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Remarks on the Asymptotic Hecke Algebra

  • Alexander Braverman
  • David Kazhdan
Chapter
Part of the Progress in Mathematics book series (PM, volume 326)

Abstract

Let G be a split reductive p-adic group, IG be an Iwahori subgroup, H(G) be the Hecke algebra and C(G) ⊃ H(G) be the Harish-Chandra Schwartz algebra. The purpose of this note is to define (in spectral terms) a subalgebra J(G) of C(G), containing H(G), which we consider as an algebraic version of C(G). We show that the subalgebra J(G)I×IJ(G) is isomorphic to the Lusztig’s asymptotic Hecke algebra J and explain a relation between the algebra J(G) and the Schwartz space of the basic affine space studied in [2].

Keywords

Hecke algebras p-adic groups 

Mathematics Subject Classification (2010):

20C11 22D10 22D20 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Alexander Braverman
    • 1
  • David Kazhdan
    • 2
  1. 1.Department of MathematicsUniversity of Toronto Perimeter Institute for Theoretical Physics and Skolkovo Institute for Science and TechnologyTorontoCanada
  2. 2.Department of MathematicsHebrew UniversityJerusalemIsrael

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