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• W1489731390 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 <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding=application/x-tex>W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Hecke algebra of double cosets of a parabolic subgroup of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper W period> <mml:semantics> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>W.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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." @default.
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• W1489731390 date "1997-01-01" @default.
• W1489731390 modified "2023-10-18" @default.
• W1489731390 title "A Lie theoretic Galois theory for the spectral curves of an integrable system. II" @default.
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• W1489731390 doi "https://doi.org/10.1090/s0002-9947-97-01853-9" @default.
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