Wednesday, 14-16, ESI, Schrödinger lecture hall
starting on October 4, 2006

 
Let F be a local field and D a central division algebra of finite dimension over F. The lecture series will be divided into three parts: the first is an overview of some general facts about reductive groups over local fields. The second is devoted to the classification of representations of GL(n,F) and GL(n,D); here a key role is played by the Jacquet-Langlands correspondence. The third is the proof of the Jacquet-Langlands correspondence when the characterisitc of the base field is zero, following Deligne, Kazhdan and Vignéras. Here is a detailed description of these three sections:

1. Local fields and reductive groups over local fields. Hecke algebra of functions. Admissible representations and their characters. Local integrability of characters. Orbital integrals. Parabolic induction and restriction. Unitary representations. Cuspidal, square integrable and tempered representations. Orthogonality of characters. Langlands classification.
2. Zelevinsky classification of smooth irreducible representations of GL(n,F). The Hopf algebra of representations. The involution. Tadic's classification of unitary smooth irreducible representations of GL(n,F). Statement of the Jacquet-Langlands correspondence. Consequences for GL(n,D). Tadic's classification of smooth irreducible representations of GL(n,D). Tadic's conjectures on the unitary dual of GL(n,D).
3. Basic facts about global representations. Simple Kazhdan-Deligne trace formula. Proof of the Jacquet-Langlands correspondence (when the characteristic of F is zero) following Deligne, Kazhdan and Vignéras.

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   ESI Senior Research Fellow Program coordinated by Prof. Joachim Schwermer, Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Wien (Joachim.Schwermer@univie.ac.at).