On modular Harish-Chandra series of finite unitary groups

Norton, Emily

doi:10.1090/ert/549

On modular Harish-Chandra series of finite unitary groups
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by Emily Norton PDF

Represent. Theory 24 (2020), 483-524 Request permission

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.

References

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 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 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 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 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, DOI 10.1017/CBO9781139172417
Richard Dipper and Jie Du, Harish-Chandra vertices, J. Reine Angew. Math. 437 (1993), 101–130. MR 1212254, DOI 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 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 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 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 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 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_beijing_2015.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 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 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 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 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 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 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 10.1007/BF02571376
Meinolf Geck, On the modular composition factors of the Steinberg representation, J. Algebra 475 (2017), 370–391. MR 3612476, DOI 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 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 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 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 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 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 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 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 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 10.1515/crll.1997.485.55
Harish-Chandra, Eisenstein series over finite fields, Functional Analysis and Related Fields (Proc. Conf. for 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 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 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 10.1006/jabr.1994.1104
Nicolas Jacon, Kleshchev multipartitions and extended Young diagrams, Adv. Math. 339 (2018), 367–403. MR 3866901, DOI 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 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 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 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 10.1016/0021-8693(77)90277-0
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, DOI 10.1017/CBO9780511994777
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 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 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 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 10.1090/jams/868