Level zero types and Hecke algebras for local central simple algebras
The theory of types for connected reductive \(p\)-adic groups was introduced by C. Bushnell and P. Kutzko. A type is a pair \((K, \tau)\) consisting of a certain open subgroup \(K\) and a certain irreducible representation \(\tau\) of \(K\). For \(G=GL_n(F)\) over the \(p\)-adic field \(F\), a classification of the irreducible smooth representations of \(G\) via types is given in terms of Hecke algebra isomorphisms [\textit{C. Bushnell} and \textit{P. Kutzko}, Ann. Math. Stud. 129 (1993; Zbl 0787.22016)]. Let \(D\) be a central division algebra and \(A^\times=GL_m(D)\) the unit group of a central simple algebra over \(F\). This article concerns the theory of types for \(A^\times\). The authors give a classification of types for all level zero Bernstein components of \(A^\times\) and prove that the Hecke algebras associated to these types are isomorphic to tensor products of Iwahori Hecke algebras, as in the split case \(G=GL_n(F)\) due to Bushnell and Kutzko. Level zero types were also studied by \textit{L. Morris} [Compos. Math. 118, No. 2, 135-157 (1999; Zbl 0937.22011)].
scientific article; zbMATH DE number 1787789
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