Prehomogeneous vector spaces and varieties
HTML articles powered by AMS MathViewer
 by Frank J. Servedio PDF
 Trans. Amer. Math. Soc. 176 (1973), 421444 Request permission
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 semiinvariant 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 Gorbits 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

Séminaire H. Cartan et C. Chevalley, 8e année: 1955/56. Géometrie algébrique, Secrétariat mathématique, Paris, 1956; P. Cartier, Exposé 13, Prop. 1, pp. 1330. MR 20 #3871.
 Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158
 Armand Borel, Linear algebraic groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 3–19. MR 0204532 David Mumford, Introduction to algebraic geometry, Preliminary version of first three chapters; Department of Mathematics, Harvard University, Cambridge, Mass.
 Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581 Jacob Eli Goodman, Thesis, Columbia University, New York, 1967.
 Armand Borel and HarishChandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535. MR 147566, DOI 10.2307/1970210
 Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New YorkLondon, 1962. MR 0144979
 Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New YorkLondon, 1962. MR 0145455
 Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New YorkLondon, 1962. MR 0143793
 E. M. Andreev, È. B. Vinberg, and A. G. Èlašvili, Orbits of highest dimension of semisimple linear Lie groups, Funkcional. Anal. i Priložen. 1 (1967), no. 4, 3–7 (Russian). MR 0267040
 Junichi Igusa, A classification of spinors up to dimension twelve, Amer. J. Math. 92 (1970), 997–1028. MR 277558, DOI 10.2307/2373406
 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 34378, DOI 10.1073/pnas.36.2.137
 Hans Freudenthal, Beziehungen der $E_7$ und $E_8$ zur Oktavenebene. I, Nederl. Akad. Wetensch. Proc. Ser. A. 57 = Indagationes Math. 16 (1954), 218–230 (German). MR 0063358
 Stephen J. Haris, Some irreducible representations of exceptional algebraic groups, Amer. J. Math. 93 (1971), 75–106. MR 279103, DOI 10.2307/2373449 Mikio Sato, The theory of prehomogeneous vector spaces, Sûgaku 151 (1970), 85157; 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
Additional Information
 © Copyright 1973 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 176 (1973), 421444
 MSC: Primary 20G05
 DOI: https://doi.org/10.1090/S00029947197303201737
 MathSciNet review: 0320173