Singularities in the nilpotent scheme of a classical group
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- by Wim Hesselink
- Trans. Amer. Math. Soc. 222 (1976), 1-32
- DOI: https://doi.org/10.1090/S0002-9947-1976-0429875-8
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Abstract:
If $(X,x)$ is a pointed scheme over a ring k, we introduce a (generalized) partition ${\text {ord}}(x,X/k)$. If G is a reductive group scheme over k, the existence of a nilpotent subscheme $N(G)$ of ${\text {Lie}}(G)$ is discussed. We prove that ${\text {ord}}(x,N(G)/k)$ characterizes the orbits in $N(G)$, their codimension and their adjacency structure, provided that G is $G{l_n}$, or $S{p_n}$ and $1/2 \in k$. For $S{O_n}$ only partial results are obtained. We give presentations of some singularities of $N(G)$. Tables for its orbit structure are added.References
- V. I. Arnol′d, On matrices depending on parameters, Uspehi Mat. Nauk 26 (1971), no. 2(158), 101–114 (Russian). MR 0301242
- Armand Borel, Linear algebraic groups, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes taken by Hyman Bass. MR 0251042
- N. Bourbaki, Éléments de mathématique. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1272, Hermann, Paris, 1959 (French). MR 0107661 —, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chaps. IV, V, VI, Actualités Sci. Indust., no. 1337, Hermann, Paris, 1968. MR 39 #1590.
- E. Brieskorn, Singular elements of semi-simple algebraic groups, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 279–284. MR 0437798 C. Chevalley, Classification des groupes de Lie algébriques, Notes polycopiées, Inst. H. Poincaré, Paris, 1956-1958.
- Michel Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301 (French). MR 342522, DOI 10.1007/BF01418790
- Schémas en groupes. I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274458
- Murray Gerstenhaber, On dominance and varieties of commuting matrices, Ann. of Math. (2) 73 (1961), 324–348. MR 132079, DOI 10.2307/1970336
- Murray Gerstenhaber, Dominance over the classical groups, Ann. of Math. (2) 74 (1961), 532–569. MR 136683, DOI 10.2307/1970297 A. Grothendieck and J. A. Dieudonné, Éléments de géométrie-algébrique. I, Die Grundlehren der math. Wissenschaften, Band 166, Springer-Verlag, Berlin, 1971. —, Éléments de géométrie algébrique. II, III, IV, Inst. Hautes Études Sci. Publ. Math. Nos. 8, 11, 17, 20, 24, 28, 32 (1961-1967). MR 29 #1212; 30 #3885; 33 #7330; 36 #177b, c; #178; 39 #220. W. H. Hesselink, Singularities in the nilpotent scheme of a classical group, Thesis, Utrecht, 1975.
- Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
- Birger Iversen, Generic local structure of the morphisms in commutative algebra, Lecture Notes in Mathematics, Vol. 310, Springer-Verlag, Berlin-New York, 1973. MR 0360575, DOI 10.1007/BFb0060790
- Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. MR 158024, DOI 10.2307/2373130
- David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 34, Springer-Verlag, Berlin-New York, 1965. MR 0214602, DOI 10.1007/978-3-662-00095-3
- Jean-Jacques Risler, Algèbre symétrique d’un idéal, Sur Quelques Problèmes d’Algèbre, Université de Montpellier, Montpellier, 1969, pp. 61–69 (French). MR 0254040
- T. A. Springer, The unipotent variety of a semi-simple group, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968) Oxford Univ. Press, London, 1969, pp. 373–391. MR 0263830
- 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. MR 0268192
- Robert Steinberg, Conjugacy classes in algebraic groups, Lecture Notes in Mathematics, Vol. 366, Springer-Verlag, Berlin-New York, 1974. Notes by Vinay V. Deodhar. MR 0352279, DOI 10.1007/BFb0067854
- F. D. Veldkamp, The center of the universal enveloping algebra of a Lie algebra in characteristic $p$, Ann. Sci. École Norm. Sup. (4) 5 (1972), 217–240. MR 308227, DOI 10.24033/asens.1225
Bibliographic Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 222 (1976), 1-32
- MSC: Primary 14B05; Secondary 14L15
- DOI: https://doi.org/10.1090/S0002-9947-1976-0429875-8
- MathSciNet review: 0429875