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Iwahori-Hecke algebras. (English) Zbl 1202.22013
Let $$F$$ denote a $$p$$-adic field with valuation ring $$\mathcal{O}$$ and prime ideal $$P$$. Further, denote by $$k$$ the residue field $$\mathcal{O} / P$$. In the paper under review, the authors consider a split connected reductive group $$G$$ over $$F$$, with split maximal torus $$A$$ and Borel subgroup $$B = AN$$. They assume that the groups $$A$$, $$N$$ and $$G$$ are defined over $$\mathcal{O}$$, and write $$I$$ for the Iwahori subgroup of $$G(\mathcal{O})$$, defined as the inverse image under the mapping $$G(\mathcal{O}) \rightarrow G(k)$$ of $$B(k)$$.
The Iwahori-Hecke algebra $$H$$ of the split $$p$$-adic group $$G$$ is defined by $$H = C_{c} (I \backslash G / I)$$. In this paper, a detailed and self-contained exposition of many basic facts about the algebra $$H$$ is given.
Influenced strongly by the work of Bernstein, the authors use the universal unramified principal series module $$M = C_{c}(A_{\mathcal{O}} N \backslash G / I)$$ to investigate the Iwahori-Hecke algebra $$H$$. We note that $$M$$ is a right $$H$$-module, which the authors use to develop the theory of the intertwining operators in a purely algebraic way. After developing this theory, many important results regarding Iwahori-Hecke algebras are proved (some of these results are already known, but here they are proved in a different way). Among others, the authors give a detailed description of the center of the algebra $$H$$, prove Macdonald’s formula, the Casselman-Shalika formula, the Lusztig-Kato formula and give an exposition of the Satake isomorphism.

##### MSC:
 22E35 Analysis on $$p$$-adic Lie groups 11F70 Representation-theoretic methods; automorphic representations over local and global fields 20C08 Hecke algebras and their representations
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