Thursday, 13:00-15:00, Schrödinger lecture hall
October 16, 2008 - February 28, 2009

 
Abstract: Let G be a reductive algebraic group over Q. Let A be the ring of ad' eles of Q. One of the goals of the Langlands program is to understand square--integrable automorphic forms which are particularly nice functions in L²(G(Q)|G(A)) that contain arithmetic information related to the absolute Galois group. According to Langlands, the formalism of Eisenstein series is used not only to construct automorphic forms but also to construct Lfunctions which appear in a constant term. Eisenstein series are meromorphic functions which have complicated singularities. We explain how one can use local representation theory to study their poles. We explain some applications of the theory of Eisenstein series. On the other hand, there are cusp forms which are mysterious and there are not so many ways to study them. We explain some methods for constructing the cusp forms.

We plan to cover the following:

• Recapitulation of some results on representation theory; unitary representation theory
• Unramified representations and partial automorphic L--functions
• Spaces of automorphic forms
• Eisenstein series and its constant term; local intertwining operators, their normalization using Langlands--Shahidi method
• An example: Eisenstein series for SL₂ and Langlands theorem on automorphic realization of representations of SL₂(A)
• Eisenstein series supported on a Borel subgroup and unipotent representations for split classical groups over R and C; automorphic realization of unipotent representations; we explain how one can use p-adic Hecke operators to draw conclusions on Q_∞=R.
• Arthur packets of square integrable automorphic representations; conjectural description of discrete spectrum for split classical groups; the unitarity of negative unramified representations for split classical p--adic groups. item Some constructions and existence of cusp forms in adelic and classical set-up; again we explain how one can use 0-adic Hecke operators to draw conclusions on Q_∞=R.