The Generalized Cayley Map from an Algebraic Group to its Lie Algebra

  • Bertram Kostant
  • Peter W. Michor
Chapter
Part of the Progress in Mathematics book series (PM, volume 213)

Abstract

Each infinitesimally faithful representation of a reductive complex connected algebraic group Ginduces a dominant morphism Φ from the group to its Lie algebra g by orthogonal projection in the endomorphism ring of the representation space. The map Φ identifies the field Q(G)of rational functions on Gwith an algebraic extension of the field Q(g)of rational functions on g. For the spin representation of Spin(V) the map Φ essentially coincides with the classical Cayley transform. In general, properties of Φ are established and these properties are applied to deal with a separation of variables (Richardson) problem for reductive algebraic groups: Find Harm(G) so that for the coordinate ring A(G) of G we have A(G) = A(G)G ® Harm(G). As a consequence of a partial solution to this problem and a complete solution for SL(n) one has in general the equality [Q(G): Q(g)] = [Q(G) G : Q(g) G ] of the degrees of extension fields. Among other results, Φ yields (for the complex case) a generalization, involving generic regular orbits, of the result of Richardson showing that the Cayley map, whenGis semisimple, defines an isomorphism from the variety of unipotent elements inGto the variety of nilpotent elements in g. In addition if G is semisimple the Cayley map establishes a diffeomorphism between the real submanifold of hyperbolic elements in G and the space of infinitesimal hyperbolic elements in g. Some examples are computed in detail.

Keywords and phrases

Cayley mappings for representations 

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References

  1. [1]
    Bardsley, P., Richardson, R.W., Etale slices for algebraic transformation groups in characteristicp Proc. Lond. Math. Soc. I. Ser. 51(1985), 295–317.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Borel, A.Linear Algebraic GroupsW. A. Benjamin, 1969.Google Scholar
  3. [3]
    Chevalley, C.The Algebraic Theory of SpinorsColumbia Univ. Press, New York, 1954.MATHGoogle Scholar
  4. [4]
    Dynkin, E.B., Semisimple subalgebras of semisimple Lie algebrasMat. Sbornik Nov. Ser. 30(1952), 349–462. English transl. inAMS Transl. II. Ser 6(1957), 111–243.MATHGoogle Scholar
  5. [5]
    Gorbatsevich, V.V., Onishchik, A.L., Vinberg, E.B., Structure of Lie groups and Lie algebras. In: Lie Groups and Lie Algebras III (eds., A.L. Onishchik, E.B. Vinberg), Encyclopedia of Mathematical Sciences41Springer-Verlag, Berlin, 1994.Google Scholar
  6. [6]
    Humphrey, J.Linear Algebraic GroupsGTM 21, Springer-Verlag, 1975, 1981.Google Scholar
  7. [7]
    Kostant, B., The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie groupAmer. J. Math. 81(1959), 973–1032.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Kostant B. Lie group representations on polynomial ringsBull. Amer. Math. Soc. 69(1963), 518–526.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Kostant, B., Lie group representations on polynomial ringsAmer. J. Math. 85(1963), 327–404.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Kostant B. Eigenvalues of a Laplacian and commutative Lie subalgebrasTopology 3(1965), 147–159.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Kostant, B., On Convexity, the Weyl group and the Iwasawa decompositionAnn. Scient. Ec. Norm. Sup. 6(1973), 413–455.MathSciNetMATHGoogle Scholar
  12. [12]
    Kostant, B., Clifford Algebra Analogue of the Hopf-Koszul-Samelson Theorem, the p-decompositionC(g)= EndV p®C(P)and the g-module structure of A g Adv. of Math. 125(1997), 275–350.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Lipschitz, R., CorrespondenceAnn. Math. 69(1959), 247–251.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Luna, D., Slices étalesBull. Soc. Math. FranceMémoire33(1973), 81–105.MATHGoogle Scholar
  15. [15]
    Richardson, R. W., The conjugating representation of a semisimple groupInvent. Math. 54 (1979)229–245.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Richardson, R. W., An application of the Serre conjecture to semisimple algebraic groups. InAlgebra Carbondale 1980Lecture Notes in Math. 848, Springer-Verlag, 1981, pp. 141–151.Google Scholar
  17. [17]
    Springer, T. A. The unipotent variety of a semisimple group. InAlgebr. Geom. Bombay Colloq. 19681969, pp. 373–391.MathSciNetGoogle Scholar
  18. [18]
    Steinberg, R., Regular elements of semisimple algebraic groupsInst. Hautes Études Sci. Publ. Math. 25(1965), 49–80.CrossRefGoogle Scholar
  19. [19]
    Steinberg, R.Conjugacy Classes in Algebraic Groups. Notes by Vinay V. DeodharLecture Notes in Mathematics 36, Springer-Verlag, 1974, pp. vi+159.Google Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Bertram Kostant
    • 1
  • Peter W. Michor
    • 2
    • 3
  1. 1.Department of MathematicsMITCambridgeUSA
  2. 2.Erwin Schrödinger Institute of Mathematical PhysicsWienAustria
  3. 3.Institut für MathematikUniversität WienWienAustria

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