Abstract
We define the (second) Adler-Gelfand-Dickey Poisson structure on differential operators over an elliptic curve and classify symplectic leaves of this structure. This problem leads to the problem of classification of coadjoint orbits for double loop algebras, conjugacy classes in loop groups, and holomorphic vector bundles over the elliptic curve. We show that symplectic leaves have a finite but (unlike the traditional case of operators on the circle) arbitrarily large codimension, and compute it explicitly.
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Etingof, P.I., Khesin, B.A. Affine Gelfand-Dickey brackets and holomorphic vector bundles. Geometric and Functional Analysis 4, 399–423 (1994). https://doi.org/10.1007/BF01896402
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DOI: https://doi.org/10.1007/BF01896402