Abstract
Let G be a connected, complex reductive group and B, a Borel subgroup. The homogeneous space G/B is called the (full) flag variety of G; for GL n this is the ordinary variety of flags of subspaces in ℂn. The properties of this variety have many implications in the study of reductive groups and their representations. For example, all the irreducible finite dimensional G-modules can be obtained as global sections of line bundles on the flag variety. More generally, any of its (G-linearized) line bundles has at most one non-vanishing sheaf-cohomology space, having a natural structure of a simple G-module if G is simply connected. This is part of the content of the Borel-Weil-Bott theorem and can be seen as a geometric interpretation of E. Cartan’s theory of highest weights.
Dedicated to Bert Kostant on his 65th birthday
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References
H.H. Andersen and J.C. Jantzen, Cohomology of induced representations for algebraic groups, Math. Ann. 269 (1984), 487–525.
A. Beilinson and J. Bernstein, Localisation de g-modules, C.R. Acad. Sci., Paris, 292 (1981), 15–18.
W. Borho and H. Kraft, Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comment. Math. Helv. 54 (1979), 61–104.
A. Broer, Hilbert series in invariant theory, Thesis, Rijksuniversiteit Utrecht (1990).
A. Broer, Line bundles on the cotangent bundle of the flag variety, Invent. Math. 113 (1993), 1–20.
R.K. Brylinski, Twisted ideals of the nullcone, In: A. Connes, M. Duflo, A. Joseph, R. Rentschier (Eds.) Operator algebras, unitary representations, enveloping algebras and invariant theory, Birkhäuser, Boston, 1990, 289–316.
R.W. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, Wiley, Chichester, 1985.
W.H. Hesselink, Cohomology and the resolution of the nilpotent variety, Math. Ann. 223 (1976), 249–252.
V. Hinich, On the singularities of nilpotent orbits, Israel J. Math. 73 (1991), 297–308.
G.R. Kempf, On the collapsing of homogeneous bundles, Invent. Math. 37 (1976), 229–239.
B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.
H. Kraft and C. Procesi, Closures of conjugacy classes of matrices are normal, Invent. Math. 53 (1979), 227–247.
H. Kraft and C. Procesi, On the geometry of conjugacy classes in classical groups, Comment. Math. Helv. 57 (1982), 539–602.
H. Kraft, Geometrische Methoden in der Invariantentheorie, Vieweg Verlag, Braunschweig, 1984.
H. Kraft, Closures of conjugacy classes in G 2, J. Algebra 126 (1989), 454–465.
D.I. Panyushev, Rationality of singularities and the Gorenstein property for nilpotent orbits, Funct. An. Appl. 25 (1991), 225–226.
J. Weyman, The equations of conjugacy classes of nilpotent matrices, Invent Math. 98 (1989), 229–245.
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Broer, B. (1994). Normality of Some Nilpotent Varieties and Cohomology of Line Bundles on the Cotangent Bundle of the Flag Variety. In: Brylinski, JL., Brylinski, R., Guillemin, V., Kac, V. (eds) Lie Theory and Geometry. Progress in Mathematics, vol 123. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0261-5_1
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DOI: https://doi.org/10.1007/978-1-4612-0261-5_1
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