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Transactions of the American Mathematical Society

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



Prehomogeneous vector spaces and varieties

Author: Frank J. Servedio
Journal: Trans. Amer. Math. Soc. 176 (1973), 421-444
MSC: Primary 20G05
MathSciNet review: 0320173
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Abstract: An affine algebraic group G over an algebraically closed field k of characteristic 0 is said to act prehomogeneously on an affine variety W over k if G has a (unique) open orbit $ o(G)$ in W. When W is the variety of points of a vector space V, $ G \subseteq GL(V)$ and G acts prehomogeneously and irreducibly on V (We say an irreducibly prehomogeneous pair (G, V).), the following conditions are shown to be equivalent: 1. the existence of a nonconstant semi-invariant P in $ k[V] \cong S({V^\ast})$, 2. $ (G',V)$ is not a prehomogeneous pair ($ G'$ is the commutator subgroup of G, a semisimple closed subgroup of G.), 3. if $ X \in o(G)$, then $ G_X^0 \subseteq G'$. ($ G_X^0$ is the connected identity component of $ {G_X}$, the stabilizer of X in G.) Further, if such a P exists, the criterion, due to Mikio Sato, ``$ o(G)$ is the principal open affine $ {U_P}$ if and only if $ G_X^0$ is reductive'' is stated.

Under the hypothesis G reductive, the condition ``there exists a Borel subgroup $ B \subseteq G$ acting prehomogeneously on W'' is shown to be sufficient for $ G\backslash W$, the set of G-orbits in the affine variety W to be finite.

These criteria are then applied to a class of irreducible prehomogeneous pairs (G, V) for which $ G'$ is simple and three further conjectures, one due to Mikio Sato, are stated.

References [Enhancements On Off] (What's this?)

  • [1] Séminaire H. Cartan et C. Chevalley, 8e année: 1955/56. Géometrie algébrique, Secrétariat mathématique, Paris, 1956; P. Cartier, Exposé 13, Prop. 1, pp. 13-30. MR 20 #3871.
  • [2] H. Weyl, The classical groups, 2nd ed., Princeton Univ. Press, Princeton, N. J., 1953, p. 250. MR 1488158 (98k:01049)
  • [3] Armand Borel, ``Linear algebraic groups,'' Algebraic groups and discontinuous subgroups, Proc. Sympos. Pure Math., vol. 9, Amer. Math. Soc., Providence, R. I., 1966, pp. 3-19. MR 34 #4371. MR 0204532 (34:4371)
  • [4] David Mumford, Introduction to algebraic geometry, Preliminary version of first three chapters; Department of Mathematics, Harvard University, Cambridge, Mass.
  • [5] Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. I, University Series in Higher Math., Van Nostrand, Princeton N. J., 1958. MR 19, 833. MR 0090581 (19:833e)
  • [6] Jacob Eli Goodman, Thesis, Columbia University, New York, 1967.
  • [7] Armand Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485-535. MR 26 #5081. MR 0147566 (26:5081)
  • [8] C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and Appl. Math., vol. 11, Interscience, New York, 1962, p. 203. [29.15] MR 26 #2519. MR 0144979 (26:2519)
  • [9] Sigurdur Helgason, Differential geometry and symmetric spaces, Pure and Appl. Math., vol. 12, Academic Press, New York, 1962, p. 110. MR 26 #2986. MR 0145455 (26:2986)
  • [10] Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Appl. Math., no. 10, Interscience, New York, 1962, p. 141. MR 26 #1345. MR 0143793 (26:1345)
  • [11] E. M. Andreev, È. B. Vinberg and A. G. Èlašvili, Orbits of greatest dimension in semi-simple linear Lie groups, Funkcional. Anal. i Priložen. 1 (1967), no. 4, 3-7 = Functional Anal. Appl. 1 (1967), 257-261. MR 42 #1942. MR 0267040 (42:1942)
  • [12] Jun-ichi Igusa, A classification of spinors up to dimension twelve, Amer. J. Math. 92 (1970), 997-1028. MR 43 #3291. MR 0277558 (43:3291)
  • [13] Claude Chevalley and R. D. Schafer, The exceptional simple Lie algebras $ {F_4}$. and $ {E_6}$, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 137-141. MR 11, 577. MR 0034378 (11:577b)
  • [14] Hans Freudenthal, Beziehungen der $ {E_7}$ und $ {E_8}$ zur Oktavenebene. I, Nederl. Akad. Wetensch. Proc. Ser. A 57 = Indag. Math. 16 (1954), 218-230. MR 16, 108. MR 0063358 (16:108b)
  • [15] Stephen J. Haris, Some irreducible representations of exceptional algebraic groups, Amer. J. Math. 93 (1971), 75-106. MR 43 #4829. MR 0279103 (43:4829)
  • [16] Mikio Sato, The theory of pre-homogeneous vector spaces, Sûgaku 15-1 (1970), 85-157; notes by T. Aratami, published by Association for Sûgaku no Ayumi, c/o S.S.S., Dr. Y. Morita, Department of Mathematics, Faculty of Science, Univ. of Tokyo, Tokyo, Japan

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Keywords: Linear algebraic group, affine variety, Zariski open orbit, semi-invariant form, reductive linear algebraic group, Borel subgroup, simple algebraic group, irreducible representation
Article copyright: © Copyright 1973 American Mathematical Society

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