Representations and Lie Theory Seminar

Schedule of talks:

Abstracts

January 23. In this talk I will go over the definition of the Yangian as an associative algebra, and two seemingly different ways to tensor its representations. I will present a method of constructing the unique twist which conjugates one tensor product to the other, based on the existence/uniqueness results for the solutions of PDEs near non-singular points. I will also show one very explicit formula for this twist, obtained in our joint work (Gautam-Toledano Laredo-Wendlandt). Time permitting, we will speculate on its applications to geometry, integrability and mathematical physics.

February 13. The elliptic stable envelopes, also known as dynamical weight functions are one of the central object in representation theory. This functions features as integrands in integral representations of solutions to quantum Knizhnik-Zamolodchikov equations and quantum dynamical equations. In this talk we consider two Nakajima varieties X and X' which are dual under "3D mirrors symmetry". We will discuss that the stable envelope functions associated to X and X' coincide after appropriate change of variables. This talk is based on joint work with R. Rimányi, A. Varchenko and Z. Zhou.

February 20. This talk is about a certain class of finite-dimensional representations (called "tilting modules") for the group GLn(k), where k is an algebraically closed field of positive characteristic. Here are some things one can do with these modules: (1) Classify the tensor ideals of tilting modules (this makes sense because the tilting modules are closed under tensor product). (2) Compute their support varieties, which are closures of nilpotent orbits in the Lie algebra of GL(n). I will explain what is known about these questions, and I will discuss a conjectural link between them, with some concrete examples. This is joint work with W. Hardesty and S. Riche.

March 8. I will discuss joint work with Ben Gammage and Ben Webster on an equivalence between coherent sheaves and microlocal sheaves on two multiplicative hypertoric varieties. I will focus on the most basic example of this equivalence, where one can define all objects in fairly elementary terms.

April 3. The Yangian Y(g) of a complex semisimple Lie algebra g is a remarkable Hopf algebra which arises as a quantization of the standard Lie bialgebra structure on g[t]. An elementary result of fundamental importance to the theory of Yangians is that Y(g) has a family of automorphisms, indexed by the complex numbers, which quantize the automorphisms of g[t] given by shifting t by a complex scalar. These are the so-called shift automorphisms of the Yangian.
A small modification to the definition of the Yangian leads to a larger algebra called the Yangian double (denoted DY(g)) which, in principle, should encode additional symmetries of the Yangian itself. In this talk, I will explain how the aforementioned shift automorphisms lead to a family of injections from DY(g) into a completion of Y(g) which can be used to exploit surprisingly deep connections between Yangians, Yangian doubles, and quantum loop algebras.

April 17 Orders in finite-dimensional algebras over number fields give rise to interesting locally symmetric spaces and algebraic varieties. Hilbert modular varieties or arithmetically defined hyperbolic 3-manifolds, compact ones as well as non-compact ones, are familiar examples. In this talk we discuss various cases related to the general linear group GL(2) over orders in division algebras of degree d defined over some number field. The underlying algebraic group is an inner form of the k-split group GL(2d). Geometry, arithmetic, and the theory of automorphic forms are interwoven in a most fruitful way in this work. We indicate a construction of automorphic representations that represent non-vanishing square-integrable cohomology classes for such arithmetically defined groups.

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