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]

Examples

When G=SL(2), the Lie algebra Springer resolution is T*P1 → n, where n are the nilpotent elements of sl(2). In this example, n are the matrices x with tr(x2)=0, which is a two dimensional conical subvariety of sl(2). n has a unique singular point 0, the fibre above which in the Springer resolution is the zero section P1 .

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, Igor; Goldstein, Norman (1984), "On the Springer resolution of the minimal unipotent conjugacy class", Journal of Pure and Applied Algebra, 32 (1): 33–47, doi:10.1016/0022-4049(84)90012-4, hdl:2027.42/24847, MR 0739636
  3. Springer, Tonny 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, vol. 514, Kluwer Acad. Publ., Dordrecht, pp. 127–183, arXiv:math/9802004, Bibcode:1998math......2004G, ISBN 0-7923-5193-2, MR 1649626
  5. Springer, Tonny A. (1976), "Trigonometric sums, Green functions of finite groups and representations of Weyl groups", Inventiones Mathematicae, 36: 173–207, Bibcode:1976InMat..36..173S, doi:10.1007/BF01390009, MR 0442103, S2CID 121820241
  6. 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
  7. Steinberg, Robert (1976), "On the desingularization of the unipotent variety", Inventiones Mathematicae, 36: 209–224, Bibcode:1976InMat..36..209S, doi:10.1007/BF01390010, MR 0430094, S2CID 120400717


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