Electronic Only Electronic Research Announcements
Representation Theory
ISSN 1088-4165
Previous volume | Table of contents | Next volume
Articles in press | Most recent volume | All volumes
Previous Article | Next Article
     

Unitary representations of rational Cherednik algebras

Author(s): Pavel Etingof; Emanuel Stoica; with an appendix by Stephen Griffeth
Journal: Represent. Theory 13 (2009), 349-370.
MSC (2000): Primary 16S99
Posted: August 18, 2009
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We study unitarity of lowest weight irreducible representations of rational Cherednik algebras. We prove several general results, and use them to determine which lowest weight representations are unitary in a number of cases.

In particular, in type A, we give a full description of the unitarity locus (justified in Subsection 5.1 and the appendix written by S. Griffeth), and resolve a question by Cherednik on the unitarity of the irreducible subrepresentation of the polynomial representation. Also, as a by-product, we establish Kasatani's conjecture in full generality (the previous proof by Enomoto assumes that the parameter $ c$ is not a half-integer).

References:

[BE]
Roman Bezrukavnikov, Pavel Etingof, Parabolic induction and restriction functors for rational Cherednik algebras, arXiv:0803.3639.

[BEG]
Yu. Berest, P. Etingof, V. Ginzburg, Finite dimensional representations of rational Cherednik algebras, Int. Math. Res. Not. 19 (2003), 1053-1088. MR 1961261 (2004h:16027)

[CEE]
D. Calaque, B. Enriquez, P. Etingof, Universal KZB equations I: the elliptic case, arXiv:math/0702670.

[Ch1]
I. Cherednik, Towards harmonic analysis for DAHA: integral formulas for canonical traces, talk, www-math.mit.edu/˜etingof/hadaha.pdf.

[Ch2]
I. Cherednik, Non-semisimple Macdonald polynomials, arXiv:0709.1742.

[Ch3]
I. Cherednik, Double affine Hecke algebras. London Mathematical Society Lecture Note Series, 319. Cambridge University Press, Cambridge, 2005. MR 2133033 (2007e:32012)

[Chm]
Chmutova, Tatyana, Representations of the rational Cherednik algebras of dihedral type. J. Algebra 297 (2006), no. 2, 542-565. MR 2209274 (2006m:16038)

[DJ1]
R. Dipper and G. D. James, Representations of Hecke algebras of the general linear groups, Proc. London Math. Soc. 52 (1986), 20-52. MR 812444 (88b:20065)

[DJ2]
Richard Dipper, Gordon James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 54 (1987), no. 1, 57-82. MR 872250 (88m:20084)

[DJO]
C.F. Dunkl, M.F.E. de Jeu and E.M. Opdam, Singular polynomials for finite reflection groups, Trans. Amer. Math. Soc. 346 (1994), 237-256. MR 1273532 (96b:33012)

[Du]
C. Dunkl, Singular polynomials and modules for the symmetric groups, math/0501494, Int. Math. Res. Not. 2005, v. 39, 2409-2436. MR 2181357 (2006j:33012)

[Du2]
C. Dunkl, Integral kernels with reflection group invariance, Can. J. Math. 43 (1991), 1213-1227. MR 1145585 (93g:33012)

[EG]
P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math., vol. 147, (2002), no. 2, 243-348. MR 1881922 (2003b:16021)

[E1]
P. Etingof, Calogero-Moser Systems and Representation Theory, Zurich lectures in advanced mathematics European Mathematical Society, 2007. MR 2296754

[En]
N. Enomoto, Composition factors of polynomial representation of DAHA and crystallized decomposition numbers, math.RT/0604369.

[FJMM]
B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials. MR 1962014 (2004g:05149)

[Gri1]
S. Griffeth, Towards a combinatorial representation theory for the rational Cherednik algebra of type $ G(r,p,n)$., to appear in Proceedings of the Edinburgh Mathematical Society. arXiv:math/0612733

[Gri2]
S. Griffeth, Orthogonal functions generalizing Jack polynomials, arXiv:0707.0251

[GGOR]
V. Ginzburg, N. Guay, E. Opdam, R. Rouquier, On the category $ {\mathcal O}$ for rational Cherednik algebras. Invent. Math. 154 (2003), no. 3, 617-651. MR 2018786 (2005f:20010)

[Ka]
M. Kasatani, Subrepresentations in the polynomial representation of the double affine Hecke algebra of type $ GL_n$ at $ t^{k+1}q^{r-1}=1$, Int. Math. Res. Notices, 2005, no. 28, 1717-1742. MR 2172339 (2007c:20012)

[Op]
E. M. Opdam, Some applications of hypergeometric shift operators. Invent. Math. 98 (1989), no. 1, 1-18. MR 1010152 (91h:33024)

[Su]
T. Suzuki, Cylindrical Combinatorics and Representations of Cherednik Algebras of type A, arXiv:math/0610029.

Similar Articles:

Retrieve articles in Representation Theory with MSC (2000): 16S99

Retrieve articles in all Journals with MSC (2000): 16S99

Additional Information:

Pavel Etingof
Affiliation: Department of Mathematics, Room 2-176, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email: etingof@math.mit.edu

Emanuel Stoica
Affiliation: Department of Mathematics, Room 2-089, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email: immanuel@math.mit.edu

Stephen Griffeth
Affiliation: School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, Minnesota 55455
Email: griffeth@math.umn.edu

DOI: 10.1090/S1088-4165-09-00356-2
PII: S 1088-4165(09)00356-2
Received by editor(s): May 5, 2009
Received by editor(s) in revised form: June 12, 2009
Posted: August 18, 2009
Copyright of article: Copyright 2009, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google