Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

The spherical Hecke algebra, partition functions, and motivic integration


Authors: William Casselman, Jorge E. Cely and Thomas Hales
Journal: Trans. Amer. Math. Soc. 371 (2019), 6169-6212
MSC (2010): Primary 22E46; Secondary 22E35, 11F70, 11S37
DOI: https://doi.org/10.1090/tran/7465
Published electronically: November 27, 2018
MathSciNet review: 3937321
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This article gives a proof of the Langlands-Shelstad fundamental lemma for the spherical Hecke algebra for every unramified $ p$-adic reductive group $ G$ in large positive characteristic. The proof is based on the transfer principle for constructible motivic integration. To carry this out, we introduce a general family of partition functions attached to the complex $ L$-group of the unramified $ p$-adic group $ G$. Our partition functions specialize to Kostant's $ q$-partition function for complex connected groups and also specialize to the Langlands $ L$-function of a spherical representation. These partition functions are used to extend numerous results that were previously known only when the $ L$-group is connected (that is, when the $ p$-adic group is split). We give explicit formulas for branching rules, the inverse of the weight multiplicity matrix, the Kato-Lusztig formula for the inverse Satake transform, the Plancherel measure, and Macdonald's formula for the spherical Hecke algebra on a nonconnected complex group (that is, nonsplit unramified $ p$-adic group).


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 22E46, 22E35, 11F70, 11S37

Retrieve articles in all journals with MSC (2010): 22E46, 22E35, 11F70, 11S37


Additional Information

William Casselman
Affiliation: Department of Mathematics, The University of British Columbia, Vancouver, British Columbia, Canada V6T172

Jorge E. Cely
Email: celyje@gmail.com

Thomas Hales
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

DOI: https://doi.org/10.1090/tran/7465
Received by editor(s): November 21, 2016
Received by editor(s) in revised form: August 3, 2017
Published electronically: November 27, 2018
Additional Notes: During the last part of this work, the second author was supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement no. 615722 MOTMELSUM
Article copyright: © Copyright 2018 William Casselman, Jorge E. Cely, and Thomas Hales