Geometric Representation Theory Seminar

Organizers：

Lin Chen, Will Donovan, Penghui Li, Peng Shan, Changjian Su, Wenbin Yan

Speaker：

Semeon Arthamonov (BIMSA)

Time：

Wed., 15:30-16:30, Nov. 6, 2024

Venue：

B627, Shuangqing Complex Building A

Title：

Genus two Double Affine Hecke Algebra and its Classical Limit

Abstract：

Double Affine Hecke Algebras were originally introduced by I.Cherednik and used in his 1995 proof of Macdonald conjecture from algebraic combinatorics. These algebras come equipped with a large automorphism group SL(2,Z) which has geometric origin, namely it is the modular group of a torus. It was subsequently shown that spherical Double Affine Hecke Algebras realize universal flat deformations of the quantum chracter variety of a torus and their existence is closely related to the fact that classical SL(n,C)-character varieties admit symplectic resolution of singularities via the Hilbert Scheme Hilb_n(\mathbb C*\times\mathbb C*).

In 2019 G. Belamy and T. Schedler have shown that SL(n,C)-character varieties of closed genus g surface admit symplectic resolutions only when g=1 or (g,n)=(2,2). In my talk I will discuss our (g,n)=(2,2) generalization of Double Affine Hecke Algebra which provide a flat deformation of quantum SL(2,C)-character variety of a closed genus two surface. I will show that solution to the word problem in our algebra has striking similarity with the Poicare-Birkhoff-Witt Theorem for the basis of Universal Enveloping Algebra of a Lie algebra. This is consistent with the philosophy formulated by A.Okounkov that resolutions of symplectic singularities should be viewed as "Lie Algebras of the XXI'st century". (joint with Sh. Shakirov)

About the Speaker：

Dr. Semeon Arthamonov has been studying Applied Mathematics and Physics at Moscow Institute of Physics and Techonology where he received his B.Sc. and M.Sc. degrees. In 2013 he entered the graduate program in Mathematics at Rutgers, The State University of New Jersey where he defended his PhD thesis in 2018 under the supervision of Prof. V. Retakh. He has then held visiting and postdoctoral positions at the University of California Berkeley, Centre de Recherces Mathematiques Montreal and University of Toronto before joining BIMSA in July 2024 as an Associate Professor.