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Unitriangular shape of decomposition matrices of unipotent blocks. (English) Zbl 07264138
Summary: We show that the decomposition matrix of unipotent \(\ell\)-blocks of a finite reductive group \(\mathbf{G}(\mathbb{F}_q)\) has a unitriangular shape, assuming \(q\) is a power of a good prime and \(\ell\) is very good for \(\mathbf{G}\). This was conjectured by Geck in 1990 as part of his PhD thesis. We establish this result by constructing projective modules using a modification of generalised Gelfand-Graev characters introduced by Kawanaka. We prove that each such character has at most one unipotent constituent which occurs with multiplicity one. This establishes a 30 year old conjecture of Kawanaka.

MSC:
20C33 Representations of finite groups of Lie type
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CHEVIE
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[1] Achar, Pramod N.; Aubert, Anne-Marie, Supports unipotents de faisceaux caract\`eres, J. Inst. Math. Jussieu. Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l’Institut de Math\'{e}matiques de Jussieu, 6, 173-207 (2007) · Zbl 1173.20033
[2] Aschbacher, M., Finite Group Theory, Cambridge Stud. Adv. Math., 10, xii+304 pp. (2000) · Zbl 0997.20001
[3] Barbasch, Dan; Vogan, Jr., David A., Unipotent representations of complex semisimple groups, Ann. of Math. (2). Annals of Mathematics. Second Series, 121, 41-110 (1985) · Zbl 0582.22007
[4] Bonnaf\'{e}, C\'{e}dric, Sur les caract\`eres des groupes r\'{e}ductifs finis \`a centre non connexe: applications aux groupes sp\'{e}ciaux lin\'{e}aires et unitaires, Ast\'{e}risque. Ast\'{e}risque, vi+165 pp. (2006) · Zbl 1157.20022
[5] Bonnaf\'{e}, C\'{e}dric; Dat, Jean-Fran\c{c}ois; Rouquier, Rapha\"{e}l, Derived categories and {D}eligne-{L}usztig varieties {II}, Ann. of Math. (2). Annals of Mathematics. Second Series, 185, 609-670 (2017) · Zbl 06701140
[6] Bonnaf\'{e}, C\'{e}dric; Rouquier, Rapha\"{e}l, Cat\'{e}gories d\'{e}riv\'{e}es et vari\'{e}t\'{e}s de {D}eligne-{L}usztig, Publ. Math. Inst. Hautes \'{E}tudes Sci.. Publications Math\'{e}matiques. Institut de Hautes \'{E}tudes Scientifiques, 1-59 (2003) · Zbl 1054.20024
[7] Borel, Armand, Linear Algebraic Groups, Grad. Texts in Math., 126, xii+288 pp. (1991) · Zbl 0726.20030
[8] Borel, Armand; Tits, Jacques, Groupes r\'{e}ductifs, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 55-150 (1965) · Zbl 0145.17402
[9] Bourbaki, N., \'{E}l\'{e}ments de math\'{e}matique. {F}asc. {XXXIV}. {G}roupes et alg\`ebres de {L}ie. {C}hapitre {IV}: {G}roupes de {C}oxeter et syst\`emes de {T}its. {C}hapitre {V}: {G}roupes engendr\'{e}s par des r\'{e}flexions. {C}hapitre {VI}: syst\`emes de racines, Act. Sci.Indust., 1337, 288 pp. (loose errata) pp. (1968) · Zbl 0186.33001
[10] Brou\'{e}, Michel, Isom\'{e}tries de caract\`eres et \'{e}quivalences de {M}orita ou d\'{e}riv\'{e}es, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 71, 45-63 (1990) · Zbl 0727.20005
[11] Cabanes, Marc; Enguehard, Michel, Representation Theory of Finite Reductive Groups, New Math. Monogr., 1, xviii+436 pp. (2004) · Zbl 1069.20032
[12] Cabanes, Marc; Sp\"{a}th, Britta, Equivariant character correspondences and inductive {M}c{K}ay condition for type {\( \tt A\)}, J. Reine Angew. Math.. Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle’s Journal], 728, 153-194 (2017) · Zbl 1456.20009
[13] Carter, Roger W., Finite Groups of {L}ie type. Conjugacy Classes and Complex Characters, Wiley Classics Library, xii+544 pp. (1993)
[14] Chaneb, R., Basic Sets for Unipotent Blocks of Finite Reductive Groups in Bad Characteristic, Int. Math. Res. Not., 2020, published 20 February 2020, issue rnaa019 (2020)
[15] Cohen, Arjeh M.; Murray, Scott H.; Taylor, D. E., Computing in groups of {L}ie type, Math. Comp.. Mathematics of Computation, 73, 1477-1498 (2004) · Zbl 1062.20049
[16] Collingwood, David H.; McGovern, William M., Nilpotent Orbits in Semisimple {L}ie Algebras, Van Nostrand Reinhold Math. Ser., xiv+186 pp. (1993) · Zbl 0972.17008
[17] Denoncin, David, Stable basic sets for finite special linear and unitary groups, Adv. Math.. Advances in Mathematics, 307, 344-368 (2017) · Zbl 1406.20015
[18] Digne, Fran\c{c}ois; Lehrer, Gustav; Michel, Jean, On character sheaves and characters of reductive groups at unipotent classes, Pure Appl. Math. Q.. Pure and Applied Mathematics Quarterly, 10, 459-512 (2014) · Zbl 1322.20035
[19] Digne, Fran\c{c}ois; Michel, Jean, Representations of Finite Groups of {L}ie Type, London Math. Soc. Stud. Texts, 21, iv+159 pp. (1991) · Zbl 0815.20014
[20] Digne, Fran\c{c}ois; Michel, Jean, Quasi-semisimple elements, Proc. Lond. Math. Soc. (3). Proceedings of the London Mathematical Society. Third Series, 116, 1301-1328 (2018) · Zbl 1429.20030
[21] Dipper, Richard; James, Gordon, The {\(q}-{S\)}chur algebra, Proc. London Math. Soc. (3). Proceedings of the London Mathematical Society. Third Series, 59, 23-50 (1989) · Zbl 0711.20007
[22] Dudas, Olivier; Malle, Gunter, Modular irreducibility of cuspidal unipotent characters, Invent. Math.. Inventiones Mathematicae, 211, 579-589 (2018) · Zbl 1427.20061
[23] Fowler, Russell; R\"{o}hrle, Gerhard, On cocharacters associated to nilpotent elements of reductive groups, Nagoya Math. J.. Nagoya Mathematical Journal, 190, 105-128 (2008) · Zbl 1185.20050
[24] Fu, Baohua; Juteau, Daniel; Levy, Paul; Sommers, Eric, Generic singularities of nilpotent orbit closures, Adv. Math.. Advances in Mathematics, 305, 1-77 (2017) · Zbl 1366.14007
[25] Geck, Meinolf, Verallgemeinerte {G}elfand–{G}raev {C}haraktere und {Z}erlegungszahlen endlicher {G}ruppen vom {L}ie-{T}yp (1990) · Zbl 0729.20005
[26] Geck, Meinolf, On the decomposition numbers of the finite unitary groups in nondefining characteristic, Math. Z.. Mathematische Zeitschrift, 207, 83-89 (1991) · Zbl 0712.20006
[27] Geck, Meinolf, Generalized {G}elfand-{G}raev characters for {S}teinberg’s triality groups and their applications, Comm. Algebra. Communications in Algebra, 19, 3249-3269 (1991) · Zbl 0756.20001
[28] Geck, Meinolf, Basic sets of {B}rauer characters of finite groups of {L}ie type. {II}, J. London Math. Soc. (2). Journal of the London Mathematical Society. Second Series, 47, 255-268 (1993) · Zbl 0797.20013
[29] Geck, Meinolf, Basic sets of {B}rauer characters of finite groups of {L}ie type. {III}, Manuscripta Math.. Manuscripta Mathematica, 85, 195-216 (1994) · Zbl 0820.20018
[30] Geck, Meinolf, Character sheaves and generalized {G}elfand-{G}raev characters, Proc. London Math. Soc. (3). Proceedings of the London Mathematical Society. Third Series, 78, 139-166 (1999) · Zbl 1035.20011
[31] Geck, Meinolf, An Introduction to Algebraic Geometry and Algebraic Groups, Oxf. Grad. Texts Math., 10, xii+307 pp. (2003) · Zbl 1037.14019
[32] Geck, Meinolf, Remarks on modular representations of finite groups of {L}ie type in non-defining characteristic. Algebraic Groups and Quantum Groups, Contemp. Math., 565, 71-80 (2012) · Zbl 1319.20012
[33] Geck, Meinolf, Generalised {G}elfand–{G}raev representations in bad characteristic?, Transform. Groups. Transformation Groups (2020) · Zbl 07239064
[34] Geck, Meinolf; Hiss, Gerhard, Modular representations of finite groups of {L}ie type in non-defining characteristic. Finite Reductive Groups, Progr. Math., 141, 195-249 (1997) · Zbl 0867.20014
[35] Geck, Meinolf; Malle, Gunter, On the existence of a unipotent support for the irreducible characters of a finite group of {L}ie type, Trans. Amer. Math. Soc.. Transactions of the Amer. Math. Soc., 352, 429-456 (2000) · Zbl 0940.20019
[36] Geck, Meinolf; Pfeiffer, G\"{o}tz, Characters of Finite {C}oxeter Groups and {I}wahori-{H}ecke Algebras, London Math. Soc. Monogr., 21, xvi+446 pp. (2000) · Zbl 0996.20004
[37] G\'{e}rardin, Paul, Weil representations associated to finite fields, J. Algebra. Journal of Algebra, 46, 54-101 (1977) · Zbl 0359.20008
[38] Gruber, Jochen; Hiss, Gerhard, Decomposition numbers of finite classical groups for linear primes, J. Reine Angew. Math.. Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle’s Journal], 485, 55-91 (1997) · Zbl 0979.20039
[39] Herpel, Sebastian, On the smoothness of centralizers in reductive groups, Trans. Amer. Math. Soc.. Transactions of the Amer. Math. Soc., 365, 3753-3774 (2013) · Zbl 1298.20057
[40] Hiss, Gerhard, On the decomposition numbers of {\(G_2(q)\)}, J. Algebra. Journal of Algebra, 120, 339-360 (1989) · Zbl 0667.20009
[41] Jantzen, Jens Carsten, Nilpotent orbits in representation theory. Lie Theory, Progr. Math., 228, 1-211 (2004) · Zbl 1169.14319
[42] Juteau, Daniel; Mautner, Carl; Williamson, Geordie, Parity sheaves, J. Amer. Math. Soc.. Journal of the Amer. Math. Soc., 27, 1169-1212 (2014) · Zbl 1344.14017
[43] Kawanaka, N., Generalized {G}el\cprime fand-{G}raev representations of exceptional simple algebraic groups over a finite field. {I}, Invent. Math.. Inventiones Mathematicae, 84, 575-616 (1986) · Zbl 0596.20028
[44] Kawanaka, Noriaki, Shintani lifting and {G}el\cprime fand-{G}raev representations. The {A}rcata {C}onference on {R}epresentations of {F}inite {G}roups, Proc. Sympos. Pure Math., 47, 147-163 (1987)
[45] Lawther, R., Unipotent classes in maximal subgroups of exceptional algebraic groups, J. Algebra. Journal of Algebra, 322, 270-293 (2009) · Zbl 1179.20041
[46] Liebeck, Martin W.; Seitz, Gary M., Unipotent and Nilpotent Classes in Simple Algebraic Groups and {L}ie Aalgebras, Math. Surveys Monogr., 180, xii+380 pp. (2012) · Zbl 1251.20001
[47] Lusztig, George, Characters of Reductive Groups over a Finite Field, Ann. of Math. Stud., 107, xxi+384 pp. (1984) · Zbl 0556.20033
[48] Lusztig, George, Character sheaves. {III}, Adv. in Math.. Advances in Mathematics, 57, 266-315 (1985) · Zbl 0594.20031
[49] Lusztig, George, Character sheaves. {V}, Adv. in Math.. Advances in Mathematics, 61, 103-155 (1986) · Zbl 0602.20036
[50] Lusztig, George, Green functions and character sheaves, Ann. of Math. (2). Annals of Mathematics. Second Series, 131, 355-408 (1990) · Zbl 0695.20024
[51] Lusztig, George, A unipotent support for irreducible representations, Adv. Math.. Advances in Mathematics, 94, 139-179 (1992) · Zbl 0789.20042
[52] Lusztig, George, Families and {S}pringer’s correspondence, Pacific J. Math.. Pacific Journal of Mathematics, 267, 431-450 (2014) · Zbl 1301.20035
[53] Lusztig, George, Restriction of a character sheaf to conjugacy classes, Bull. Math. Soc. Sci. Math. Roumanie (N.S.). Bulletin Math\'{e}matique de la Soci\'{e}t\'{e} des Sciences Math\'{e}matiques de Roumanie. Nouvelle S\'{e}rie, 58(106), 297-309 (2015) · Zbl 1399.20051
[54] Malle, Gunter; Testerman, Donna, Linear Algebraic Groups and Finite Groups of {L}ie Type, Cambridge Stud. Adv. Math., 133, xiv+309 pp. (2011) · Zbl 1256.20045
[55] McNinch, George J., Nilpotent orbits over ground fields of good characteristic, Math. Ann.. Mathematische Annalen, 329, 49-85 (2004) · Zbl 1141.17017
[56] McNinch, George J., Optimal {\({\rm SL}(2)\)}-homomorphisms, Comment. Math. Helv.. Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society, 80, 391-426 (2005) · Zbl 1097.20040
[57] McNinch, George J.; Sommers, Eric, Component groups of unipotent centralizers in good characteristic, J. Algebra. Journal of Algebra, 260, special issue celebrating the 80th birthday of Robert Steinberg, 323-337 (2003) · Zbl 1026.20026
[58] Michel, Jean, The development version of the {{\tt CHEVIE}} package of {{\tt{GAP3}}}, J. Algebra. Journal of Algebra, 435, 308-336 (2015) · Zbl 1322.20002
[59] Navarro, Gabriel; Tiep, Pham Huu, A reduction theorem for the {A}lperin weight conjecture, Invent. Math.. Inventiones Mathematicae, 184, 529-565 (2011) · Zbl 1234.20010
[60] Okuyama, Tetsuro; Waki, Katsushi, Decomposition numbers of {\({\rm SU}(3,q^2)\)}, J. Algebra. Journal of Algebra, 255, 258-270 (2002) · Zbl 1023.20005
[61] Premet, Alexander, Nilpotent orbits in good characteristic and the {K}empf-{R}ousseau theory, J. Algebra. Journal of Algebra, 260, special issue celebrating the 80th birthday of Robert Steinberg, 338-366 (2003) · Zbl 1020.20031
[62] Richardson, R. W., Finiteness theorems for orbits of algebraic groups, Nederl. Akad. Wetensch. Indag. Math.. Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae, 47, 337-344 (1985) · Zbl 0595.20039
[63] Rotman, Joseph J., An Introduction to Homological Algebra, Universitext, xiv+709 pp. (2009) · Zbl 1157.18001
[64] Shoji, Toshiaki, Lusztig’s conjecture for finite special linear groups, Represent. Theory. Representation Theory. An Electronic Journal of the Amer. Math. Soc., 10, 164-222 (2006) · Zbl 1134.20057
[65] Shoji, Toshiaki, On the computation of unipotent characters of finite classical groups. Computational Methods in Lie Theory, Appl. Algebra Engrg. Comm. Comput., 7, 165-174 (1996) · Zbl 0845.20033
[66] Sobaje, Paul, Unipotent elements and generalized exponential maps, Adv. Math.. Advances in Mathematics, 333, 463-496 (2018) · Zbl 1405.20041
[67] Sommers, Eric, A generalization of the {B}ala-{C}arter theorem for nilpotent orbits, Internat. Math. Res. Notices. International Mathematics Research Notices, 539-562 (1998) · Zbl 0909.22014
[68] Sommers, Eric, Lusztig’s canonical quotient and generalized duality, J. Algebra. Journal of Algebra, 243, 790-812 (2001) · Zbl 1050.13006
[69] Spaltenstein, Nicolas, Classes Unipotentes et Sous-Groupes de {B}orel, Lecture Notes in Math., 946, ix+259 pp. (1982) · Zbl 0486.20025
[70] Spaltenstein, Nicolas, On the generalized {S}pringer correspondence for exceptional groups. Algebraic Groups and Related Topics, Adv. Stud. Pure Math., 6, 317-338 (1985)
[71] Springer, T. A., Linear Algebraic Groups, Modern Birkh\"{a}user Classics, xvi+334 pp. (2009) · Zbl 1202.20048
[72] Springer, T. A.; Steinberg, R., Conjugacy classes. Seminar on {A}lgebraic {G}roups and {R}elated {F}inite {G}roups, Lecture Notes in Mathematics, Vol. 131, 167-266 (1970)
[73] Steinberg, Robert, Endomorphisms of Linear Algebraic Groups, Mem. Amer. Math. Soc., 80, 108 pp. (1968) · Zbl 0164.02902
[74] Taylor, Jay, On unipotent supports of reductive groups with a disconnected centre, J. Algebra. Journal of Algebra, 391, 41-61 (2013) · Zbl 1286.20060
[75] Taylor, Jay, Generalized {G}elfand-{G}raev representations in small characteristics, Nagoya Math. J.. Nagoya Mathematical Journal, 224, 93-167 (2016) · Zbl 1372.20042
[76] Taylor, Jay, The structure of root data and smooth regular embeddings of reductive groups, Proc. Edinb. Math. Soc. (2). Proceedings of the Edinburgh Mathematical Society. Series II, 62, 523-552 (2019) · Zbl 07047473
[77] Tits, J., Normalisateurs de tores. {I}. {G}roupes de {C}oxeter \'{e}tendus, J. Algebra. Journal of Algebra, 4, 96-116 (1966) · Zbl 0145.24703
[78] Wings, E., \"{U}ber die unipotenten {C}haraktere der {C}hevalley-{G}ruppen vom {T}yp {\( \tt F_4\)} in guter {C}harakteristik (1995) · Zbl 0858.20010
[79] Winter, David L., The automorphism group of an extraspecial {\(p\)}-group, Rocky Mountain J. Math.. The Rocky Mountain Journal of Mathematics, 2, 159-168 (1972) · Zbl 0242.20023
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