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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Support varieties for modules over Chevalley groups and classical Lie algebras

Authors: Jon F. Carlson, Zongzhu Lin and Daniel K. Nakano
Journal: Trans. Amer. Math. Soc. 360 (2008), 1879-1906
MSC (2000): Primary 17B55, 20Gxx; Secondary 17B50
Published electronically: November 9, 2007
MathSciNet review: 2366967
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Abstract: Let $ G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $ p>0$, $ G_{1}$ be the first Frobenius kernel, and $ G({\mathbb{F}}_{p})$ be the corresponding finite Chevalley group. Let $ M$ be a rational $ G$-module. In this paper we relate the support variety of $ M$ over the first Frobenius kernel with the support variety of $ M$ over the group algebra $ kG({\mathbb{F}}_{p})$. This provides an answer to a question of Parshall. Applications of our new techniques are presented, which allow us to extend results of Alperin-Mason and Janiszczak-Jantzen, and to calculate the dimensions of support varieties for finite Chevalley groups.

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  • [A] J.L. Alperin, Diagrams for modules, J. Pure and Appl. Algebra 16 (1980), 111-119. MR 556154 (81h:16047)
  • [AM1] J.L. Alperin, G. Mason, On simple modules for SL$ (2,q)$, Bull. London Math. Soc. 25 (1993), no. 1, 17-22. MR 1190358 (93j:20033)
  • [AM2] J.L. Alperin, G. Mason, Partial Steinberg modules for finite groups of Lie type, Bull. London Math. Soc. 25 (1993), no. 6, 553-557. MR 1245081 (94i:20024)
  • [AJ] H. H. Andersen, J. C. Jantzen, Cohomology of induced representations for algebraic groups, Math. Ann. 269, (1984), 487-525. MR 766011 (86g:20057)
  • [AS] G.S. Avrunin, L.L. Scott, Quillen stratification for modules, Invent. Math. 66 (1982), 277-286. MR 656624 (83h:20048)
  • [ABS] H. Azad, M. Barry, G. Seitz, On the structure of parabolic groups, Comm. Algebra 18 (2) (1990), 551-562. MR 1047327 (91d:20048)
  • [Ben1] D.J. Benson, Representations and cohomology. II. Cohomology of groups and modules. Cambridge Studies in Advanced Mathematics, 31. Cambridge University Press, Cambridge, 1991. MR 1156302 (93g:20099)
  • [Ben2] D.J. Benson, Modular representation theory: new trends and methods. Lecture Notes in Mathematics, 1081. Springer-Verlag, Berlin, 1984. MR 765858 (86g:20013)
  • [Bo] A. Borel, Linear Algebraic Groups, Second Enlarged Edition, Springer-Verlag, New York, 1991. MR 1102012 (92d:20001)
  • [B] N. Bourbaki, Groupes et algèbres de Lie, 4-6, Hermann, 1968. MR 0240238 (39:1590)
  • [CLNP] J.F. Carlson, Z. Lin, D.K. Nakano, B.J. Parshall, The restricted nullcone, Contemp. Math., 325 (2003), 51-75. MR 1988985 (2004g:17017)
  • [C] R.W. Carter, Finite groups of Lie type, Wiley-Interscience, 1985. MR 794307 (87d:20060)
  • [Ch] L.G. Chouinard, Projectivity and relative projectivity over group rings. J. Pure Appl. Algebra 7 (1976), no. 3, 287-302. MR 0401943 (53:5769)
  • [CPSvdK] E. Cline, B. Parshall, L. Scott, W. van der Kallen, Rational and generic cohomology Invent. Math. 39 (1977), 143-163. MR 0439856 (55:12737)
  • [CM] D.H. Collingwood, W.M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold, 1993. MR 1251060 (94j:17001)
  • [D] S. Donkin, The normality of closures of conjugacy classes of matrices, Invent. Math. 101 (1990), 717-736 MR 1062803 (91j:14040)
  • [FP1] E.M. Friedlander, B.J. Parshall, On the cohomology of algebraic and related finite groups, Invent. Math. 74 (1983), 85-117. MR 722727 (85d:20035)
  • [FP2] E.M. Friedlander, B.J. Parshall, Support varieties for restricted Lie algebras, Invent. Math. 86 (1986), 553-562. MR 860682 (88f:17018)
  • [FPe] E.M. Friedlander, J. Pevtsova, Representation theoretic support spaces for finite group schemes, Amer. J. Math 127 (2005), 379-420. MR 2130619 (2005k:14096)
  • [FS] E.M. Friedlander, A.A. Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209-270. MR 1427618 (98h:14055a)
  • [GLR] D. Gorenstein, R. Lyons, R. Solomon,The classification of the finite simple groups, Number 3. Part I. Chapter A, Almost simple $ K$-groups. Mathematical Surveys and Monographs, 40.3. Amer. Math. Soc., Providence, 1998. MR 1490581 (98j:20011)
  • [HS] D.F. Holt, N. Spaltenstein, Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristics, J. Australian Math. Soc. Ser. A 38 (1985), 330-350. MR 779199 (86g:17007)
  • [Hum1] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972. MR 0323842 (48:2197)
  • [Hum2] J.E. Humphreys, Linear Algebraic Groups, Springer-Verlag, 1975. MR 0396773 (53:633)
  • [Hum3] J.E. Humphreys, Ordinary and Modular Representations of Chevalley Groups, Lecture Notes in Math. 528, Springer-Verlag, 1976. MR 0453884 (56:12137)
  • [Hum4] J.E. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups, Mathematical Surveys and Monographs 43, Amer. Math. Soc., Providence, 1995. MR 1343976 (97i:20057)
  • [JJ] I. Janiszczak, J.C. Jantzen, Simple periodic modules over Chevalley groups, J. London Math. Soc. 41 (1990), 217-230. MR 1067263 (91f:20010)
  • [Jan1] J.C. Jantzen, Representations of Algebraic Groups, Academic Press, 1987. MR 899071 (89c:20001)
  • [Jan2] J.C. Jantzen, Representations of Chevalley groups in their own characteristic. The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), 127-146, Proc. Sympos. Pure Math., 47, Part 1, Amer. Math. Soc., Providence, RI, 1987. MR 933356 (89g:20076)
  • [Jan3] J.C. Jantzen, Support varieties of Weyl modules, Bull. London Math. Soc. 19 (1987), 238-244. MR 879510 (88e:17008)
  • [Jan4] J.C. Jantzen, Nilpotent Orbits in Representation Theory, Progr. Math. 228, Birkhäuser, Boston, 2004.
  • [KLT] S. Kumar, N. Lauritzen, J.F. Thomsen, Frobenius splitting of cotangent bundles of flag varieties, Invent. Math. 136 (1999), 603-621. MR 1695207 (2000g:20088)
  • [LN] Z. Lin, D.K. Nakano, Complexity for modules over finite Chevalley groups and classical Lie algebras, Invent. Math. 138 (1999), 85-101. MR 1714337 (2000m:20077)
  • [LT] R. Lawther, D. Testerman, $ A_1$-subgroups of exceptional algebraic groups, Memoirs of Amer. Math. Soc. No. 674 (1999).
  • [McN1] G.J. McNinch, Abelian unipotent subgroups of reductive groups, J. Pure and Applied Algebra 167 (2002), 269-300. MR 1874545 (2002i:20064)
  • [McN2] G.J. McNinch, Optimal SL$ (2)$-Homomorphisms, Comment. Math. Helv. 80 (2005), 391-426. MR 2142248 (2006f:20055)
  • [McN3] G.J. McNinch, Sub-principal homomorphisms in positive characteristics, Math. Z. 244 (2003), 433-455. MR 1992546 (2004c:20080)
  • [NPV] D.K. Nakano, B.J. Parshall, D.C. Vella, Support varieties for algebraic groups, J. Reine Angew. Math. 547 (2002), 15-47. MR 1900135 (2003b:20063)
  • [P] B.J. Parshall, Cohomology of algebraic groups, Proc. Symp. Pure Math., 47 (1987), 233-248. MR 933362 (89b:20095)
  • [PV] V.L. Popov, E.B. Vinberg, Invariant theory, Algebraic Geometry IV (Encyclopaedia of Math. Sci. Vol. 55), Springer-Verlag, 1994, 123-278.
  • [Sei] G. Seitz, Unipotent elements, tilting modules, and saturation, Invent. Math. 141 (2000) 3, 467-502. MR 1779618 (2001j:20074)
  • [Ser] J.P. Serre, Lie Algebras and Lie Groups, LNM 1500, Springer-Verlag, 1992. MR 1176100 (93h:17001)
  • [Sh] I. Shafarevich, Basic Algebraic Geometry, Vol. 1, 2nd Edition, Springer-Verlag, Berlin/Heidelberg, 1994.
  • [Sp1] T.A. Springer, The unipotent variety of a semisimple group, Proc. Coll. in Alg. Geom. (Tata Institute) (1969), 373-391. MR 0263830 (41:8429)
  • [Sp2] T.A. Springer, Linear Algebraic Groups, Progress in Math. 9, Birkhäuser, 1998. MR 1642713 (99h:20075)
  • [Sp3] T.A. Springer, Linear algebraic groups, Algebraic Geometry IV (Encyclopaedia of Math. Sci.) Vol. 55, Springer-Verlag, (1994), 1-121. MR 1309681 (95g:14002)
  • [SS] T.A. Springer, R. Steinberg, Conjugacy Classes, Seminar on Algebraic Groups and Related Finite Groups, LNM 131, Springer-Verlag, (1970) 167-266. MR 0268192 (42:3091)
  • [St] R. Steinberg, Lectures on Chevalley Groups, Yale University, 1967. MR 0466335 (57:6215)
  • [SFB] A. Suslin, E. Friedlander, C. Bendel, Support varieties for infinitesimal group schemes, Journal of AMS 10, (1997), 729-759. MR 1443547 (98h:14055c)
  • [T1] D. M. Testerman, The construction of the maximal $ A_1$'s in the exceptional algebraic groups, Proc. Amer. Math. Soc., 116 (1992), 635-644. MR 1100666 (93a:20073)
  • [T2] D. M. Testerman, $ A_1$-type overgroups of elements of order $ p$ in semisimple algebraic groups and the associated finite groups. J. Algebra 177 (1995), 34-76. MR 1356359 (96j:20067)
  • [Th] J.F. Thomsen, Normality of certain nilpotent varieties in positive characteristic, J. Algebra 226 (2000), 865-874. MR 1759837 (2001g:14079)
  • [UGA] University of Georgia VIGRE Algebra Group, Varieties of nilpotent elements of simple Lie algebras I: good primes, J. Algebra 280 (2004), 719-737. MR 2090060 (2005h:17016)

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Additional Information

Jon F. Carlson
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

Zongzhu Lin
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506

Daniel K. Nakano
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602

Received by editor(s): April 21, 2005
Received by editor(s) in revised form: October 17, 2005
Published electronically: November 9, 2007
Additional Notes: The research of the first author was supported in part by NSF grant DMS-0100662 and DMS-0401431
The research of the second author was supported in part by NSF grant DMS-0200673
The research of the third author was supported in part by NSF grant DMS-0102225 and DMS-0400548
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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