## The Satake isomorphism for special maximal parahoric Hecke algebras

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- by Thomas J. Haines and Sean Rostami
- Represent. Theory
**14**(2010), 264-284 - DOI: https://doi.org/10.1090/S1088-4165-10-00370-5
- Published electronically: March 8, 2010
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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 $\mathcal {H}_{\widetilde {K}}$.## References

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## Bibliographic Information

**Thomas J. Haines**- Affiliation: University of Maryland, Department of Mathematics, College Park, Maryland 20742-4015
- MR Author ID: 659516
- Email: tjh@math.umd.edu
**Sean Rostami**- Affiliation: University of Maryland, Department of Mathematics, College Park, Maryland 20742-4015
- Email: srostami@math.umd.edu
- 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.
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2602034