SCIENTIFIC PROGRAMS AND ACTIVITIES

December  6, 2013
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
 Geometric Representation Theory Seminar 2013-14 Room 332 Organizing Committee: Joel Kamnitzer (jkamnitzmath.toronto.edu) Peter Samuelson (peter.samuelsongmail.com) Oded Yacobi (oyacobi math.toronto. edu)
 Starting September 19 Seminars will be held at the Fields Institute Dec. 5 Room 332 TBA Dec. 12 Room 332 Past Seminars Nov. 28 Room 332 Joel Kamnitzer 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. Nov. 21 Daniele Rosso 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. Nov. 14 Room 332 Gwyn Bellamy Generalizing Kashiwara's equivalence to conic quantized symplectic manifolds 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. Nov. 7 Tsao-Hsien Chen 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.) Oct. 31 Peter Samuelson 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.) Oct. 24 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 schemes. Oct. 17 No Seminar Oct. 10 Dinakar Muthiah 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. Oct. 3 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. Sept 26 Leonid Rybnikov 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. Sept 19 Yanagida Shintaro 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. Tuesday July 30 Bahen 6180 Chia-Cheng Liu 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.