ℤ/𝕞ℤ-graded Lie algebras and perverse sheaves, III: Graded double affine Hecke algebra

Lusztig, George; Yun, Zhiwei

doi:10.1090/ert/515

$\mathbb {Z}/m\mathbb {Z}$-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

Joseph Bernstein and Valery Lunts, Equivariant sheaves and functors, Lecture Notes in Mathematics, vol. 1578, Springer-Verlag, Berlin, 1994. MR 1299527, DOI 10.1007/BFb0073549
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
G. Lusztig, Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), no. 2, 205–272. MR 732546, DOI 10.1007/BF01388564
George Lusztig, Fourier transforms on a semisimple Lie algebra over $\textbf {F}_q$, Algebraic groups Utrecht 1986, Lecture Notes in Math., vol. 1271, Springer, Berlin, 1987, pp. 177–188. MR 911139, DOI 10.1007/BFb0079237
George Lusztig, Cuspidal local systems and graded Hecke algebras. I, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 145–202. MR 972345, DOI 10.1007/BF02699129
George Lusztig, Classification of unipotent representations of simple $p$-adic groups, Internat. Math. Res. Notices 11 (1995), 517–589. MR 1369407, DOI 10.1155/S1073792895000353
George Lusztig, Study of perverse sheaves arising from graded Lie algebras, Adv. Math. 112 (1995), no. 2, 147–217. MR 1327095, DOI 10.1006/aima.1995.1031
G. Lusztig, Classification of unipotent representations of simple $p$-adic groups. II, Represent. Theory 6 (2002), 243–289. MR 1927955, DOI 10.1090/S1088-4165-02-00173-5
George Lusztig and Zhiwei Yun, $\mathbf {Z}/m$-graded Lie algebras and perverse sheaves, I, Represent. Theory 21 (2017), 277–321. MR 3697026, DOI 10.1090/ert/500
George Lusztig and Zhiwei Yun, $\mathbf {Z}/m$-graded Lie algebras and perverse sheaves, II, Represent. Theory 21 (2017), 322–353. MR 3698042, DOI 10.1090/ert/501
Alexei Oblomkov and Zhiwei Yun, Geometric representations of graded and rational Cherednik algebras, Adv. Math. 292 (2016), 601–706. MR 3464031, DOI 10.1016/j.aim.2016.01.015
Robert Steinberg, Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR 0230728
Eric Vasserot, Induced and simple modules of double affine Hecke algebras, Duke Math. J. 126 (2005), no. 2, 251–323. MR 2115259, DOI 10.1215/S0012-7094-04-12623-5