scholarly journals $\mathbb {Z}/m\mathbb {Z}$-graded Lie algebras and perverse sheaves, III: Graded double affine Hecke algebra

2018 ◽  
Vol 22 (4) ◽  
pp. 87-118
Author(s):  
George Lusztig ◽  
Zhiwei Yun
Keyword(s):  
Lie Algebras ◽  
Hecke Algebra ◽  
Perverse Sheaves ◽  
Graded Lie Algebras ◽  
Affine Hecke Algebra ◽  
Double Affine Hecke Algebra

scholarly journals Elliptic Double Affine Hecke Algebras

2020 ◽  
Author(s):  
Eric M. Rains ◽  
Keyword(s):  
Hilbert Scheme ◽  
Hecke Algebra ◽  
Rational Surface ◽  
Symmetric Power ◽  
Affine Hecke Algebra ◽  
Double Affine Hecke Algebra ◽  
Affine Weyl Group ◽  
Double Affine Hecke Algebras ◽  
Hilbert Scheme Of Points ◽  
Affine Hecke Algebras

We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the C<sub>n</sub> version of the construction to construct a flat noncommutative deformation of the nth symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points.

scholarly journals Wall-crossings and a categorification of K-theory stable bases of the Springer resolution

2021 ◽  
Vol 157 (11) ◽  
pp. 2341-2376
Author(s):  
Changjian Su ◽  
Gufang Zhao ◽  
Changlong Zhong
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Positive Characteristic ◽  
Quantum Cohomology ◽  
Hecke Algebra ◽  
Verma Modules ◽  
Affine Hecke Algebra ◽  
Wall Crossing ◽  
K Theory ◽  
Monodromy Matrices

Abstract We compare the $K$ -theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure–Lusztig operators in the affine Hecke algebra. We then show that these bases are categorified by the Verma modules of the Lie algebra, under the localization of Lie algebras in positive characteristic of Bezrukavnikov, Mirković, and Rumynin. As an application, we prove that the wall-crossing matrices of the $K$ -theory stable bases coincide with the monodromy matrices of the quantum cohomology of the Springer resolution.

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