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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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