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


Authors: George Lusztig and Zhiwei Yun
Journal: Represent. Theory 22 (2018), 87-118
MSC (2010): Primary 20G99, 20C08
DOI: https://doi.org/10.1090/ert/515
Published electronically: July 19, 2018
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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
Email: gyuri@math.mit.edu

Zhiwei Yun
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email: zyun@mit.edu

DOI: https://doi.org/10.1090/ert/515
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.
Article copyright: © Copyright 2018 American Mathematical Society

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