Title：Representation theory of the general linear group and its finite central covers over a non-archimedean local field
Speaker: Jiandi Zou（Technion-Israel Institute of Technology）
Time: Friday, Nov3, 15:45-16:45
Location：Mingde Building, B201-1
Abstract: Let F be a non-archimedean local field of residual characteristic p and let G=GL_r(F). Motivated by the local Langlands correspondence, one would like to study
1. the category of (equivalence classes of) smooth complex representations Rep(G) of G;
2. the subset of irreducible representations Irr(G);
3. the subset of cuspidal representations Cusp(G).
The answer to the first question is given by Bernstein. He considered the block decomposition of Rep(G) into the product of Rep_{s}(G), where s ranges over the set of inertial cuspidal supports of G. Then the first question reduces to studying the subcategory Rep_{s}(G) for each s, which is somehow equivalent to studying modules of an affine Hecke algebra of type A. The answer to the second question is due to Bernstein-Zelevinsky and Zelevinsky, where they gave a combinatorial classification of irreducible representations via cuspidal representations. Finally, the answer to the third question is given by Bushnell-Kutzko, where they gave an explicit and exhaustive construction of all cuspidal representations via compact induction.
In this talk, I will first focus on explaining the above known theories, which lead to the answer to the three questions.Then I will explain my work (partially joint with Erez Lapid and Eyal Kaplan) on exploring the same questions for Rep(G'), Irr(G'), Cusp(G'), where G' is a certain good finite central cover of G=GL_r(F).
