Representations of Reductive Groups

This course consists of two parts. In the first we will study representations of reductive groups over local non-archimedian fields [ such as Qp and Fq((s))]. In this part I’ll closely follow the notes of the course of J.Bernstein. Moreover I’ll often copy big chanks from these notes. In the second the representations of reductive groups over 2-dimensional local fields [ such as Qp((s))]. In the first part we explain the basics of a) induction from parabolic and parahoric subgroups, b) Jacquet functors, c) cuspidal representations d) the second adjointness and e) Affine Hecke algebras. In the second we discuss the generalization these concepts to the case of representations of reductive groups over 2-dimensional local fields. Prerequisites. The familiarity with the following subjects will be helpful. a) P -adic numbers, [see first few chapters of the book ”p-adic numbers, p-adic analysis, and zeta-functions” by N.Koblitz or sections 4-5 in the book ”Number theory” of Borevich and Shafarevich]. b) Basics of the theory of split reductive groups G [Bruhat decomposition, Weyl groups, parabolic and Levi subgroups] of reductive groups, [ One who does not know this this theory can restrict oneself to the case when G = GL(n) when Bruhat decomposition= Gauss decomposition.] c) Basics of the category theory: adjoint functors, Abelian categories. [ see the chapter 2 of book ”Methods of homological algebra”