize our proof to the p-adic case, where Casselman's theorem does not hold, and the finite multiplicity theorem is known only for certain kinds of spherical pairs (due to Delorme and to Sakellaridis-Venkatesh). The talk is based on an ongoing work with A. Aizenbud and D. Gourevitch.