Orthogonal functions generalizing Jack polynomials
Author:
Stephen Griffeth
Journal:
Trans. Amer. Math. Soc. 362 (2010), 61316157
MSC (2010):
Primary 05E05, 05E10, 05E15, 16S35, 20C30; Secondary 16D90, 16S38, 16T30
Published electronically:
June 21, 2010
MathSciNet review:
2661511
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Abstract: The rational Cherednik algebra is a certain algebra of differentialreflection operators attached to a complex reflection group and depending on a set of central parameters. Each irreducible representation of corresponds to a standard module for . This paper deals with the infinite family of complex reflection groups; our goal is to study the standard modules using a commutative subalgebra of discovered by Dunkl and Opdam. In this case, the irreducible modules are indexed by certain sequences of partitions. We first show that acts in an upper triangular fashion on each standard module , with eigenvalues determined by the combinatorics of the set of standard tableaux on . As a consequence, we construct a basis for consisting of orthogonal functions on with values in the representation . For with these functions are the nonsymmetric Jack polynomials. We use intertwining operators to deduce a norm formula for our orthogonal functions and give an explicit combinatorial description of the lattice of submodules of in the case in which the orthogonal functions are all welldefined. A consequence of our results is the construction of a number of interesting finite dimensional modules with intricate structure. Finally, we show that for a certain choice of parameters there is a cyclic group of automorphisms of so that the rational Cherednik algebra for is the fixed subalgebra. Our results therefore descend to the rational Cherednik algebra for by Clifford theory.
 [Ari]
Susumu
Ariki, Representation theory of a Hecke algebra of
𝐺(𝑟,𝑝,𝑛), J. Algebra
177 (1995), no. 1, 164–185. MR 1356366
(96j:20021), http://dx.doi.org/10.1006/jabr.1995.1292
 [ArKo]
Susumu
Ariki and Kazuhiko
Koike, A Hecke algebra of
(𝑍/𝑟𝑍)≀𝔖_{𝔫} and
construction of its irreducible representations, Adv. Math.
106 (1994), no. 2, 216–243. MR 1279219
(95h:20006), http://dx.doi.org/10.1006/aima.1994.1057
 [BEG]
Yuri
Berest, Pavel
Etingof, and Victor
Ginzburg, Finitedimensional representations of rational Cherednik
algebras, Int. Math. Res. Not. 19 (2003),
1053–1088. MR 1961261
(2004h:16027), http://dx.doi.org/10.1155/S1073792803210205
 [Che1]
I.
V. Cherednik, Calculation of the monodromy of some
𝑊invariant local systems of type 𝐵,𝐶 and
𝐷, Funktsional. Anal. i Prilozhen. 24 (1990),
no. 1, 88–89 (Russian); English transl., Funct. Anal. Appl.
24 (1990), no. 1, 78–79. MR 1052280
(91i:17019), http://dx.doi.org/10.1007/BF01077930
 [Che2]
Ivan
Cherednik, Double affine Hecke algebras and difference Fourier
transforms, Invent. Math. 152 (2003), no. 2,
213–303. MR 1974888
(2005h:20005), http://dx.doi.org/10.1007/s0022200202400
 [Chm]
Tatyana
Chmutova, Representations of the rational Cherednik algebras of
dihedral type, J. Algebra 297 (2006), no. 2,
542–565. MR 2209274
(2006m:16038), http://dx.doi.org/10.1016/j.jalgebra.2005.12.024
 [Dez]
C. Dezélée, Generalized graded Hecke algebra for complex reflection group of type , arXiv:math.RT/0605410v2.
 [Dri]
V.
G. Drinfel′d, Degenerate affine Hecke algebras and
Yangians, Funktsional. Anal. i Prilozhen. 20 (1986),
no. 1, 69–70 (Russian). MR 831053
(87m:22044)
 [DuOp]
C.
F. Dunkl and E.
M. Opdam, Dunkl operators for complex reflection groups, Proc.
London Math. Soc. (3) 86 (2003), no. 1, 70–108.
MR
1971464 (2004d:20040), http://dx.doi.org/10.1112/S0024611502013825
 [Dun1]
Charles
F. Dunkl, Singular polynomials and modules for the symmetric
groups, Int. Math. Res. Not. 39 (2005),
2409–2436. MR 2181357
(2006j:33012), http://dx.doi.org/10.1155/IMRN.2005.2409
 [Dun2]
Charles
F. Dunkl, Singular polynomials for the symmetric groups, Int.
Math. Res. Not. 67 (2004), 3607–3635. MR 2129695
(2006k:20022), http://dx.doi.org/10.1155/S1073792804140610
 [EtGi]
Pavel
Etingof and Victor
Ginzburg, Symplectic reflection algebras, CalogeroMoser space, and
deformed HarishChandra homomorphism, Invent. Math.
147 (2002), no. 2, 243–348. MR 1881922
(2003b:16021), http://dx.doi.org/10.1007/s002220100171
 [EtMo]
Pavel
Etingof and Silvia
Montarani, Finite dimensional representations of
symplectic reflection algebras associated to wreath products, Represent. Theory 9 (2005), 457–467
(electronic). MR
2167902 (2007d:16024), http://dx.doi.org/10.1090/S1088416505002888
 [Gan]
Wee
Liang Gan, Reflection functors and symplectic reflection algebras
for wreath products, Adv. Math. 205 (2006),
no. 2, 599–630. MR 2258267
(2007g:16022), http://dx.doi.org/10.1016/j.aim.2005.08.002
 [GGOR]
Victor
Ginzburg, Nicolas
Guay, Eric
Opdam, and Raphaël
Rouquier, On the category 𝒪 for rational Cherednik
algebras, Invent. Math. 154 (2003), no. 3,
617–651. MR 2018786
(2005f:20010), http://dx.doi.org/10.1007/s0022200303138
 [Gor]
Iain
Gordon, On the quotient ring by diagonal invariants, Invent.
Math. 153 (2003), no. 3, 503–518. MR 2000467
(2004f:20075), http://dx.doi.org/10.1007/s0022200302965
 [Gor2]
I.
G. Gordon, Quiver varieties, category 𝒪 for rational
Cherednik algebras, and Hecke algebras, Int. Math. Res. Pap. IMRP
3 (2008), Art. ID rpn006, 69. MR 2457847
(2010c:16032)
 [Gri1]
S. Griffeth, Rational Cherednik algebras and coinvariant rings, Ph.D. thesis, University of Wisconsin, Madison, Madison, WI 53704, August 2006.
 [Gri2]
S. Griffeth, Towards a combinatorial representation theory for the national Cherednik algebra of type , to appear in Proceedings of the Edinburgh Mathematical Society, arXiv:math/0612733.
 [Gri3]
S. Griffeth, The complex representations of , http://www.math.umn.edu/ griffeth/notes/WreathProducts.pdf.
 [JaKe]
Gordon
James and Adalbert
Kerber, The representation theory of the symmetric group,
Encyclopedia of Mathematics and its Applications, vol. 16,
AddisonWesley Publishing Co., Reading, Mass., 1981. With a foreword by P.
M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
(83k:20003)
 [KnSa]
Friedrich
Knop and Siddhartha
Sahi, A recursion and a combinatorial formula for Jack
polynomials, Invent. Math. 128 (1997), no. 1,
9–22. MR
1437493 (98k:33040), http://dx.doi.org/10.1007/s002220050134
 [Mon]
Silvia
Montarani, On some finite dimensional representations of symplectic
reflection algebras associated to wreath products, Comm. Algebra
35 (2007), no. 5, 1449–1467. MR 2317620
(2009h:16015), http://dx.doi.org/10.1080/00927870601168855
 [OkVe]
Andrei
Okounkov and Anatoly
Vershik, A new approach to representation theory of symmetric
groups, Selecta Math. (N.S.) 2 (1996), no. 4,
581–605. MR 1443185
(99g:20024), http://dx.doi.org/10.1007/PL00001384
 [Opd]
Eric
M. Opdam, Harmonic analysis for certain representations of graded
Hecke algebras, Acta Math. 175 (1995), no. 1,
75–121. MR
1353018 (98f:33025), http://dx.doi.org/10.1007/BF02392487
 [Ram]
A. Ram, The wreath products , http://www.math.wisc.edu/ ram/Notes2005/ GH1k7.22.05.pdf.
 [RaRa]
Arun
Ram and Jacqui
Ramagge, Affine Hecke algebras, cyclotomic Hecke algebras and
Clifford theory, A tribute to C. S. Seshadri (Chennai, 2002) Trends
Math., Birkhäuser, Basel, 2003, pp. 428–466. MR 2017596
(2004i:20009)
 [RaSh]
Arun
Ram and Anne
V. Shepler, Classification of graded Hecke algebras for complex
reflection groups, Comment. Math. Helv. 78 (2003),
no. 2, 308–334. MR 1988199
(2004d:20007), http://dx.doi.org/10.1007/s000140300013
 [Rou]
Raphaël
Rouquier, 𝑞Schur algebras and complex reflection
groups, Mosc. Math. J. 8 (2008), no. 1,
119–158, 184 (English, with English and Russian summaries). MR 2422270
(2010b:20081)
 [Suz]
T. Suzuki, Cylindrical combinatorics and representations of Cherednik algebras of type A, arXiv:math/0610029.
 [Val]
Richard
Vale, Rational Cherednik algebras and diagonal coinvariants of
𝐺(𝑚,𝑝,𝑛), J. Algebra
311 (2007), no. 1, 231–250. MR 2309886
(2008c:20077), http://dx.doi.org/10.1016/j.jalgebra.2006.12.017
 [Ari]
 S. Ariki, Representation theory of a Hecke algebra of , J. Algebra 177 (1995), no. 1, 164185. MR 1356366 (96j:20021)
 [ArKo]
 S. Ariki and K. Koike, A Hecke algebra of and construction of its irreducible representations, Adv. Math. 106 (1994), no. 2, 216243. MR 1279219 (95h:20006)
 [BEG]
 Y. Berest, P. Etingof, and V. Ginzburg, Finitedimensional representations of rational Cherednik algebras, Int. Math. Res. Not. (2003), no. 19, 10531088. MR 1961261 (2004h:16027)
 [Che1]
 I. Cherednik, Calculation of the monodromy of some invariant local systems of type and . (Russian) Funktsional. Anal. i Prilozhen. 24 (1990), no. 1, 8889; translation in Funct. Anal. Appl. 24 (1990), no. 1, 7879 MR 1052280 (91i:17019)
 [Che2]
 I. Cherednik, Double affine Hecke algebra and difference Fourier transforms, Invent. Math. 152 (2003), no. 2, 213303. MR 1974888 (2005h:20005)
 [Chm]
 T. Chmutova, Representations of the rational Cherednik algebras of dihedral type. J. Algebra 297 (2006), no. 2, 542565. MR 2209274 (2006m:16038)
 [Dez]
 C. Dezélée, Generalized graded Hecke algebra for complex reflection group of type , arXiv:math.RT/0605410v2.
 [Dri]
 V. G. Drinfel'd, Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 6970. MR 831053 (87m:22044)
 [DuOp]
 C. F. Dunkl and E. M. Opdam, Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), no. 1, 70108. MR 1971464 (2004d:20040)
 [Dun1]
 C.F. Dunkl, Singular polynomials and modules for the symmetric groups. Int. Math. Res. Not. 2005, no. 39, 24092436. MR 2181357 (2006j:33012)
 [Dun2]
 C.F. Dunkl, Singular polynomials for the symmetric groups. Int. Math. Res. Not. 2004, no. 67, 36073635. MR 2129695 (2006k:20022)
 [EtGi]
 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)
 [EtMo]
 P. Etingof and S. Montarani, Finite dimensional representations of symplectic reflection algebras associated to wreath products, Represent. Theory 9 (2005), 457467 (electronic). MR 2167902 (2007d:16024)
 [Gan]
 W.L. Gan, Reflection functors and symplectic reflection algebras for wreath products, Adv. Math. 205 (2006), no. 2, 599630. MR 2258267 (2007g:16022)
 [GGOR]
 V. Ginzburg, N. Guay, E. Opdam, and R. Rouquier, On the category for rational Cherednik algebras, Invent. Math. 154 (2003), no. 3, 617651. MR 2018786 (2005f:20010)
 [Gor]
 I. Gordon, On the quotient ring by diagonal invariants, Invent. Math. 153 (2003), no. 3, 503518. MR 2000467 (2004f:20075)
 [Gor2]
 I. Gordon, Quiver varieties, category O for rational Cherednik algebras, and Hecke algebras, arXiv:math/0703150. MR 2457847
 [Gri1]
 S. Griffeth, Rational Cherednik algebras and coinvariant rings, Ph.D. thesis, University of Wisconsin, Madison, Madison, WI 53704, August 2006.
 [Gri2]
 S. Griffeth, Towards a combinatorial representation theory for the national Cherednik algebra of type , to appear in Proceedings of the Edinburgh Mathematical Society, arXiv:math/0612733.
 [Gri3]
 S. Griffeth, The complex representations of , http://www.math.umn.edu/ griffeth/notes/WreathProducts.pdf.
 [JaKe]
 G. James and A. Kerber, The representation theory of the symmetric group, With a foreword by P. M. Cohn. With an introduction by Gilbert de B. Robinson. Encyclopedia of Mathematics and its Applications, 16. AddisonWesley Publishing Co., Reading, Mass., 1981. MR 644144 (83k:20003)
 [KnSa]
 F. Knop and S. Sahi, A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), no. 1, 922. MR 1437493 (98k:33040)
 [Mon]
 S. Montarani, On some finite dimensional representations of symplectic reflection algebras associated to wreath products, Comm. Algebra 35 (2007), no. 5, 14491467. MR 2317620 (2009h:16015)
 [OkVe]
 A. Okounkov and A. Vershik, A new approach to representation theory of symmetric groups. Selecta Math. (N.S.) 2 (1996), no. 4, 581605. MR 1443185 (99g:20024)
 [Opd]
 E.M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math., 175 (1995), 75121. MR 1353018 (98f:33025)
 [Ram]
 A. Ram, The wreath products , http://www.math.wisc.edu/ ram/Notes2005/ GH1k7.22.05.pdf.
 [RaRa]
 A. Ram and J. Ramagge, Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory. A tribute to C. S. Seshadri (Chennai, 2002), 428466, Trends Math., Birkhäuser, Basel, 2003. MR 2017596 (2004i:20009)
 [RaSh]
 A. Ram and A.V. Shepler, Classification of graded Hecke algebras for complex reflection groups, Comment. Math. Helv. 78 (2003), no. 2, 308334. MR 1988199 (2004d:20007)
 [Rou]
 R. Rouquier, Schur algebras and complex reflection groups, Mosc. Math. J. 8 (2008), no. 1, 119158, 184. MR 2422270
 [Suz]
 T. Suzuki, Cylindrical combinatorics and representations of Cherednik algebras of type A, arXiv:math/0610029.
 [Val]
 R. Vale, Rational Cherednik algebras and diagonal coinvariants of , J. Algebra 311 (2007), no. 1, 231250. MR 2309886 (2008c:20077)
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Additional Information
Stephen Griffeth
Affiliation:
School of Mathematics, University of Minnesota, 127 Church Street, Minneapolis, Minnesota 55455
Address at time of publication:
School of Mathematics, James Clerk Maxwell Building, University of Edinburgh, Edinburgh, EH9 3JZ, United Kingeom
Email:
griffeth@math.umn.edu, S.Griffeth@ed.ac.uk
DOI:
http://dx.doi.org/10.1090/S000299472010051566
PII:
S 00029947(2010)051566
Received by editor(s):
November 20, 2008
Received by editor(s) in revised form:
July 3, 2009
Published electronically:
June 21, 2010
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
