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The Satake isomorphism for special maximal parahoric Hecke algebras


Authors: Thomas J. Haines and Sean Rostami
Journal: Represent. Theory 14 (2010), 264-284
MSC (2010): Primary 11E95, 20G25; Secondary 22E20
DOI: https://doi.org/10.1090/S1088-4165-10-00370-5
Published electronically: March 8, 2010
MathSciNet review: 2602034
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Abstract: Let $ G$ denote a connected reductive group over a nonarchimedean local field $ F$. Let $ K$ denote a special maximal parahoric subgroup of $ G(F)$. We establish a Satake isomorphism for the Hecke algebra $ \mathcal{H}_K$ of $ K$-bi-invariant compactly supported functions on $ G(F)$. The key ingredient is a Cartan decomposition describing the double coset space $ K\backslash G(F)/K$. As an application we define a transfer homomorphism $ t: \mathcal{H}_{K^*}(G^*) \rightarrow \mathcal{H}_K(G)$ where $ G^*$ is the quasi-split inner form of $ G$. We also describe how our results relate to the treatment of Cartier [Car], where $ K$ is replaced by a special maximal compact open subgroup $ \widetilde{K} \subset G(F)$ and where a Satake isomorphism is established for the Hecke algebra $ \mathcal{H}_{\widetilde{K}}$.


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Additional Information

Thomas J. Haines
Affiliation: University of Maryland, Department of Mathematics, College Park, Maryland 20742-4015
Email: tjh@math.umd.edu

Sean Rostami
Affiliation: University of Maryland, Department of Mathematics, College Park, Maryland 20742-4015
Email: srostami@math.umd.edu

DOI: https://doi.org/10.1090/S1088-4165-10-00370-5
Received by editor(s): October 17, 2009
Received by editor(s) in revised form: November 29, 2009
Published electronically: March 8, 2010
Additional Notes: The first author was partially supported by NSF Focused Research Grant DMS-0554254 and NSF Grant DMS-0901723, and by a University of Maryland GRB Semester Award.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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