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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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$\mathbb {Z}/m\mathbb {Z}$-graded Lie algebras and perverse sheaves, III: Graded double affine Hecke algebra
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by George Lusztig and Zhiwei Yun PDF
Represent. Theory 22 (2018), 87-118 Request permission

Abstract:

In this paper we construct representations of certain graded double affine Hecke algebras (DAHA) with possibly unequal parameters from geometry. More precisely, starting with a simple Lie algebra $\mathfrak {g}$ together with a $\mathbb {Z}/m\mathbb {Z}$-grading $\bigoplus _{i\in \mathbb {Z}/m\mathbb {Z}}\mathfrak {g}_{i}$ and a block of $\mathcal {D}_{G_{\underline 0}}(\mathfrak {g}_{i})$ as introduced in [J. Represent. Theory 21 (2017), pp. 277-321], we attach a graded DAHA and construct its action on the direct sum of spiral inductions in that block. This generalizes results of Vasserot [Duke Math J. 126 (2005), pp. 251-323] and Oblomkov-Yun [Adv. Math 292 (2016), pp. 601-706] which correspond to the case of the principal block.
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Additional Information
  • George Lusztig
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
  • MR Author ID: 117100
  • Email: gyuri@math.mit.edu
  • Zhiwei Yun
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
  • MR Author ID: 862829
  • Email: zyun@mit.edu
  • Received by editor(s): April 2, 2017
  • Received by editor(s) in revised form: June 4, 2018
  • Published electronically: July 19, 2018
  • Additional Notes: The first author was partially supported by the NSF grant DMS-1566618.
    The second author was supported by the NSF grant DMS-1302071 (with extension as DMS-1736600) and the Packard Foundation.
  • © Copyright 2018 American Mathematical Society
  • Journal: Represent. Theory 22 (2018), 87-118
  • MSC (2010): Primary 20G99, 20C08
  • DOI: https://doi.org/10.1090/ert/515
  • MathSciNet review: 3829497