Algebra and Representation Theory Seminar
Automorphic L-functions in positive characteristic and applications to representation theory of p-adic reductive groups
Luis Lomeli, University of Oklahoma
Friday, March 30, 2012
Building upon previous work, the study of automorphic L-functions in positive characteristic in the case of a Siegel Levi subgroup of a quasi-split classical group is concluded by adding the remaining case of a quasi-split orthogonal group to the list of examples. Besides being of importance by themselves, the resulting L-functions play an important role in the study of globally generic cuspidal automorphic representations of classical groups in general. Having L-functions and root numbers defined for every smooth representation of a Siegel Levi subgroup of a classical group over any non-archimedean local field, gives rise to interesting applications that are new in characteristic p. Among other tools in p-adic harmonic analysis, intertwining operators between parabolically induced representations play a crucial role in Harish-Chandra's study of the Plancherel theorem for p-adic reductive groups. When the representation is generic, the Langlands-Shahidi theory of the local coefficient is defined via intertwining operators and the uniqueness property of Whittaker models. It is then possible to obtain a formula for the Plancherel measure in terms of local L-factors and root numbers. It is also possible to normalize intertwining operators in a way predicted by the theory of endoscopy. Furthermore, one can obtain a reducibility criterion in order to determine the reducibility points of an induced representation in terms of L-functions. Time permitting, we will discuss additional applications.