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A Lie theoretic Galois theory
for the spectral curves
of an integrable system. II

Authors: Andrew McDaniel and Lawrence Smolinsky
Journal: Trans. Amer. Math. Soc. 349 (1997), 713-746
MSC (1991): Primary 58F05, 58F07
MathSciNet review: 1401779
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Abstract: In the study of integrable systems of ODE's arising from a Lax pair with a parameter, the constants of the motion occur as spectral curves. Many of these systems are algebraically completely integrable in that they linearize on the Jacobian of a spectral curve. In an earlier paper the authors gave a classification of the spectral curves in terms of the Weyl group and arranged the spectral curves in a hierarchy. This paper examines the Jacobians of the spectral curves, again exploiting the Weyl group action. A hierarchy of Jacobians will give a basis of comparison for flows from various representations. A construction of V. Kanev is generalized and the Jacobians of the spectral curves are analyzed for abelian subvarieties. Prym-Tjurin varieties are studied using the group ring of the Weyl group $W$ and the Hecke algebra of double cosets of a parabolic subgroup of $W.$ For each algebra a subtorus is identified that agrees with Kanev's Prym-Tjurin variety when his is defined. The example of the periodic Toda lattice is pursued.

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  • [A] Robert D.M. Accola, Vanishing Properties of Theta Functions for Abelian Covers of Riemann Surfaces, Advances in the Theory of Riemann Surfaces, Proceedings of the 1969 Stony Brook Conference, Princeton University Press, Princeton, N.J., 1971, pp. 7-18. MR 44:5448
  • [AvM1] M. Adler and P. van Moerbeke, Completely Integrable Systems, Euclidean Lie Algebras, and Curves, Advances in Math. 38 (1980), 267-317. MR 83m:58041
  • [AvM2] -, Linearization of Hamiltonian Systems, Jacobi Varieties, and Representation Theory, Advances in Math. 38 (1980), 318-379. MR 83m:58042
  • [BE] Robert J. Baston and Michael G. Eastwood, The Penrose Transform: Its Interaction with Representation Theory, Oxford University Press, New York, N.Y., 1989. MR 92j:32112
  • [BL] Ch. Birkenhake and H. Lange, Norm-endomorphisms of abelian subvarieties, Classification of Irregular Varieties, Lecture Notes in Mathematics, vol. 1515, Springer-Verlag, Berlin, 1992, pp. 21-32. MR 93g:14051
  • [B] Armand Borel, et al., Seminar on Transformation Groups, Annals of Mathematics Studies, Number 46, Princeton University Press, Princeton, N.J., 1960. MR 22:7129
  • [BMP] M.R. Bremner, R.V. Moody, and J. Patera, Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras, Pure and Applied Mathematics, vol. 90, Marcel Dekker, Inc., New York, N.Y., 1985. MR 86f:17002
  • [BD] Theodor Bröcker and Tammo tom Dieck, Representations of Compact Lie Groups, GTM, vol. 98, Springer-Verlag, New York, N.Y., 1985. MR 86i:22023
  • [CR] Charles W. Curtis and Irving Reiner, Methods of Representation Theory, vol. I, John Wiley & Sons, New York, N.Y., 1981. MR 82i:20001
  • [G] Phillip A. Griffiths, Linearizing Flows and a Cohomological Interpretation of Lax Equations, American Journal of Mathematics 107 (1985), 1445-1483. MR 87e:58048
  • [GH] Phillip A. Griffiths and Joseph Harris, Principles of Algebraic Geometry, John Wiley & Sons, New York, N.Y., 1978. MR 80b:14001
  • [Ha] Morton Hamermesh, Group Theory and its Application to Physical Problems, Dover Publications, Inc., New York, N.Y., 1962. MR 25:132
  • [H1] James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge, Great Britain, 1990.
  • [H2] -, Introduction to Lie algebras and representation theory, GTM, vol. 9, Springer-Verlag, New York, N.Y., 1972. MR 48:2197
  • [K] Vassil I. Kanev, Spectral Curves, Simple Lie Algebras, and Prym-Tjurin Varieties, Proceedings of Symposia in Pure Mathematics 49 (1989), 627-645. MR 91b:14028
  • [Kr] Aloys Krieg, Hecke Algebras, Memoirs of the American Mathematical Society 87 (435) (1990). MR 90m:16024
  • [M] William Massey, Algebraic Topology: An Introduction, Harcourt-Brace, New York, N.Y., 1967. MR 35:2271
  • [Mc1] Andrew McDaniel, Lie Algebra Representations and the Toda lattice, thesis, Brandeis University (1985).
  • [Mc2] A. McDaniel, Representations of $sl(n,\mathbf {C})$ and the Toda lattice, Duke Math. Journal 56 (1988), 47-99. MR 89e:58061
  • [MS] A. McDaniel and L. Smolinsky, A Lie Theoretic Galois Theory for the Spectral Curves of an Integrable System: I, Communications in Mathematical Physics 149 (1992), 127-148. MR 94f:58065
  • [MS1] A. McDaniel and L. Smolinsky, The Flow of the $G_2$ Periodic Toda Lattice, preprint.
  • [OP] M.A. Olshanetsky and A.M. Perelomov, Classical Integrable Systems Related to Lie Algebras, Physics Reports 71 (1981), 313-400. MR 83d:58032
  • [P] Stefanos Pantazis, Prym Varieties and the Geodesic Flow on SO(n), Mathematische Annalen 273 (1986), 297-315. MR 87g:58054
  • [R] John F. X. Ries, Splittable Jacobi Varieties, Curves, Jacobians, and Abelian Varieties, Contemporary Mathematics, vol. 136, American Mathematical Society, Providence, R.I., 1992, pp. 305-326. MR 93j:14031
  • [vMM] P. van Moerbeke and D. Mumford, The Spectrum of Difference Operators and Algebraic Curves, Acta Math. 143 (1979), 93-154. MR 80e:58028

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Additional Information

Andrew McDaniel
Affiliation: Department of Mathematics, Georgetown University, Washington, D.C. 20057

Lawrence Smolinsky
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803

Received by editor(s): February 22, 1995
Additional Notes: The second author was partially supported by a Louisiana Education Quality Support Fund grant LEQSF(90-93)-RD-A-10.
Article copyright: © Copyright 1997 American Mathematical Society

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