Springer resolution

In mathematics, the Springer resolution is a resolution of the variety of nilpotent elements in a semisimple Lie algebra, or the unipotent elements of a reductive algebraic group, introduced by Springer (1969). The fibers of this resolution are called Springer fibers.

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

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.

References

• Chriss, Neil; Ginzburg, Victor (1997), Representation theory and complex geometry, Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-3792-3, MR 1433132
• Dolgachev, I.; Goldstein, N. (1984), "On the Springer resolution of the minimal unipotent conjugacy class", J. Pure Appl. Algebra, 32 (1): 33–47, doi:10.1016/0022-4049(84)90012-4, MR 0739636
• 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, vol. 514, Kluwer Acad. Publ., Dordrecht, pp. 127–183, arXiv:math/9802004, ISBN 0-7923-5193-2, MR 1649626
• 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
• Springer, T. A. (1976), "Trigonometric sums, Green functions of finite groups and representations of Weyl groups", Invent. Math., 36: 173–207, doi:10.1007/BF01390009, MR 0442103
• Steinberg, Robert (1974), Conjugacy classes in algebraic groups., Lecture Notes in Mathematics, vol. 366, Berlin-New York: Springer-Verlag, doi:10.1007/BFb0067854, ISBN 978-3-540-06657-6, MR 0352279
• Steinberg, Robert (1976), "On the desingularization of the unipotent variety", Invent. Math., 36: 209–224, doi:10.1007/BF01390010, MR 0430094