Remarks on the asymptotic hecke algebra

Alexander Braverman; David Kazhdan

doi:10.1007/978-3-030-02191-7_4

Remarks on the asymptotic hecke algebra

Alexander Braverman^*, David Kazhdan

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

3 Scopus citations

Abstract

Let G be a split reductive p-adic group, I ⊂ G 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×I ⊂ J(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].

Original language	English
Title of host publication	Progress in Mathematics
Publisher	Springer Basel
Pages	91-108
Number of pages	18
DOIs	https://doi.org/10.1007/978-3-030-02191-7_4
State	Published - 2018

Publication series

Name	Progress in Mathematics
Volume	326
ISSN (Print)	0743-1643
ISSN (Electronic)	2296-505X

Bibliographical note

Publisher Copyright:
© 2018, Springer Nature Switzerland AG.

Keywords

Hecke algebras
p-adic groups

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10.1007/978-3-030-02191-7_4

Cite this

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AB - Let G be a split reductive p-adic group, I ⊂ G 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×I ⊂ J(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].

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Remarks on the asymptotic hecke algebra

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