Recruitment Talk -- Double Affine Hecke Algebra beyond genus one

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January 26, 2023
10:00AM - 11:00AM
Location
Zoom

Date Range
Add to Calendar 2023-01-26 10:00:00 2023-01-26 11:00:00 Recruitment Talk -- Double Affine Hecke Algebra beyond genus one Speaker:  Semeon Artamonov Title:  Double Affine Hecke Algebra beyond genus one Abstract:  Double Affine Hecke Algebras were introduced by I.Cherednik in his classical 1995 proof of the Macdonald constant term conjecture. These algebras have soon proved to be useful well-beyond the original context of Algebraic Combinatorics and Representation Theory. Most notably, they are closely related to the geometry of G-character varieties of a torus and resolutions of its symplectic singularities. Following the idea suggested by A.Okounkov, resolutions of symplectic singularities should be viewed as "Lie Algebras of the XXI'st century". Symplectic resolutions of character varieties of surface groups were classified recently and it turns out that aside from the well-known genus one case, there is only one example, namely an SL(2,C)-character variety of a closed genus two surface which admits symplectic resolution.  In my talk I will discuss our genus two generalization of Double Affine Hecke Algebra and its classical limit. I will show that solution to the word problem in our algebra, in line with Okounkov’s philosophy, has striking similarity with the Poincare-Birkhoff-Witt Theorem for Universal Enveloping Algebra of a Lie algebra. Zoom Department of Mathematics math@osu.edu America/New_York public
Description

Speaker:  Semeon Artamonov

Title:  Double Affine Hecke Algebra beyond genus one

Abstract:  Double Affine Hecke Algebras were introduced by I.Cherednik in his classical 1995 proof of the Macdonald constant term conjecture. These algebras have soon proved to be useful well-beyond the original context of Algebraic Combinatorics and Representation Theory. Most notably, they are closely related to the geometry of G-character varieties of a torus and resolutions of its symplectic singularities. Following the idea suggested by A.Okounkov, resolutions of symplectic singularities should be viewed as "Lie Algebras of the XXI'st century". Symplectic resolutions of character varieties of surface groups were classified recently and it turns out that aside from the well-known genus one case, there is only one example, namely an SL(2,C)-character variety of a closed genus two surface which admits symplectic resolution.  In my talk I will discuss our genus two generalization of Double Affine Hecke Algebra and its classical limit. I will show that solution to the word problem in our algebra, in line with Okounkov’s philosophy, has striking similarity with the Poincare-Birkhoff-Witt Theorem for Universal Enveloping Algebra of a Lie algebra.

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