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CARTAN DECOMPOSITION FOR COMPLEX LOOP GROUPS

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Abstract

In this note we formulate and prove a version of Cartan decomposition for holomorphic loop groups, similar to Cartan decomposition for p-adic loop groups, discussed in [3], [6]. The main technical tool that we use is the (well-known) interpretation of twisted conjugacy classes in the holomorphic loop group in terms of principal holomorphic bundles on an elliptic curve.

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References

  1. M. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414–452.

  2. V. Balaji, Lectures on principal bundles, in: Moduli Spaces and Vector Bundles, London Math. Soc. Lecture Note Ser., Vol. 359, Cambridge Univ. Press, Cambridge, 2009, pp. 2–28.

  3. A. Braverman, D. Kazhdan, The spherical Hecke algebra for affine Kac–Moody groups I, Ann. of Math. (2) 174 (2011), no. 3, 1603–1642.

  4. R. Friedman, J. Morgan, Holomorphic principal G-bundles over elliptic curves, preprint.

  5. R. Friedman, J. Morgan, E. Witten, Principal G-bundles over elliptic curves, Math. Res. Lett. 5 (1998), no. 1-2, 97–118.

    Article  MATH  MathSciNet  Google Scholar 

  6. H. Garland, The arithmetic theory of loop groups, Publ. Math. IH ÉS 52 (1980), 5–136.

    Article  MATH  MathSciNet  Google Scholar 

  7. A. H. W. Schmitt, Geometric Invariant Theory and Decorated Principal Bundles, Zürich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008.

  8. N. Wallach, Real Reductive Groups I, Pure and Applied Mathematics, Vol. 132, Academic Press, Boston, MA, 1988.

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Authors and Affiliations

  1. Department of Mathematics, Brown University, 151 Thayer street, Providencde, RI, 02912, USA

    ALEXANDER BRAVERMAN

  2. Einstein Institute of Mathematics Edmond J. Safra Campus, The Hebrew University of Jerusalem Givat Ram, Jerusalem, 91904, Israel

    DAVID KAZHDAN

Authors
  1. ALEXANDER BRAVERMAN
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  2. DAVID KAZHDAN
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Correspondence to ALEXANDER BRAVERMAN.

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To E. B. Dynkin on his 90th birthday

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BRAVERMAN, A., KAZHDAN, D. CARTAN DECOMPOSITION FOR COMPLEX LOOP GROUPS. Transformation Groups 19, 303–311 (2014). https://doi.org/10.1007/s00031-014-9268-7

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  • DOI: https://doi.org/10.1007/s00031-014-9268-7

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