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Inventiones mathematicae

, Volume 204, Issue 2, pp 347–442 | Cite as

Iwahori–Hecke algebras for p-adic loop groups

  • Alexander BravermanEmail author
  • David Kazhdan
  • Manish M. Patnaik
Article

Abstract

This paper is a continuation of Braverman and Kazhdan (Ann Math (2) 174(3):1603–1642, 2011) in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac–Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori–Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik’s double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald’s formula for the spherical function in the finite-dimensional case. The results of this paper have been previously announced in Braverman and Kazhdan (European Congress of Mathematics. European Mathematical Society, Zürich, 2014).

Notes

Acknowledgments

A.B. was partially supported by the NSF Grant DMS-1200807 and by Simons Foundation. D.K. was partially supported by the European Research Council Grant 247049. M.P. was supported by an NSF Postdoctoral Fellowship, DMS-0802940 and an University of Alberta startup grant while this work was being completed. We would like to thank I. Cherednik, P. Etingof, H. Garland, and E. Vasserot for useful discussions. We are very grateful to the referee for a careful reading of our paper, for pointing out a number of inaccuracies, and for offering several helpful and clarifying suggestions.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Alexander Braverman
    • 1
    Email author
  • David Kazhdan
    • 2
  • Manish M. Patnaik
    • 3
  1. 1.Mathematics DepartmentBrown UniversityProvidenceUSA
  2. 2.Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael
  3. 3.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

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