Strongly positive representations of odd $GSpin$ groups

by Dr. Yeansu Kim (University of Iowa, USA,)

Tuesday, August 5, 2014 from 16:00 to 17:00 (Asia/Kolkata)
at Colaba Campus ( AG-77 )

Description

The classfication of discrete series representations of connected reductive groups $G$ over a non-Archimedean local field $F$ of characteristic zero is an imPortant step in establishing local Langlands correspondence. Briefly, the local Langlands correspondence asserts that there exists a 'natural'
bijection between two different sets of objects: an Arithmetic (Galois or Weil-Deligne) side and an analytic (representation theoretic) side.
On the analytic side, the objects are irreducible admissible
representations of connected reductive group over a local field. To study admissible representations, the following filtration of admissible
representation according to growth properties of matrix coefficients is useful: (1) supercuspidal $\subseteq$ discrete series $\subseteq$ tempered $\subseteq$ admissible: One of the strategy to study properties or theorems of admissible representations is first to prove those in the case of supercuspidal representations. If it works, we generalize those proofs to the case of discrete series, tempered and admissible representations following the filtration (1). The natural question is how we generalize the theorems in the case of previous class to the case of next class. The last step (from tempered representations to admissible representations) is well constructed in general and is called 'Langlands classification'. The second
step (from discrete series representations to tempered representations) is called R-groups which is studied by Goldberg and others in the case of classical groups. In this talk, I will explain the first step (from supercuspidal representations to discrete series representations) which is called 'classification of discrete series representations' in the case of $GSpin$ groups. More precisely, we consider one more important class between supercuspidal representations and discrete series which is called strongly positive discrete series representations. I will explain the classification of strongly positive discrete series of odd $GSpin$ groups over $F$ and describe the general discrete series representations of odd $GSpin$ groups over $F$ using the classification of strongly positive representations. One of the applications of this classification results is to show the equality of $L$-functions from Langlands-Shahidi method and Artin $L$-functions through local Langlands correspondence. Furthermore, I will explain one of the applications of the equality of $L$-functions, which is so-called the generic Arthur packet conjecture.