Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Hochschild cohomology and graded Hecke algebras

Authors: Anne V. Shepler and Sarah Witherspoon
Journal: Trans. Amer. Math. Soc. 360 (2008), 3975-4005
MSC (2000): Primary 16E40, 16S80
Published electronically: March 20, 2008
MathSciNet review: 2395161
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We develop and collect techniques for determining Hochschild cohomology of skew group algebras $ S(V)\char93 G$ and apply our results to graded Hecke algebras. We discuss the explicit computation of certain types of invariants under centralizer subgroups, focusing on the infinite family of complex reflection groups $ G(r,p,n)$ to illustrate our ideas. Resulting formulas for Hochschild two-cocycles give information about deformations of $ S(V)\char93 G$ and, in particular, about graded Hecke algebras. We expand the definition of a graded Hecke algebra to allow a nonfaithful action of $ G$ on $ V$, and we show that there exist nontrivial graded Hecke algebras for $ G(r,1,n)$, in contrast to the case of the natural reflection representation. We prove that one of these graded Hecke algebras is equivalent to an algebra that has appeared before in a different form.

References [Enhancements On Off] (What's this?)

  • 1. S. Ariki and K. Koike, ``A Hecke Algebra of $ (\mathbb{Z}/r\mathbb{Z})\wr\mathfrak{S}_n$ and Construction of Its Irreducible Representations,'' Advances in Mathematics, 106 (1994), 216-243. MR 1279219 (95h:20006)
  • 2. M. Broué and G. Malle, ``Zyklotomische Heckealgebren'', Astérisque, 212 (1993), 119-189. MR 1235834 (94m:20095)
  • 3. M. Broué, G. Malle, and J. Michel, ``Towards Spetses I'', Transform. Groups 4 (1999), no. 2-3, 157-218. MR 1712862 (2001b:20082)
  • 4. M. Broué, G. Malle, and R. Rouquier, ``Complex Reflection Groups, Braid Groups, Hecke Algebras,'' J. reine angew. Math., 500 (1998), 127-190. MR 1637497 (99m:20088)
  • 5. A. Căldăraru, A. Giaquinto, and S. Witherspoon, ``Algebraic deformations arising from orbifolds with discrete torsion,'' J. Pure Appl. Algebra 187 (2004), no. 1-3, 51-70. MR 2027895 (2005c:16013)
  • 6. T. Chmutova, ``Twisted symplectic reflection algebras,'' to appear in J. Pure Appl. Algebra.
  • 7. C. Dezélée, ``Une généralisation de l'algèbre de Hecke graduée de type $ B$,'' math.RT/0304484.
  • 8. C. Dezélée, ``Generalized graded Hecke algebra for complex reflection group of type $ G(r,1,n)$,'' math.RT/0605410.
  • 9. V. G. Drinfeld, ``Degenerate affine Hecke algebras and Yangians,'' Funct. Anal. Appl. 20 (1986), 58-60. MR 831053 (87m:22044)
  • 10. P. Etingof, ``Exploring noncommutative algebras via deformation theory,'' math. QA/0506144.
  • 11. P. Etingof and V. Ginzburg, ``Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism,'' Invent. Math. 147 (2002), no. 2, 243-348. MR 1881922 (2003b:16021)
  • 12. M. Farinati, ``Hochschild duality, localization, and smash products,'' J. Algebra 284 (2005), no. 1, 415-434. MR 2115022 (2005j:16009)
  • 13. M. Gerstenhaber, ``On the deformation of rings and algebras,'' Ann. Math. 79 (1964), 59-103. MR 0171807 (30:2034)
  • 14. V. Ginzburg and D. Kaledin, ``Poisson deformations of symplectic quotient singularities,'' Adv. Math. 186 (2004), no. 1, 1-57. MR 2065506 (2005h:32072)
  • 15. D. Kazhdan and G. Lusztig, ``Proof of the Deligne-Langlands conjecture for Hecke algebras,'' Invent. Math. 87 (1987), no. 1, 153-215. MR 862716 (88d:11121)
  • 16. G. Kemper and H. Derksen, ``Computational invariant theory,'' Invariant Theory and Algebraic Transformation Groups, I. Encyclopaedia of Mathematical Science, 130. Springer-Verlag, Berlin, 2002. MR 1918599 (2003g:13004)
  • 17. G. Lusztig, ``Cuspidal local systems and graded Hecke algebras I,'' Inst. Hautes Études Sci. Publ. Math. 67 (1988), 145-202. MR 972345 (90e:22029)
  • 18. G. Lusztig, ``Affine Hecke algebras and their graded version,'' J. Amer. Math. Soc. 2 (1989), no. 3, 599-635. MR 991016 (90e:16049)
  • 19. I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. MR 1354144 (96h:05207)
  • 20. A. Mathas, ``The representation theory of the Ariki-Koike and cyclotomic $ q$-Schur algebras,'' Representation Theory of Algebraic Groups and Quantum Groups, 17-25, Adv. Stud. Pure Math. 40, Math. Soc. Japan, Tokyo, 2004. MR 2074597 (2005f:20014)
  • 21. P. Orlik and H. Terao, Arrangements of Hyperplanes, Grundlehren der Mathematischen Wissenschaften 300, Springer-Verlag, Berlin, 1992. MR 1217488 (94e:52014)
  • 22. A. Ram and A. V. Shepler, ``Classification of graded Hecke algebras for complex reflection groups,'' Comment. Math. Helv. 78 (2003), 308-334. MR 1988199 (2004d:20007)
  • 23. G. C. Shephard and J. A. Todd, ``Finite unitary reflection groups,'' Canad. J. Math. 6 (1954), 274-304. MR 0059914 (15:600b)
  • 24. D. Ştefan, ``Hochschild cohomology on Hopf Galois extensions,'' J. Pure Appl. Algebra 103 (1995), 221-233. MR 1358765 (96h:16013)
  • 25. C. A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Adv. Math. 38, Cambridge Univ. Press, Cambridge, 1994. MR 1269324 (95f:18001)
  • 26. S. Witherspoon, ``Skew derivations and deformations of a family of group crossed products,'' Commun. Algebra 34 (2006), no. 11, 4187-4206. MR 2267580 (2007j:16016)
  • 27. S. Witherspoon, ``Twisted graded Hecke algebras,'' J. Algebra 317 (2007), 30-42.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 16E40, 16S80

Retrieve articles in all journals with MSC (2000): 16E40, 16S80

Additional Information

Anne V. Shepler
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203

Sarah Witherspoon
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Keywords: Graded Hecke algebra, degenerate affine Hecke algebra, deformation, Hochschild cohomology, reflection group, hyperplane arrangement, Ariki-Koike algebra
Received by editor(s): January 20, 2006
Published electronically: March 20, 2008
Additional Notes: The first author was partially supported by NSF grant #DMS-0402819
The second author was partially supported by NSF grant #DMS-0443476 and the Alexander von Humboldt Foundation
Dedicated: We dedicate this article to Sergey Yuzvinsky on the occasion of his 70th birthday.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society