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J(X; L, d) and the Langlands Duality

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2072)

Abstract

This section corresponds to the discussion in §1.4 of the Introduction. For \(\Gamma \) an admissible, simple component of C r(L, d), we consider the relative Infinite Grassmannian over \(\breve{\Gamma }\) associated to the sheaf of Lie algebras \(\boldsymbol{{\mathcal{G}}}^{{\prime}}_{\mbox{ $\Gamma $}}\) (see (11.5) for notation).

Keywords

  • Fibre Bundle
  • Weyl Group
  • Cartan Subalgebra
  • Tensor Category
  • Perverse Sheave

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    These are the Lie algebras \(\mathbf{sl}({H}^{0}(\mathcal{O}_{{Z}^{{\prime}}}))\), for \([Z] \in \breve{ \Gamma }\).

  2. 2.

    Recall from (11.1): \(\boldsymbol{\mathcal{G}}_{\mbox{ $\Gamma $}} = \mbox{ ${\pi }^{{\ast}}\mathbf{sl}({\mathcal{F}}^{{\prime}})$}\).

  3. 3.

    See §6 for details about sl 2-triples associated to d  + (v).

  4. 4.

    See [P-S], Ch8, or [Gi], Proposition 1.2.2.

References

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  3. I. Mirkovič, K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. Math. (2) 166(1), 95–143 (2007)

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  4. A. Presley, G. Segal, in Loop Groups. Oxford Mathematical Monographs (Oxford Science Publications/The Clarendon Press, Oxford University Press/New York, 1986)

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Reider, I. (2013). J(X; L, d) and the Langlands Duality. In: Nonabelian Jacobian of Projective Surfaces. Lecture Notes in Mathematics, vol 2072. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35662-9_12

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