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Decomposition Varieties in Semisimple Lie Algebras

Published online by Cambridge University Press:  20 November 2018

Abraham Broer
Abraham Broer*
Affiliation:
Département de mathématiques et de statistique Université de Montréal C.P. 6128, succursale Centre-ville Montréal, Québec H3C 3J7 email: broera@DMS.UMontreal.CA
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Abstract

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The notion of decompositon class in a semisimple Lie algebra is a common generalization of nilpotent orbits and the set of regular semisimple elements.We prove that the closure of a decomposition class has many properties in common with nilpotent varieties, e.g., its normalization has rational singularities.

The famous Grothendieck simultaneous resolution is related to the decomposition class of regular semisimple elements. We study the properties of the analogous commutative diagrams associated to an arbitrary decomposition class.

Keywords

14L3014M1715A3017B45

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 50 , Issue 5 , 01 October 1998 , pp. 929 - 971
Copyright
Copyright © Canadian Mathematical Society 1998

References

1. Altman, A. and Kleiman, S., Introduction to Grothendieck duality theory. Lecture Notes in Math. 146, Springer-Verlag, New York, 1970.Google Scholar
2. Bardsley, P. and Richardson, R.W., Etale slices for algebraic transformation groups in characteristic p. Proc. London. Math. Soc. (3) 51(1985), 295–317.Google Scholar
3. Beynon, W.M. and Spaltenstein, N., Green functions of finite Chevalley groups of type En (n = 6Ò7Ò8). J. Algebra 88(1984), 584–614.Google Scholar
4. Bongartz, K., Schichten von Matrizen sind rationale Varietäten. Math. Ann. 283(1989), 53–64.Google Scholar
5. Borho, W., Über Schichten halbeinfacher Lie-Algebren. Invent. Math. 65(1981), 283–317.Google Scholar
6. Borho, W. and Brylinski, J.-L., Differential operators on homogeneous spaces. I. Irreducibility of the associated variety for annihilators of induced modules. Invent. Math. 69(1982), 437–476.Google Scholar
7. Borho, W., Differential operators on homogeneous spaces. II. Relative enveloping algebras. Bull. Soc. Math. France 117(1989), 167–210.Google Scholar
8. Borho, W. and Kraft, H., Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen. Comment. Math. Helv. 54(1979), 61–104.Google Scholar
9. Boutot, J.-F., Singularités rationnelles et quotients par les groupes réductifs. Invent. Math. 88(1987), 65–68.Google Scholar
10. Broer, A., Line bundles on the cotangent bundle of the flag variety. Invent. Math. 113(1993), 1–20.Google Scholar
11. Broer, A., Normality of some nilpotent varieties and cohomology of line bundles on the cotangent bundle of the flag variety. In: J.-L. Brylinski et al. Lie theory and geometry, In honor of Bertrant Kostant, Progress in Math. 123(1994), 1–19.Google Scholar
12. Broer, A., Normal nilpotent varieties in F4, J. Algebra, to appear.Google Scholar
13. Broer, A., Lectures on decomposition classes. Representation theories and algebraic geometry, (ed. Broer, A.), NATO ASI Series C, Kluwer Academic Publishers, Dordrecht, 1998. to appear.Google Scholar
14. Brylinski, R. and Kostant, B., On the variety associated to a TDS. December 1992, preprint.Google Scholar
15. Carter, R.W., Conjugacy classes in the Weyl group. In: Borel et al, Seminar on algebraic groups and related finite groups, Lecture Notes in Math. 131, Springer-Verlag, New York, 1970.Google Scholar
16. Carter, R.W., Finite groups of Lie type. Conjugacy classes and complex characters. Wiley, Chichester, 1985.Google Scholar
17. Collingwood, D.H. and McGovern, W.M., Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold, New York, 1993.Google Scholar
18. Elkik, R., Singularités rationnelles et déformations. Invent. Math. 47(1978), 139–147.Google Scholar
19. Flenner, H., Rationale quasihomogene Singularitäten. Arch.Math. 36(1981), 35–44.Google Scholar
20. Gel, I.M.’fand and Kirillov, A.A., The structure of the Lie field connected with a split semisimple Lie algebra. Functional Anal. Appl. 3(1969), 6–21.Google Scholar
21. Grothendieck, A., Éléments de géométrie algébrique, IV. Publ. Math. I.H.E.S. 20, 24, 28, 32(1964–67).Google Scholar
22. Hartshorne, R., Algebraic Geometry. Graduate Texts in Math. 52, Springer-Verlag, New York, 1977.Google Scholar
23. Hesselink, W.H., Polarizations in the classical groups. Math. Z. 160(1978), 217–234.Google Scholar
24. Hinich, V., On the singularities of nilpotent orbits. Israel J. Math. 73(1991), 297–308.Google Scholar
25. Howlett, R. B., Normalizers of parabolic subgroups of reflection groups. J. LondonMath. Soc. 21(1980), 62–80.Google Scholar
26. Jósefiak, T., Pragacz, P. and Weyman, J., Resolutions of determinantal varieties. In: Tableaux de Young et foncteurs de Schur en algèbre et géométrie, Torún, Astérisque 87–88, Société Mathématique de France, (1981), 109–189.Google Scholar
27. Katsylo, P.I., Sections of sheets in a reductive Lie algebra. Math. USSR-Izv. 20(1983), 449–457.Google Scholar
28. Klimek, J., W. Kráskiewicz and Weyman, J., A free resolution of a symplectic rank variety. J. Algebra, to appear.Google Scholar
29. Knop, F., Weylgruppe und Momentabbildung. Invent. Math. 99(1990), 1–23.Google Scholar
30. Knop, F., A Harish-Chandra homomorphism for reductive group actions. Ann. of Math. 140(1994), 253–288.Google Scholar
31. Kraft, H., Paramatrisierung von Konjugationsklassen in sin. Math. Ann. 234(1978), 209–220.Google Scholar
32. Kraft, H., Geometrische Methoden in der Invariantentheorie. Vieweg Verlag, Braunschweig, 1984.Google Scholar
33. Kraft, H., Closures of conjugacy classes in G2. J. Algebra 126(1989), 454–465.Google Scholar
34. Kraft, H. and Procesi, C., Closures of conjugacy classes of matrices are normal. Invent. Math. 53(1979), 227–247.Google Scholar
35. Kraft, H. , On the geometry of conjugacy classes in classical groups. Comment. Math. Helv. 57(1982), 539–602.Google Scholar
36. Lusztig, G., Intersection cohomology complexes on a reductive group. Invent. Math. 75(1984), 205–272.Google Scholar
37. Lusztig, G., Cuspidal local systems and graded Hecke algebras, I. Publ. Math. I.H.E.S. 67(1988), 145–202.Google Scholar
38. Lusztig, G. and Spaltenstein, N., Induced unipotent classes. J. London Math. Soc. 19(1979), 41–52.Google Scholar
39. Matsumura, H., Commutative ring theory. Cambridge Univ. Press, Cambridge, 1986.Google Scholar
40. Mehta, V.B. and van, W. der Kallen, A simultaneous Frobenius splitting for closures of conjugacy classes of nilpotent matrices. Compositio Math. 84(1992), 169–178.Google Scholar
41. Panyushev, D.I., Rationality of singularities and the Gorenstein property for nilpotent orbits. Functional Anal. Appl. 25(1991), 225–226.Google Scholar
42. Peterson, D., Geometry of the adjoint representation of a complex semisimple Lie algebra. thesis, Harvard University, 1978.Google Scholar
43. Richardson, R.W., Normality of G-stable subvarieties of a semisimple Lie algebra. In: A. Cohen et al, Algebraic groups, Utrecht 1986, Lecture Notes in Math. 1271, Springer-Verlag, New York, 1987.Google Scholar
44. Richardson, R.W., Derivatives of invariant polynomials on a semisimple Lie algebra. In: Harmonic analysis and operator theory, Proc. Cent. Math. Anal. Austral. Nat. Univ. 15(1987).Google Scholar
45. Rubenthaler, H., Paramétrisation d’orbites dans les nappes de Dixmier admissibles. In: Duflo et al. Analyse harmonique sur les groupes de Lie et les espaces symétriques, Mém. Soc. Math. France (N.S.) 15(1984), 255–275.Google Scholar
46. Shafarevich, I.R., Basic algebraic geometry 1, Varieties in projective space, Second, revised and expanded version. Springer-Verlag, New York, 1994.Google Scholar
47. Slodowy, P., Simple singularities and simple algebraic groups. Lecture Notes in Math. 815, Springer- Verlag, New York, 1980.Google Scholar
48. Slodowy, P., Four lectures on simple groups and singularities. Communications of the Mathematical Institute 11-1980, Rijksuniversiteit Utrecht, 1980.Google Scholar
49. Soergel, W., Universelle versus relative Einhüllende: Eine geometrische Untersuchung von Quotienten von universellen Einhüllenden halbeinfacher Lie-Algebren. Math. Ann. 284(1989), 177–198.Google Scholar
50. Spaltenstein, N., Nilpotent classes and sheets of Lie algebras in bad characteristic. Math. Z. 181(1982), 31–48.Google Scholar
51. Spaltenstein, N., Classes unipotents et sous-groupes de Borel. Lecture Notes in Math. 946, Springer-Verlag, New York, 1982.Google Scholar