Level Zero Types and Hecke Algebras for Local Central Simple Algebras

✍ Scribed by Martin Grabitz; Allan J Silberger; Ernst-Wilhelm Zink

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 296 KB
• Volume: 91
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2001.2684

✦ Synopsis

Let D be a central division algebra and A × =GL m (D) the unit group of a central simple algebra over a p-adic field F. The purpose of this paper is to give types (in the sense of Bushnell and Kutzko) for all level zero Bernstein components of A × and to establish that the Hecke algebras associated to these types are isomorphic to tensor products of Iwahori Hecke algebras. The types which we consider are lifted from cuspidal representations y of M(k D ), where M is a standard Levi subgroup of GL m and k D is the residual field of D. Two types are equivalent if and only if the corresponding pairs (M(k D ), y) are conjugate with respect to A × . The results are basically the same as in the split case A × =GL n (F) due to Bushnell and Kutzko. In the non-split case there are more equivalent types and the proofs are technically more complicated.