Local Langlands Correspondence and Vogan's Conjecture

In this talk, I will provide an elementary introduction to the Local Langlands Correspondence, focusing on the key concepts and definitions of the objects involved. We will build intuition by examining some simpler instances of the correspondence, including local class field theory, the case of GLn , and the split case, before presenting the general statement. In the second part, we will explore a conjecture proposed by Vogan, which suggests a reduction of the Local Langlands Correspondence from p-adic fields to finite fields. Particular emphasis will be placed on the GLn case, where the conjecture has an easier formulation and has been established through the work of Macdonald, Silberger, and Zink. Finally, I will briefly discuss the conjecture for SLn, talking about my research contributions to this area.