Representations of infinitesimal Cherednik algebras
Authors:
Fengning Ding and Alexander Tsymbaliuk
Journal:
Represent. Theory 17 (2013), 557583
MSC (2010):
Primary 17B10
Published electronically:
November 5, 2013
MathSciNet review:
3123740
Fulltext PDF Free Access
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Abstract: Infinitesimal Cherednik algebras are continuous analogues of rational Cherednik algebras, and in the case of , are deformations of universal enveloping algebras of the Lie algebras . In the first half of this paper, we compute the determinant of the Shapovalov form, enabling us to classify all irreducible finite dimensional representations of . In the second half, we investigate Poissonanalogues of the infinitesimal Cherednik algebras and generalize various results to , including Kostant's theorem.
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 V. G. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 6970 (Russian). MR 0831053 (87m:22044)
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 P. Etingof and V. Ginzburg, Symplectic reflection algebras, CalogeroMoser space, and deformed HarishChandra homomorphism, Invent. Math. 147 (2002), no. 2, 243348. MR 1881922 (2003b:16021), http://dx.doi.org/10.1007/s002220100171
 [EGG]
 P. Etingof, W. L. Gan, and Victor Ginzburg, Continuous Hecke algebras, Transform. Groups 10 (2005), no. 34, 423447. MR 2183119 (2006h:20006), http://dx.doi.org/10.1007/s0003100504042
 [FH]
 W. Fulton and J. Harris, Representation theory. A first course, Graduate Texts in Mathematics, vol. 129, Readings in Mathematics, SpringerVerlag, New York, 1991. MR 1153249 (93a:20069)
 [K]
 H. Kaneta, The invariant polynomial algebras for the groups and , Nagoya Math. J. 94 (1984), 6173. MR 748092 (86f:17012b)
 [KK]
 V. G. Kac and D. A. Kazhdan, Structure of representations with highest weight of infinitedimensional Lie algebras, Adv. in Math. 34 (1979), no. 1, 97108. MR 547842 (81d:17004), http://dx.doi.org/10.1016/00018708(79)900665
 [KT]
 A. Tikaradze and A. Khare, Center and representations of infinitesimal Hecke algebras of , Comm. Algebra 38 (2010), no. 2, 405439. MR 2598890 (2011g:17024), http://dx.doi.org/10.1080/00927870903448740
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 G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599635. MR 0991016 (90e:16049), http://dx.doi.org/10.2307/1990945
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 I. Losev and A. Tsymbaliuk, Infinitesimal Cherednik algebras as Walgebras, arXiv: 1305.6873.
 [S]
 S.P. Smith, A class of algebras similar to the enveloping algebra of , Trans. Amer. Math. Soc. 222 (1990), no. 1, 285314. MR 0972706 (91b:17013)
 [T1]
 A. Tikaradze, Center of infinitesimal Cherednik algebras of , Represent. Theory 14 (2010), 18. MR 2577654 (2011b:16111), http://dx.doi.org/10.1090/S1088416510003638
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 A. Tikaradze, On maximal primitive quotients of infinitesimal Cherednik algebras of , J. Algebra 355 (2012), 171175. MR 2889538, http://dx.doi.org/10.1016/j.jalgebra.2012.01.013
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Additional Information
Fengning Ding
Affiliation:
Phillips Academy, 180 Main St., Andover, Massachusetts 01810
Address at time of publication:
Department of Mathematics, Harvard College, Cambridge, Massachusetts 02138
Email:
fding@college.harvard.edu
Alexander Tsymbaliuk
Affiliation:
Independent University of Moscow, 11 Bol’shoy Vlas’evskiy per., Moscow 119002, Russia
Address at time of publication:
Department of Mathematics, MIT, 77 Massachusetts Ave., Cambridge, Massachusetts 02139
Email:
sasha{\textunderscore}ts@mit.edu
DOI:
http://dx.doi.org/10.1090/S108841652013004430
PII:
S 10884165(2013)004430
Received by editor(s):
October 21, 2012
Received by editor(s) in revised form:
February 26, 2013, March 30, 2013, and July 31, 2013
Published electronically:
November 5, 2013
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
