On modular Harish-Chandra series of finite unitary groups
Author:
Emily Norton
Journal:
Represent. Theory 24 (2020), 483-524
MSC (2010):
Primary 20C33; Secondary 17B65, 05E10, 20C20
DOI:
https://doi.org/10.1090/ert/549
Published electronically:
October 7, 2020
MathSciNet review:
4159154
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In the modular representation theory of finite unitary groups when the characteristic $\ell$ of the ground field is a unitary prime, the $\widehat {\mathfrak {sl}}_e$-crystal on level $2$ Fock spaces graphically describes the Harish-Chandra branching of unipotent representations restricted to the tower of unitary groups. However, how to determine the cuspidal support of an arbitrary unipotent representation has remained an open question. We show that for $\ell$ sufficiently large, the $\mathfrak {sl}_\infty$-crystal on the same level $2$ Fock spaces provides the remaining piece of the puzzle for the full Harish-Chandra branching rule.
- Bernd Ackermann, The Loewy series of the Steinberg-PIM of finite general linear groups, Proc. London Math. Soc. (3) 92 (2006), no. 1, 62–98. MR 2192385, DOI https://doi.org/10.1017/S0024611505015443
- Olivier Brunat, Olivier Dudas, and Jay Taylor, Unitriangular shape of decomposition matrices of unipotent blocks, Ann. of Math. (2) 192 (2020), no. 2, 583–663. MR 4151085, DOI https://doi.org/10.4007/annals.2020.192.2.7
- Jonathan Brundan, Alistair Savage, and Ben Webster, On the definition of quantum Heisenberg category, Algebra Number Theory 14 (2020), no. 2, 275–321. MR 4104410, DOI https://doi.org/10.2140/ant.2020.14.275
- Jonathan Brundan, Alistair Savage, and Ben Webster, Heisenberg and Kac-Moody categorification, arXiv:1907.11988 (2019).
- Joseph Chuang and Raphaël Rouquier, Derived equivalences for symmetric groups and $\mathfrak {sl}_2$-categorification, Ann. of Math. (2) 167 (2008), no. 1, 245–298. MR 2373155, DOI https://doi.org/10.4007/annals.2008.167.245
- François Digne and Jean Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. MR 1118841
- Richard Dipper and Jie Du, Harish-Chandra vertices, J. Reine Angew. Math. 437 (1993), 101–130. MR 1212254, DOI https://doi.org/10.1515/crll.1993.437.101
- Richard Dipper and Jie Du, Harish-Chandra vertices and Steinberg’s tensor product theorems for finite general linear groups, Proc. London Math. Soc. (3) 75 (1997), no. 3, 559–599. MR 1466661, DOI https://doi.org/10.1112/S0024611597000439
- Richard Dipper and Jochen Gruber, Generalized $q$-Schur algebras and modular representation theory of finite groups with split $(BN)$-pairs, J. Reine Angew. Math. 511 (1999), 145–191. MR 1695794, DOI https://doi.org/10.1515/crll.1999.511.145
- Olivier Dudas and Nicolas Jacon, Alvis-Curtis duality for finite general linear groups and a generalized Mullineux involution, SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 007, 18. MR 3755684, DOI https://doi.org/10.3842/SIGMA.2018.007
- Olivier Dudas and Gunter Malle, Decomposition matrices for low-rank unitary groups, Proc. Lond. Math. Soc. (3) 110 (2015), no. 6, 1517–1557. MR 3356813, DOI https://doi.org/10.1112/plms/pdv008
- Olivier Dudas and Gunter Malle, Modular irreducibility of cuspidal unipotent characters, Invent. Math. 211 (2018), no. 2, 579–589. MR 3748314, DOI https://doi.org/10.1007/s00222-017-0753-1
- Olivier Dudas and Jean Michel, Lectures on finite reductive groups and their representations, https://webusers.imj-prg.fr/~jean.michel/papiers/lectures_{b}eijing_{2}015.pdf, 2015.
- Olivier Dudas and Raphäel Rouquier, Macdonald polynomials and decomposition numbers for finite unitary groups, Oberwolfach Report 52/2018, Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, Germany, 2018.
- O. Dudas, M. Varagnolo, and E. Vasserot, Categorical actions on unipotent representations of finite unitary groups, Publ. Math. Inst. Hautes Études Sci. 129 (2019), 129–197. MR 3949029, DOI https://doi.org/10.1007/s10240-019-00104-x
- Olivier Dudas, Michela Varagnolo, and Eric Vasserot, Categorical actions on unipotent representations of finite classical groups, Categorification and higher representation theory, Contemp. Math., vol. 683, Amer. Math. Soc., Providence, RI, 2017, pp. 41–104. MR 3611711, DOI https://doi.org/10.1090/conm/683
- Naoya Enomoto, Composition factors of polynomial representation of DAHA and $q$-decomposition numbers, J. Math. Kyoto Univ. 49 (2009), no. 3, 441–473. MR 2583598, DOI https://doi.org/10.1215/kjm/1260975035
- Omar Foda, Bernard Leclerc, Masato Okado, Jean-Yves Thibon, and Trevor A. Welsh, Branching functions of $A^{(1)}_{n-1}$ and Jantzen-Seitz problem for Ariki-Koike algebras, Adv. Math. 141 (1999), no. 2, 322–365. MR 1671762, DOI https://doi.org/10.1006/aima.1998.1783
- O. Foda and T. A. Welsh, Cylindric partitions, $\mathcal W_r$ characters and the Andrews-Gordon-Bressoud identities, J. Phys. A 49 (2016), no. 16, 164004, 37. MR 3491322, DOI https://doi.org/10.1088/1751-8113/49/16/164004
- Paul Fong and Bhama Srinivasan, The blocks of finite general linear and unitary groups, Invent. Math. 69 (1982), no. 1, 109–153. MR 671655, DOI https://doi.org/10.1007/BF01389188
- Meinolf Geck, On the decomposition numbers of the finite unitary groups in nondefining characteristic, Math. Z. 207 (1991), no. 1, 83–89. MR 1106814, DOI https://doi.org/10.1007/BF02571376
- Meinolf Geck, On the modular composition factors of the Steinberg representation, J. Algebra 475 (2017), 370–391. MR 3612476, DOI https://doi.org/10.1016/j.jalgebra.2015.11.005
- Meinolf Geck, Gerhard Hiss, and Gunter Malle, Cuspidal unipotent Brauer characters, J. Algebra 168 (1994), no. 1, 182–220. MR 1289097, DOI https://doi.org/10.1006/jabr.1994.1226
- Meinolf Geck, Gerhard Hiss, and Gunter Malle, Towards a classification of the irreducible representations in non-describing characteristic of a finite group of Lie type, Math. Z. 221 (1996), no. 3, 353–386. MR 1381586, DOI https://doi.org/10.1007/PL00004253
- Thomas Gerber, Cylindric multipartitions and level-rank duality, arXiv:1809.09519, (2018).
- Thomas Gerber, Triple crystal action in Fock spaces, Adv. Math. 329 (2018), 916–954. MR 3783431, DOI https://doi.org/10.1016/j.aim.2018.02.030
- Thomas Gerber, Heisenberg algebra, wedges and crystals, J. Algebraic Combin. 49 (2019), no. 1, 99–124. MR 3908358, DOI https://doi.org/10.1007/s10801-018-0820-8
- Thomas Gerber and Gerhard Hiss, Branching graphs for finite unitary groups in nondefining characteristic, Comm. Algebra 45 (2017), no. 2, 561–574. MR 3562522, DOI https://doi.org/10.1080/00927872.2016.1175610
- Thomas Gerber, Gerhard Hiss, and Nicolas Jacon, Harish-Chandra series in finite unitary groups and crystal graphs, Int. Math. Res. Not. IMRN 22 (2015), 12206–12250. MR 3456719, DOI https://doi.org/10.1093/imrn/rnv058
- Thomas Gerber, Nicolas Jacon, and Emily Norton, Generalized Mullineux involution and perverse equivalences, arXiv:1808.06087 (2018).
- Thomas Gerber and Emily Norton, The $\mathfrak {sl}_\infty $-crystal combinatorics of higher level Fock spaces, J. Comb. Algebra 2 (2018), no. 2, 103–145. MR 3802250, DOI https://doi.org/10.4171/JCA/2-2-1
- I. G. Gordon, Quiver varieties, category $\scr O$ for rational Cherednik algebras, and Hecke algebras, Int. Math. Res. Pap. IMRP 3 (2008), Art. ID rpn006, 69. MR 2457847
- Jochen Gruber, Green vertex theory, Green correspondence, and Harish-Chandra induction, J. Algebra 186 (1996), no. 2, 476–521. MR 1423273, DOI https://doi.org/10.1006/jabr.1996.0384
- Jochen Gruber and Gerhard Hiss, Decomposition numbers of finite classical groups for linear primes, J. Reine Angew. Math. 485 (1997), 55–91. MR 1442189, DOI https://doi.org/10.1515/crll.1997.485.55
- Harish-Chandra, Eisenstein series over finite fields, Functional analysis and related fields (Proc. Conf. M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York, 1970, pp. 76–88. MR 0457579
- Gerhard Hiss, Harish-Chandra series of Brauer characters in a finite group with a split $BN$-pair, J. London Math. Soc. (2) 48 (1993), no. 2, 219–228. MR 1231711, DOI https://doi.org/10.1112/jlms/s2-48.2.219
- R. B. Howlett and G. I. Lehrer, Induced cuspidal representations and generalised Hecke rings, Invent. Math. 58 (1980), no. 1, 37–64. MR 570873, DOI https://doi.org/10.1007/BF01402273
- R. B. Howlett and G. I. Lehrer, On Harish-Chandra induction and restriction for modules of Levi subgroups, J. Algebra 165 (1994), no. 1, 172–183. MR 1272585, DOI https://doi.org/10.1006/jabr.1994.1104
- Nicolas Jacon, Kleshchev multipartitions and extended Young diagrams, Adv. Math. 339 (2018), 367–403. MR 3866901, DOI https://doi.org/10.1016/j.aim.2018.09.038
- Nicolas Jacon and Cédric Lecouvey, A combinatorial decomposition of higher level Fock spaces, Osaka J. Math. 50 (2013), no. 4, 897–920. MR 3161420
- N. Jacon and C. Lecouvey, Crystal isomorphisms and wall crossing maps for rational Cherednik algebras, Transform. Groups 23 (2018), no. 1, 101–117. MR 3763943, DOI https://doi.org/10.1007/s00031-016-9402-9
- G. D. James, Some combinatorial results involving Young diagrams, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 1, 1–10. MR 463280, DOI https://doi.org/10.1017/S0305004100054220
- Ivan Losev, Supports of simple modules in cyclotomic Cherednik categories O, arXiv:1509.00526 (2015).
- G. Lusztig, Irreducible representations of finite classical groups, Invent. Math. 43 (1977), no. 2, 125–175. MR 463275, DOI https://doi.org/10.1007/BF01390002
- George Lusztig and Bhama Srinivasan, The characters of the finite unitary groups, J. Algebra 49 (1977), no. 1, 167–171. MR 453886, DOI https://doi.org/10.1016/0021-8693%2877%2990277-0
- Gunter Malle and Donna Testerman, Linear algebraic groups and finite groups of Lie type, Cambridge Studies in Advanced Mathematics, vol. 133, Cambridge University Press, Cambridge, 2011. MR 2850737
- Tomasz Przeździecki, The combinatorics of $\Bbb C^\ast $-fixed points in generalized Calogero-Moser spaces and Hilbert schemes, J. Algebra 556 (2020), 936–992. MR 4089565, DOI https://doi.org/10.1016/j.jalgebra.2020.04.003
- Peng Shan, Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 1, 147–182 (English, with English and French summaries). MR 2760196, DOI https://doi.org/10.24033/asens.2141
- P. Shan and E. Vasserot, Heisenberg algebras and rational double affine Hecke algebras, J. Amer. Math. Soc. 25 (2012), no. 4, 959–1031. MR 2947944, DOI https://doi.org/10.1090/S0894-0347-2012-00738-3
- William A. Stein et al, Sage Mathematics Software (Version 8.9). The Sage Development Team, 2019. http://www.sagemath.org.
- Geordie Williamson, Schubert calculus and torsion explosion, J. Amer. Math. Soc. 30 (2017), no. 4, 1023–1046. With a joint appendix with Alex Kontorovich and Peter J. McNamara. MR 3671935, DOI https://doi.org/10.1090/jams/868
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Additional Information
Emily Norton
Affiliation:
Department of Mathematics, TU Kaiserslautern, Gottlieb-Daimler-Strasse 48, 67663 Kaiserslautern, Germany
Email:
norton@mathematik.uni-kl.de
Received by editor(s):
November 4, 2019
Received by editor(s) in revised form:
May 29, 2020
Published electronically:
October 7, 2020
Additional Notes:
During the first two weeks of work on this paper, the author was supported by Max Planck Institute for Mathematics, Bonn. The rest of the time the author was supported at TU Kaiserslautern by the grant SFB-TRR 195
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