Representations and Lie Theory SeminarFall 2017
Time: Wednesday, 16:30  17:30


TIME  SPEAKER  TITLE 
August 23  Oleksandr Tsymbaliuk  Shifted quantum affine algebras and shifted Yangians 
August 30  Staff  YangBaxter Equation 
September 22 (Friday 34PM) 
Monica Vazirani  An elliptic SchurWeyl construction of the rectangular representation of the DAHA 
October 11  No seminar  Fall break 
October 18  Noah Arbesfeld  A geometric Rmatrix for a general surface 
October 25  No seminar  Talk in Invitations Seminar 
November 1  Robin Walters  The FinkelbergGinzburg Mirabolic Monodromy Conjecture 
November 22  No seminar  Thanksgiving 
November 29  Curtis Wendlandt  The Rmatrix formalism for Yangians 
August 23:
In this talk, I will speak about shifted quantum affine algebras and shifted Yangians, as well as their incarnations through geometry of parabolic Laumon spaces, affine Grassmannians, and Toda lattice.
The shifted Yangians were originally introduced by BrundanKleshchev in the gl(n) case with a dominant shift and were later generalized by KamnitzerWebsterWeekesYacobi to any simple Lie algebra with an arbitrary shift. These algebras attracted recently a new interest due to their interplay with the Coulomb branches.
In the first half of the talk, I will remind those results about shifted Yangians, while the second part will be devoted to the multiplicative analogue of this story. On the algebraic side this leads to the notion of shifted quantum affine algebras, while on the geometric side we replace cohomology by Ktheory.
This is a joint work with M. Finkelberg.
September 22: Building on the work of CalaqueEnriquezEtingof, LyubashenkoMajid, and ArakawaSuzuki, Jordan constructed a functor from quantum Dmodules on general linear groups to representations of the double affine Hecke algebra (DAHA) in type A. When we input quantum functions on GL(N) the output is L(k^N), the irreducible DAHA representation indexed by an N by k rectangle. For the specified parameters, L(k^N) is Ysemisimple, i.e. one can diagonalize the Dunkl operators. We give an explicit combinatorial description of this module via its Yweight basis. This is joint work with David Jordan.
October 18 Generalizing work of Maulik and Okounkov, we produce a solution to the YangBaxter equation from the Hilbert scheme of points on a general surface using an intertwiner of certain highest weight Virasoro modules. We also explain how to modify this construction to produce formulas for multiplication by Chern classes of tautological bundles on the Hilbert scheme.
November 1 This is a joint workinprogress with Valerio Toledano Laredo. We study the monodromy of the trigonometric KZ connection associated to the covariant representation J _{\theta} of the trigonometric Cherednik algebra H_c. This is motivated by Finkelberg and Ginzburg's study of a mirabolic version G_{\theta,c} of the HarishChandra Dmodule defined by Hotta and Kashiwara. Through Hamiltonian reduction and the KZ functor, G_{\theta,c} can be understood by the above monodromy problem. The monodromy of J_{\theta} was computed by Opdam, Heckman, and Cherednik for values of \theta and c when the connection is nonresonant. Our aim is to compute monodromy for all values of the parameters. Our tools are Opdam's shift operators in c as well as shift operators in \theta arising from a commuting copy of the difference Cherednik algebra. In this talk, I will focus on the example of the rank one case, which is easier to illustrate and fully proved.