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Long root tori in Chevalley groups. (English. Russian original) Zbl 1304.20062
St. Petersbg. Math. J. 24, No. 3, 387-430 (2013); translation from Algebra Anal. 24, No. 3, 22-83 (2012).
This article studies a certain class of long root type semisimple elements in Chevalley groups. Specifically, let \(G=G(\Phi,K)\) be a Chevalley group with root system \(\Phi\) over the field \(K\); then for each \(\alpha\in\Phi\) and \(\varepsilon\in K^*\) one can define a semisimple element \(h_\alpha(\varepsilon)\) which commutes with a given maximal torus of \(G\) and acts in a prescribed way on the root subgroups of \(G\). The long root tori of this paper are the tori in \(G\) which are \(G\)-conjugate to \(\{h_\alpha(\varepsilon)\mid\varepsilon\in K^*\}\) where \(\alpha\in\Phi\) is a long root.
Fix a Borel subgroup \(B\) of \(G\) and let its unipotent radical be \(U\). The first two main results (parts of which have appeared previously in various forms) are as follows: 1) Any long root torus of \(G\) is \(U\)-conjugate to a subtorus of a subsystem subgroup \(G(\Delta,K)\), where \(\Delta\) is a subsystem of \(\Phi\) isomorphic to a “twisted” subsystem of \(D_4\); 2) Barring finitely many exceptions, all elements of a long root torus lie in the same Bruhat cell (which is explicitly described). The other three main results give more information about the Bruhat decomposition of a typical long root torus element. A lot of detail is given in the exposition, together with a comprehensive list of references, and the paper includes a lengthy account of the historical development of the work contained within it.

MSC:
20G15 Linear algebraic groups over arbitrary fields
20G35 Linear algebraic groups over adèles and other rings and schemes
20G40 Linear algebraic groups over finite fields
20E07 Subgroup theorems; subgroup growth
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[1] Armand Borel, Properties and linear representations of Chevalley groups, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 1 – 55. · Zbl 0793.01013
[2] Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4 – 6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. · Zbl 1120.17002
[3] Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 7 – 9, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2005. Translated from the 1975 and 1982 French originals by Andrew Pressley. · Zbl 0319.17002
[4] N. A. Vavilov, Subgroups of the special linear group which contain the group of diagonal matrices. I, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 4 (1985), 3 – 7, 120 (Russian, with English summary). N. A. Vavilov, Subgroups of the special linear group which contain the group of diagonal matrices. II, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 1 (1986), 10 – 15, 133 (Russian, with English summary). N. A. Vavilov, Subgroups of the special linear group which contain the group of diagonal matrices. III, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 2 (1987), 3 – 8, 132 (Russian, with English summary). N. A. Vavilov, Subgroups of the special linear group which contain the group of diagonal matrices. IV, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 3 (1988), 10 – 15, 123 – 124 (Russian, with English summary); English transl., Vestnik Leningrad Univ. Math. 21 (1988), no. 3, 7 – 15. N. A. Vavilov, On subgroups of the special linear group which contain the group of diagonal matrices. V, Vestnik S.-Peterburg. Univ. Mat. Mekh. Astronom. vyp. 2 (1993), 10 – 15, 124 (Russian, with English and Russian summaries); English transl., Vestnik St. Petersburg Univ. Math. 26 (1993), no. 2, 6 – 9.
[5] N. A. Vavilov, Bruhat decomposition of one-dimensional transformations, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 3 (1986), 14 – 20, 123 (Russian, with English summary). · Zbl 0613.20028
[6] N. A. Vavilov, Weight elements of Chevalley groups, Dokl. Akad. Nauk SSSR 298 (1988), no. 3, 524 – 527 (Russian); English transl., Soviet Math. Dokl. 37 (1988), no. 1, 92 – 95. · Zbl 0727.20034
[7] N. A. Vavilov, Conjugacy theorems for subgroups of extended Chevalley groups that contain split maximal tori, Dokl. Akad. Nauk SSSR 299 (1988), no. 2, 269 – 272 (Russian); English transl., Soviet Math. Dokl. 37 (1988), no. 2, 360 – 363.
[8] N. A. Vavilov, Bruhat decomposition for long root semisimple elements in Chevalley groups, Rings and modules. Limit theorems of probability theory, No. 2 (Russian), Leningrad. Univ., Leningrad, 1988, pp. 18 – 39, 212 (Russian).
[9] N. A. Vavilov, Bruhat decomposition of two-dimensional transformations, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 3 (1989), 3 – 7, 120 (Russian, with English summary); English transl., Vestnik Leningrad Univ. Math. 22 (1989), no. 3, 1 – 6.
[10] N. A. Vavilov, Root semisimple elements and triples of root unipotent subgroups in Chevalley groups, Problems in algebra, No. 4 (Russian) (Gomel\(^{\prime}\), 1986) ”Universitet\cdot skoe”, Minsk, 1989, pp. 162 – 173 (Russian). · Zbl 0711.20027
[11] N. A. Vavilov, Subgroups of Chevalley groups that contain a maximal torus, Trudy Leningrad. Mat. Obshch. 1 (1990), 64 – 109, 245 – 246 (Russian).
[12] N. A. Vavilov, Unipotent elements in subgroups of extended Chevalley groups that contain a split maximal torus, Dokl. Akad. Nauk 328 (1993), no. 5, 536 – 539 (Russian); English transl., Russian Acad. Sci. Dokl. Math. 47 (1993), no. 1, 112 – 116.
[13] N. A. Vavilov, Subgroups of the group \?\?_{\?} over a semilocal ring, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 343 (2007), no. Vopr. Teor. Predts. Algebr. i Grupp. 15, 33 – 53, 272 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 147 (2007), no. 5, 6995 – 7004.
[14] N. Vavilov, Geometry of 1-tori in \?\?_{\?}, Algebra i Analiz 19 (2007), no. 3, 119 – 150 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 19 (2008), no. 3, 407 – 429.
[15] N. A. Vavilov, How is one to view the signs of structure constants?, Algebra i Analiz 19 (2007), no. 4, 34 – 68 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 19 (2008), no. 4, 519 – 543.
[16] N. A. Vavilov, On subgroups of a symplectic group containing a subsystem subgroup, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 349 (2007), no. Voprosy Teorii Predstavleniĭ Algebr i Grupp. 16, 5 – 29, 242 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 151 (2008), no. 3, 2937 – 2948. · Zbl 1312.20046
[17] N. Vavilov, Weight elements of Chevalley groups, Algebra i Analiz 20 (2008), no. 1, 34 – 85 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 20 (2009), no. 1, 23 – 57.
[18] N. A. Vavilov and E. V. Dybkova, Subgroups of the general symplectic group containing the group of diagonal matrices, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 103 (1980), 31 – 47, 155 – 156 (Russian). Modules and linear groups. N. A. Vavilov and E. V. Dybkova, Subgroups of the general symplectic group containing the group of diagonal matrices. II, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 132 (1983), 44 – 56 (Russian). Modules and algebraic groups, 2. · Zbl 0468.20041
[19] N. A. Vavilov and M. Yu. Mitrofanov, Intersection of two Bruhat cells, Dokl. Akad. Nauk 377 (2001), no. 1, 7 – 10 (Russian). · Zbl 1069.20036
[20] N. A. Vavilov and V. V. Nesterov, Geometry of microweight tori, Vladikavkaz. Mat. Zh. 10 (2008), no. 1, 10 – 23 (Russian, with Russian summary). · Zbl 1324.20026
[21] -, Geometry of \( 2\)-tori in \( \mathrm {GL}_n\) (to appear).
[22] -, Geometry of \( \varpi _1\)-tori in \( \mathrm {SO}_{2l}\) (to appear).
[23] -, Pairs of microweight tori in Chevalley group of type \( \mathrm E_6\) (to appear).
[24] -, Pairs of microweight tori in Chevalley group of type \( \mathrm E_7\) (to appear).
[25] -, Pairs of long root tori in Chevalley groups (to appear).
[26] N. A. Vavilov and I. M. Pevzner, Triples of long root subgroups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 343 (2007), no. Vopr. Teor. Predts. Algebr. i Grupp. 15, 54 – 83, 272 – 273 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 147 (2007), no. 5, 7005 – 7020.
[27] N. A. Vavilov and A. A. Semenov, Bruhat decomposition for long root tori in Chevalley groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 175 (1989), no. Kol\(^{\prime}\)tsa i Moduli. 3, 12 – 23, 162 (Russian); English transl., J. Soviet Math. 57 (1991), no. 6, 3453 – 3458. · Zbl 0746.20024
[28] N. A. Vavilov and A. A. Semenov, Long root semisimple elements in Chevalley groups, Dokl. Akad. Nauk 338 (1994), no. 6, 725 – 727 (Russian); English transl., Russian Acad. Sci. Dokl. Math. 50 (1995), no. 2, 325 – 329. · Zbl 0859.20035
[29] E. V. Dybkova, Form nets and the lattice of upper-diagonal subgroups of the symplectic group over a field of characteristic 2, Algebra i Analiz 10 (1998), no. 4, 113 – 129 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 10 (1999), no. 4, 651 – 661.
[30] -, Subgroups of hyperbolic unipotent groups, Dokt. diss., S.-Peterburg. Gos. Univ., St. Petersburg, 2006, c. 1-182. (Russian)
[31] E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349 – 462 (3 plates) (Russian). · Zbl 0048.01701
[32] A. E. Zalesskiĭ, Semisimple root elements of algebraic groups, Preprint no. 13, Inst. Mat. Akad. Nauk BSSR, Minsk, 1980, pp. 1-24. (Russian)
[33] A. E. Zalesskiĭ, Linear groups, Algebra. Topology. Geometry, Vol. 21, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983, pp. 135 – 182 (Russian).
[34] V. V. Kashin, Orbits of an adjoint and co-adjoint action of Borel subgroups of a semisimple algebraic group, Problems in group theory and homological algebra (Russian), Matematika, Yaroslav. Gos. Univ., Yaroslavl\(^{\prime}\), 1990, pp. 141 – 158 (Russian). · Zbl 0765.20018
[35] A. S. Kondrat\(^{\prime}\)ev, Subgroups of finite Chevalley groups, Uspekhi Mat. Nauk 41 (1986), no. 1(247), 57 – 96, 240 (Russian).
[36] V. V. Nesterov, Pairs of short root subgroups in Chevalley group, Kand. diss., S.-Peterburg. Gos. Univ., St. Petersburg, 1995, pp. 1-72. (Russian)
[37] V. V. Nesterov, Pairs of short root subgroups in Chevalley groups, Dokl. Akad. Nauk 357 (1997), no. 3, 302 – 305 (Russian). · Zbl 0963.20022
[38] V. V. Nesterov, Arrangement of long and short root subgroups in a Chevalley group of type \?\(_{2}\), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 272 (2000), no. Vopr. Teor. Predst. Algebr i Grupp. 7, 273 – 285, 349 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 116 (2003), no. 1, 3035 – 3041. · Zbl 1069.20035
[39] V. V. Nesterov, Pairs of short root subgroups in a Chevalley group of type \?\(_{2}\), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 281 (2001), no. Vopr. Teor. Predst. Algebr. i Grupp. 8, 253 – 273, 284 – 285 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 120 (2004), no. 4, 1630 – 1641. · Zbl 1078.20048
[40] V. V. Nesterov, Generation of pairs of short root subgroups in Chevalley groups, Algebra i Analiz 16 (2004), no. 6, 172 – 208 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 16 (2005), no. 6, 1051 – 1077.
[41] I. M. Pevzner, The geometry of root elements in groups of type \?\(_{6}\), Algebra i Analiz 23 (2011), no. 3, 261 – 309 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 23 (2012), no. 3, 603 – 635.
[42] I. M. Pevzner, The width of groups of type \?\(_{6}\) with respect to root elements. I, Algebra i Analiz 23 (2011), no. 5, 155 – 198 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 23 (2012), no. 5, 891 – 919.
[43] I. M. Pevzner, The width of groups of type \?\(_{6}\) with respect to the set of root elements. II, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 386 (2011), no. Voprosy Teorii Predstvaleniĭ Algebr i Grupp. 20, 242 – 264, 290 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 180 (2012), no. 3, 338 – 350. · Zbl 1306.20051
[44] A. A. Semenov, Bruhat decomposition of root semisimple subgroups in the special linear group, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 160 (1987), no. Anal. Teor. Chisel i Teor. Funktsiĭ. 8, 239 – 246, 302 (Russian); English transl., J. Soviet Math. 52 (1990), no. 3, 3178 – 3185. · Zbl 0632.20028
[45] -, Bruhat decomposition of long root tori in Chevalley groups, Kand. diss., S.-Peterburg. Gos. Univ., St. Petersburg, 1991, pp. 1-143. (Russian)
[46] T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 167 – 266. · Zbl 0249.20024
[47] Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. · Zbl 1196.22001
[48] James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. · Zbl 0447.17001
[49] C. Chevalley, Sur certains groupes simples, Tôhoku Math. J. (2) 7 (1955), 14 – 66 (French). · Zbl 0066.01503
[50] H. Azad, M. Barry, and G. Seitz, On the structure of parabolic subgroups, Comm. Algebra 18 (1990), no. 2, 551 – 562. · Zbl 0717.20029
[51] M. Brion, Représentations exceptionnelles des groupes semi-simples, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 345 – 387 (French). · Zbl 0588.22010
[52] Hartmut Bürgstein and Wim H. Hesselink, Algorithmic orbit classification for some Borel group actions, Compositio Math. 61 (1987), no. 1, 3 – 41. · Zbl 0612.17005
[53] N. Cantarini, G. Carnovale, and M. Costantini, Spherical orbits and representations of \?_{\?}(\?), Transform. Groups 10 (2005), no. 1, 29 – 62. · Zbl 1101.17006
[54] Giovanna Carnovale, Spherical conjugacy classes and the Bruhat decomposition, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2329 – 2357 (English, with English and French summaries). · Zbl 1195.20051
[55] Roger W. Carter, Simple groups of Lie type, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989. Reprint of the 1972 original; A Wiley-Interscience Publication. · Zbl 0723.20006
[56] R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1 – 59. · Zbl 0254.17005
[57] Kei Yuen Chan, Jiang-Hua Lu, and Simon Kai-Ming To, On intersections of conjugacy classes and Bruhat cells, Transform. Groups 15 (2010), no. 2, 243 – 260. · Zbl 1207.20042
[58] Jean-Louis Clerc, Special prehomogeneous vector spaces associated to \?\(_{4}\),\?\(_{6}\),\?\(_{7}\),\?\(_{8}\) and simple Jordan algebras of rank 3, J. Algebra 264 (2003), no. 1, 98 – 128. · Zbl 1034.17006
[59] A. M. Cohen, H. Cuypers, and H. Sterk, Linear groups generated by reflection tori, Canad. J. Math. 51 (1999), no. 6, 1149 – 1174. Dedicated to H. S. M. Coxeter on the occasion of his 90th birthday. · Zbl 0952.20042
[60] Mauro Costantini, On the coordinate ring of spherical conjugacy classes, Math. Z. 264 (2010), no. 2, 327 – 359. · Zbl 1195.20049
[61] Erich W. Ellers and Nikolai Gordeev, Intersection of conjugacy classes with Bruhat cells in Chevalley groups, Pacific J. Math. 214 (2004), no. 2, 245 – 261. · Zbl 1062.20050
[62] Erich W. Ellers and Nikolai Gordeev, Intersection of conjugacy classes with Bruhat cells in Chevalley groups: the cases \?\?_{\?}(\?),\?\?_{\?}(\?), J. Pure Appl. Algebra 209 (2007), no. 3, 703 – 723. · Zbl 1128.20034
[63] N. L. Gordeev and E. W. Ellers, Big and small elements in Chevalley groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 386 (2011), no. Voprosy Teorii Predstvaleniĭ Algebr i Grupp. 20, 203 – 226, 289 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 180 (2012), no. 3, 315 – 329. · Zbl 1306.20049
[64] Alan Harebov and Nikolai Vavilov, On the lattice of subgroups of Chevalley groups containing a split maximal torus, Comm. Algebra 24 (1996), no. 1, 109 – 133. · Zbl 0857.20023
[65] Stephen J. Haris, Some irreducible representations of exceptional algebraic groups, Amer. J. Math. 93 (1971), 75 – 106. · Zbl 0215.39603
[66] Wim H. Hesselink, A classification of the nilpotent triangular matrices, Compositio Math. 55 (1985), no. 1, 89 – 133. · Zbl 0579.15011
[67] Noriaki Kawanaka, Unipotent elements and characters of finite Chevalley groups, Osaka J. Math. 12 (1975), no. 2, 523 – 554. · Zbl 0314.20031
[68] Sergei Krutelevich, On a canonical form of a 3\times 3 Hermitian matrix over the ring of integral split octonions, J. Algebra 253 (2002), no. 2, 276 – 295. · Zbl 1034.15008
[69] Sergei Krutelevich, Jordan algebras, exceptional groups, and Bhargava composition, J. Algebra 314 (2007), no. 2, 924 – 977. · Zbl 1163.17032
[70] Martin W. Liebeck and Gary M. Seitz, Subgroups generated by root elements in groups of Lie type, Ann. of Math. (2) 139 (1994), no. 2, 293 – 361. · Zbl 0824.20041
[71] Martin W. Liebeck and Gary M. Seitz, Subgroups of simple algebraic groups containing elements of fundamental subgroups, Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 3, 461 – 479. · Zbl 0939.20049
[72] G. Lusztig, From conjugacy classes in the Weyl group to unipotent classes. I-III arXiv:1003. 0412v5 [math.RT] (23 Aug. 2010), 1-41; arXiv:1104.0196v2 [math.RT] (30 Apr. 2011), 1-26; arXiv:1104.3112v1 [math.RT] (15 Apr. 2011), 1-22.
[73] -On \( C\)-small conjugacy classes in a reductive group, arXiv:1005.4313v2 [math.RT] (10 Dec. 2010), 1-19.
[74] -Bruhat decomposition and applications, arXiv:1006.5004v1 [math.RT] (25 Jun. 2010), 1-4.
[75] -From unipotent classes to conjugacy classes in the Weyl group, arXiv:1008.2692v1 [math.RT] (16 Aug. 2010), 1-10.
[76] J. G. M. Mars, Les nombres de Tamagawa de certains groupes exceptionnels, Bull. Soc. Math. France 94 (1966), 97 – 140 (French). · Zbl 0146.04601
[77] Hideya Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1 – 62 (French). · Zbl 0261.20025
[78] Eugene Plotkin, Andrei Semenov, and Nikolai Vavilov, Visual basic representations: an atlas, Internat. J. Algebra Comput. 8 (1998), no. 1, 61 – 95. · Zbl 0957.17006
[79] Roger Richardson, Gerhard Röhrle, and Robert Steinberg, Parabolic subgroups with abelian unipotent radical, Invent. Math. 110 (1992), no. 3, 649 – 671. · Zbl 0786.20029
[80] Gerhard Röhrle, Orbits in internal Chevalley modules, Groups, combinatorics & geometry (Durham, 1990) London Math. Soc. Lecture Note Ser., vol. 165, Cambridge Univ. Press, Cambridge, 1992, pp. 311 – 315. · Zbl 0827.20054
[81] Gerhard E. Röhrle, On the structure of parabolic subgroups in algebraic groups, J. Algebra 157 (1993), no. 1, 80 – 115. · Zbl 0829.20069
[82] Gerhard E. Röhrle, On extraspecial parabolic subgroups, Linear algebraic groups and their representations (Los Angeles, CA, 1992) Contemp. Math., vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 143 – 155. · Zbl 0832.20071
[83] A. A. Semenov, N. A. Vavilov, Halbeinfache Wurzelelemente in Chevalley-Gruppen, Preprint Universität Bielefeld, 1994, no. 2, 1-11.
[84] T. A. Springer, Linear algebraic groups, Progress in Mathematics, vol. 9, Birkhäuser, Boston, Mass., 1981. · Zbl 0453.14022
[85] Michael R. Stein, Stability theorems for \?\(_{1}\), \?\(_{2}\) and related functors modeled on Chevalley groups, Japan. J. Math. (N.S.) 4 (1978), no. 1, 77 – 108. · Zbl 0403.18010
[86] Robert Steinberg, Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 49 – 80.
[87] Nikolai A. Vavilov, Structure of Chevalley groups over commutative rings, Nonassociative algebras and related topics (Hiroshima, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 219 – 335. · Zbl 0799.20042
[88] Nikolai Vavilov, Intermediate subgroups in Chevalley groups, Groups of Lie type and their geometries (Como, 1993) London Math. Soc. Lecture Note Ser., vol. 207, Cambridge Univ. Press, Cambridge, 1995, pp. 233 – 280. · Zbl 0879.20020
[89] Nikolai Vavilov, Unipotent elements in subgroups which contain a split maximal torus, J. Algebra 176 (1995), no. 2, 356 – 367. · Zbl 0865.20030
[90] Nikolai Vavilov, A third look at weight diagrams, Rend. Sem. Mat. Univ. Padova 104 (2000), 201 – 250. · Zbl 1016.20029
[91] Nikolai Vavilov and Eugene Plotkin, Chevalley groups over commutative rings. I. Elementary calculations, Acta Appl. Math. 45 (1996), no. 1, 73 – 113. · Zbl 0861.20044
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