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 G×X such that u is in the Borel subgroup B. The map to U is the projection to the first factor.

References

• Chriss, Neil; Ginzburg, Victor (1997), Representation theory and complex geometry, Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-3792-3, MR1433132, http://books.google.com/books?id=lwS59rR78eIC&dq 
• 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, MR0739636 
• 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, arXiv:math/9802004, ISBN 0-7923-5193-2, MR1649626 
• 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, MR0263830, http://books.google.com/books?ei=m5T1Tbz_A8TmiALD2KWUBw |isbn=978-0196352817}}
• 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, MR0442103 
• Steinberg, Robert (1974), Conjugacy classes in algebraic groups., Lecture Notes in Mathematics, 366, Berlin-New York: Springer-Verlag, doi:10.1007/BFb0067854, ISBN 978-3540066576, MR0352279 
• Steinberg, Robert (1976), "On the desingularization of the unipotent variety", Invent. Math. 36: 209–224, doi:10.1007/BF01390010, MR0430094