Summary
Let a reductive groupG act linearly on a vectorspaceV, and letO∋V be an orbit. For each irreducible representation ω ofG, the multiplicity with which ω occurs in the ring of regular functions onO (or on its closureŌ) provides an interesting numerical invariant ofO. We introduce an algebraic notion of a “deformation” of an orbit into another one. The main goal of this paper is to give sufficient conditions on an orbit, in order that an arbitrary deformation of this orbit has to preserve all the multiplicites mentioned above.
In the special case whereG=PSL n acts on its Lie-algebra in the usual way, every (nonzero) nilpotent orbit may be deformed into a semisimple orbit, and a conjecture of Dixmier amounts to saying that these deformations should preserve multiplicities. We prove this conjecture to be true in the case where the closure of the niloptent orbit is a normal variety.
In development of an idea of Dixmier [6], we also introduce a notion of “sheets” ofV (maximal irreducible subsets consisting of orbits of a fixed dimension), and we give a useful description (5.4) for all sheets of a semi-simple Lie-algebra.
This paper is strongly influenced by some investigations of B. Kostant [10] on similar problems and extends some of his results.
Similar content being viewed by others
Literaturverzeichnis
Borho W.,Definition einer Dixmier-Abbildung für sl(n, ℂ: Inv. math.40, 143–169 (1977).
Borho W. undJantzen J.C.,Über primitive Ideale in Einhüllenden halbeinfacher Lie-Algebren; Inv. math.39, 1–53 (1977).
Borho W. undKraft H. Über die Gelfand-Kirillov Dimension, Math. Ann.220, 1–24 (1976).
Demazure M. undGabriel P.,Groupes algébriques; North-Holland, Amsterdam 1970.
Dieudonné J. undGrothendieck A.,Eléments de géométrie algébrique III, IV; Publ. Math. IHESn 0 11, 17, 20, 24, 28, 32.
Dixmier J.,Polarisations dans les algébres de Lie semi-simples complexes; Bull. Sci. Math.99, 45–63 (1975).
Hesselink W.,The normality of closures of orbits in a Lie algebra; Comment. Math. Helv.54, (1979).
Humphreys J. E.,Linear algebraic groups, Springer GT, New York-Heidelberg-Berlin 1976.
Johnston D. S. undRichardson R. W.,Conjugacy classes in parabolic subgroups of semisimple algebraic groups II; preprint Durham 1976.
Kostant B.,Lie group representations on polynomial rings; Amer. J. Math.85, 327–404 (1963).
Kraft H.,Parametrisierung der Konjugationsklassen in sln; Math. Ann.234, 209–220 (1978).
Mumford D.,Geometric Invarient Theory; Ergebnisse der Math.34, Springer, Berlin 1965.
Richardson R. W.,Principal Orbit Types for Algebraic Transformation Spaces in Characteristic Zero; Inv. math.16, 6–14 (1972).
Springer T. A. undSteinberg R.,Conjugacy Classes; in “Seminar on algebraic groups and related finite groups”, Springer LN131, (1970).
Steinberg R.,Conjugacy Classes in Algebraic Groups; Springer LN366 (1974).
Vinberg E. B. undPopov V. L.,On a class of quasihomogeneus affine varieties; Math. USSR Izvestija.6 (1972),n 0 4.
Borho W.,Recent advances in enveloping algebras of semisimple Lie algebras; Sém. Bourbaki n 0 489 (1976).
Esnault H.,Singularités rationelles et groupes algébriques; Thèse de troisième cycle, Paris VII (1976).
Ozeki H. undWakimoto M.,On polarisations of certain homogeneous spaces; Hiroshima Math. J.2, 445–482 (1972).
Matsushima Y.,Espaces homogènes de Stein des groupes de Lie complexes; Nagoya Math. J.16, 205–218 (1960).
Bialynicki-Birula A.,On homogeneous affine spaces of linear algebraic groups; Amer. J. Math.85, 577–582 (1963).
Richardson R. W.,Deformations of Lie Subgroups and the Variation of Isotropy Subgroups; Acta math.129, 35–73 (1972).
Grauert H., undRiemenschneider O.,Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen; Inv. math.11 263–292 (1970).
Kempf G., Knudson F., Mumford D. undSaint-Donat B.,Toroidal Embeddings I; Springer LN339 (1973).
Hartshorne R. undOgus A., On the factoriality of local rings of small embedding codimension; Communications in A.1 415–437 (1974).
Luna D.,Slices étales; Bull. Soc. math. France, Mémoire33, 81–105 (1973).
Luna D., Adhérences d'orbite et invariants; Inv. math.29, 231–238 (1975).
Slodowy P., Einfache Singularitäten und einfache algebraische Gruppen; Regensburger Mathematische Schriften2, (1978).
Rights and permissions
About this article
Cite this article
Borho, W., Kraft, H. Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen. Commentarii Mathematici Helvetici 54, 61–104 (1979). https://doi.org/10.1007/BF02566256
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02566256