Springer resolution

In mathematics, the Springer resolution is a resolution of the variety of nilpotent elements in a semisimple Lie algebra,[1][2] or the unipotent elements of a reductive algebraic group, introduced by Tonny Albert Springer in 1969.[3] The fibers of this resolution are called Springer fibers.[4]

If U is the variety of unipotent elements in a reductive group G, and X the variety of Borel subgroups B, then the Springer resolution of U is the variety of pairs (u,B) of U×X such that u is in the Borel subgroup B. The map to U is the projection to the first factor. The Springer resolution for Lie algebras is similar, except that U is replaced by the nilpotent elements of the Lie algebra of G and X replaced by the variety of Borel subalgebras.[5]

The Grothendieck–Springer resolution is defined similarly, except that U is replaced by the whole group G (or the whole Lie algebra of G). When restricted to the unipotent elements of G it becomes the Springer resolution.[6][7]

References

  1. ↑ Chriss, Neil; Ginzburg, Victor (1997), Representation theory and complex geometry, Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-3792-3, MR 1433132
  2. ↑ Dolgachev, I.; Goldstein, N. (1984), "On the Springer resolution of the minimal unipotent conjugacy class", J. Pure Appl. Algebra, 32 (1): 33–47, MR 0739636, doi:10.1016/0022-4049(84)90012-4
  3. ↑ Springer, T. A. (1969), "The unipotent variety of a semi-simple group", Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, pp. 373–391, ISBN 978-0-19-635281-7, MR 0263830
  4. ↑ Ginzburg, Victor (1998), "Geometric methods in the representation theory of Hecke algebras and quantum groups", Representation theories and algebraic geometry (Montreal, PQ, 1997), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 514, Kluwer Acad. Publ., Dordrecht, pp. 127–183, ISBN 0-7923-5193-2, MR 1649626, arXiv:math/9802004 Freely accessible
  5. ↑ Springer, T. A. (1976), "Trigonometric sums, Green functions of finite groups and representations of Weyl groups", Invent. Math., 36: 173–207, MR 0442103, doi:10.1007/BF01390009
  6. ↑ Steinberg, Robert (1974), Conjugacy classes in algebraic groups., Lecture Notes in Mathematics, 366, Berlin-New York: Springer-Verlag, ISBN 978-3-540-06657-6, MR 0352279, doi:10.1007/BFb0067854
  7. ↑ Steinberg, Robert (1976), "On the desingularization of the unipotent variety", Invent. Math., 36: 209–224, MR 0430094, doi:10.1007/BF01390010


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