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 Representation Theory
Representation Theory
ISSN 1088-4165

     

Double affine Hecke algebras at roots of unity

Author(s): M. Varagnolo; E. Vasserot
Journal: Represent. Theory 14 (2010), 510-600.
MSC (2000): Primary 17B37; Secondary 16W35, 20C08
Posted: August 3, 2010
MathSciNet review: 2672950
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Abstract | References | Similar articles | Additional information

Abstract: We study double affine Hecke algebras at roots of unity and their relations with deformed Hilbert schemes. In particular their categories of finitely generated modules are derived equivalent to some category of coherent sheaves.

References:

[AM]
Alekseev, A.Y., Malkin, A.Z., Symplectic structures associated to Lie-Poisson groups, Comm. Math. Phys. 162 (1994), 147-173. MR 1272770 (95g:58074)

[AZ]
Artin, M., Zhang, J.J., Noncommutative projective schemes, Adv. Math. 109 (1994), 228-287. MR 1304753 (96a:14004)

[APW]
Andersen, H. H., Polo, P., Wen, K., Representations of quantum algebras, Invent. Math. 104 (1991), 1-59. MR 1094046 (92e:17011)

[B1]
Bourbaki, N., Algèbre commutative, Eléments de mathématiques, Hermann, 1961-1965.

[B2]
Bourbaki, N., Algèbre, Eléments de mathématiques, Masson, 1980.

[BFG]
Bezrukavnikov, R., Finkelberg, M., Ginzburg, V., Cherednik algebras and Hilbert schemes in characteristic $ p$, Represent. Theory 10 (2006), 254-298 (electronic). MR 2219114 (2007j:14004)

[BG1]
Brown, K., A., Goodearl, I., Homological aspects of Noetherian PI Hopf algebras and irreducible modules of maximal dimension, J. Algebra 198 (1997), 240-265. MR 1482982 (99c:16036)

[BG2]
Brown, K., A., Gordon, I., Poisson orders, symplectic reflexion algebras, and representation theory, J. Reine Angew. Math 559 (2003), 193-216. MR 1989650 (2004i:16025)

[BG3]
Brown, K., A., Goodearl, I., Lectures on algebraic quantum groups, Advanced Courses in Mathematrics CRM Barcelona, Birkhäuser, 2002. MR 1898492 (2003f:16067)

[BGS]
Brown, K., A., Gordon, I., Stafford, $ \mathcal{O} _{\epsilon }[G]$ is a free module over $ \mathcal{O}[G]$, Preprint math.QA/0007179.

[BK1]
Bezrukavnikov, R., Kaledin, D., MacKay equivalence for symplectic quotient singularities, Proc. Steklov Inst. Math. 246 (2004), 13-33. MR 2101282 (2006e:14006)

[BK2]
Brion, M., Kumar, S., Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, vol. 231, Birkhäuser, 2004. MR 2107324 (2005k:14104)

[BK3]
Backelin, E., Kremnitzer, K., Quantum flag varieties, equivariant quantum $ \mathcal{D}$-modules, and localization of quantum groups, Adv. Math. 203 (2006), no. 2, 408-429. MR 2227727 (2007b:17021)

[BS]
Bauman, P., Schmitt, F., Classification of bicovariant differential calculi on quantum groups (a representation theoretic approach), Commun. Math. Phys. 194 (1998), 71-96. MR 1628278 (99f:58012)

[C]
Cherednik, I., Double affine Hecke algebras, Cambridge University Press, 2005. MR 2133033 (2007e:32012)

[CV]
Crawley-Boevey, W., Van Den Bergh, M., Absolutely indecomposable representations and Kac-Moody Lie algebras, Invent. Math. 155 (2004), 537-559. MR 2038196 (2004m:17032)

[CH]
Crawley-Boevey, W., Holland, M. P., Noncommutative deformations of Kleinian singularities, Duke Math. J.. MR 1620538 (99f:14003)

[D1]
Dixmier, J., Enveloping algebras, Graduate Studies in Math., vol. 11, Amer. Math. Soc., 1996. MR 1393197 (97c:17010)

[DM]
Donin, J., Mudrov, A., Reflection equation, twist, and equivariant quantization, Isr. J. Math. 136 (2003), 11-28. MR 1998103 (2004g:16046)

[DKP]
De Concini, C., Kac, V., Procesi, C., Quantum coadjoint action, J. Amer. Math. Soc. 5 (1992), 151-190. MR 1124981 (93f:17020)

[DL]
De Concini, C., Lyubashenko, V., Quantum function algebra at roots of 1, Adv. Math. 108 (1994), 205-262. MR 1296515 (95m:17014)

[DP]
De Concini, C., Procesi, C., Quantum groups, Lecture Notes in Math., vol. 1565, Springer-Verlag, 1993, pp. 31-140. MR 1288995 (95j:17012)

[E]
Eisenbud, D., Commutative algebra with a view toward algebraic geometry, Graduate Texts in Math., vol. 150, Springer-Verlag, 1994. MR 1322960 (97a:13001)

[EG]
Etingof, P., Ginzburg, V., Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), 243-348. MR 1881922 (2003b:16021)

[EGA3]
Grothendieck, A., Étude cohomologique des faisceaux cohérents, Publications Mathématiques de l'IHES, vol. 17, 1963.

[EK]
Etingof, P., Kirillov, A., Macdonald polynomials and representations of quantum groups, Math. Res. Letters 1 (1994), 279-296. MR 1302644 (96m:17025)

[FR]
Fock, V.V., Rosly, A.A., Poisson structure on moduli of flat connections on Riemann surfaces and the $ r$-matrix, Amer. Math. Soc. Transl. Ser. 2 1991 (1999). MR 1730456 (2001k:53167)

[G]
Gabriel, P., Des catégories Abéliennes, Bulletin de la S.M.F. 90 (1962), 323-448. MR 0232821 (38:1144)

[GG]
Gan, W.L., Ginzburg, V., Almost-commuting variety, $ \mathcal{D}$-modules, and Cherednik algebras, Int. Math. Res. Publ. 2006, 26439, 1-54. MR 2210660 (2008c:14062)

[J1]
Jantzen, J. C., Representations of algebraic groups (2nd edition), Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, 2003. MR 2015057 (2004h:20061)

[J2]
Joseph, A., Quantum groups and their primitive ideals, Ergeb. Math. Grenzgeb. (3), vol. 29, Springer-Verlag, 1995. MR 1315966 (96d:17015)

[K2]
Kirillov, A., Lectures on affine Hecke algebras and Macdonald conjectures, Bull. Amer. Math. Soc. (N.S.) 34 (1997), 251-292. MR 1441642 (99c:17015)

[L1]
Lin, Z., Induced representations of Hopf algebras: applications to quantum groups at roots of 1, J. Algebra 154 (1993), 152-187. MR 1201918 (94e:16046)

[L2]
Lusztig, G., Introduction to quantum groups, Birkhäuser, 1993. MR 1227098 (94m:17016)

[L3]
Lu, J.-H., Momentum mappings and reduction of Poisson actions, Symplectic geometry, groupoids, and integrable systems (Berkeley, 1989), Math. Sci. Res. Inst. Publ., vol. 20, 1991, pp. 209-226. MR 1104930 (92j:58036)

[L4]
Lu, J.-H., Moment maps at the quantum level, Commun. Math. Phys. 157 (1993), 389-404. MR 1244874 (94m:58096)

[LR]
Lunts, V.A., Rosenberg, A. L., Differential operators on noncommutative rings, Selecta Math. New. Ser. 3 (1997), 335-359. MR 1481133 (98k:16038)

[LW]
Lu, J.H., Weinstein, A., Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geometry 31 (1990), 501-526. MR 1037412 (91c:22012)

[M1]
Majid, S., Foundations of quantum group theory, Cambridge University Press, 1995. MR 1381692 (97g:17016)

[M2]
Milne, J.-S., Étale cohomology, Princeton University Press, 1980. MR 559531 (81j:14002)

[MR]
McConnell, J.-C., Robson, J.-C., Noncommutative Noetherian rings, Pure Appl. Math., John Wiley and Sons, 1987. MR 934572 (89j:16023)

[N]
Nakajima, H., Lectures on Hilbert schemes of points on surfaces, University Lecture Series, vol. 18, Amer. Math. Soc., 1999. MR 1711344 (2001b:14007)

[NV1]
Nastasescu, C., Van Oystaeyen, F., Graded and filtered rings and modules, Lectures Notes in Math., vol. 758, Springer-Verlag, 1979. MR 551625 (80k:16002)

[NV2]
Nastasescu, C., Van Oystaeyen, F., Graded ring theory, North-Holland, 1982. MR 676974 (84i:16002)

[P]
Paradowski, J., Filtrations of modules over the quantum algebra, Algebraic groups and their generalizations: classical methods (Haboush, W.F., Parshall, B.J., eds.), Proc. Sympos. Pure Math., vol. 56:2, 1994, pp. 93-108. MR 1278729 (95c:20060)

[O]
Oblomkov, A., Double affine Hecke algebras and Calogero Moser spaces, Int. Math. Res. Not. 18 (2004), 877-912. MR 2037756 (2005j:20005)

[R]
Ryom-Hansen, S., A $ q$-analogue of Kempf's vanishing theorem, Mosc. Math. J. 3 (2003), 173-187. MR 1996807 (2004e:17014)

[S]
Seshadri, C. S., Geometric reductivity over arbitrary base, Adv. Math. 26 (1977), 225-274. MR 0466154 (57:6035)

[T1]
Tanisaki, T., Killing forms, Harish-Chandra isomorphisms, and universal R-matrices for quantum algebras, Int. J. Mod. Phys. A7, Suppl. 1B (1992), 941-961. MR 1187582 (93k:17040)

[V]
Vasserot, E., On induced and simple modules of double affine Hecke algebras, Duke Math. J. 126 (2005), 251-323. MR 2115259 (2006j:20011)

[VV1]
Van den Bergh, M., Van Oystaeyen, F., Lifting maximal orders, Comm. Algebra 17 (1989), 341-349. MR 978479 (90a:16006)

[VV2]
Varagnolo, M., Vasserot, E., From double affine Hecke algebras to quantum affine Schur algebras, Int. Math. Res. Not. 26 (2004), 1299-1333. MR 2046511 (2005c:20009)

[VV3]
Varagnolo, M., Vasserot, E., Finite-dimensional representations of DAHA and affine Springer fibers: the spherical case, Duke Math. J. 147 (2009), no. 3, 439-540. MR 2510742 (2010e:20011)

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Additional Information:

M. Varagnolo
Affiliation: Département de Mathématiques, Université de Cergy-Pontoise, 2 av. A. Chauvin, BP 222, 95302 Cergy-Pontoise Cedex, France
Email: michela.varagnolo@math.u-cergy.fr

E. Vasserot
Affiliation: Département de Mathématiques, Université Paris 7, 175 rue du Chevaleret, 75013 Paris, France
Email: vasserot@math.jussieu.fr

DOI: 10.1090/S1088-4165-2010-00384-2
PII: S 1088-4165(2010)00384-2
Received by editor(s): June 14, 2006
Received by editor(s) in revised form: January 21, 2009
Posted: August 3, 2010
Copyright of article: Copyright 2010, American Mathematical Society




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