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

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- by George Lusztig and Zhiwei Yun
- Represent. Theory
**22**(2018), 87-118 - 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.## References

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## Bibliographic 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