Introduction. These notes are based on a series of lectures given by the author at the Central China Normal University in Wuhan (July 2007). The aim of the lectures was to provide an introduction to the Langlands philosophy. According to a theorem of Chevalley, connected reductive split algebraic groups over a fixed field A are in bijection with certain combinatorial objects called root systems...

We carry out (with technical modifications) some cases of a procedure proposed by R. Langlands in Beyond Endoscopy. This gives a new proof of the classification of “dihedral forms” on GL(2), avoiding endoscopic methods: instead, it uses an infinite limiting process in the trace formula. We also explain the relationship of Langlands’ idea to the results of [10] that count automorphic forms assoc...

Geometric Langlands duality is usually formulated as a statement about Riemann surfaces, but it can be naturally understood as a consequence of electric-magnetic duality of four-dimensional gauge theory. This duality in turn is naturally understood as a consequence of the existence of a certain exotic supersymmetric conformal field theory in six dimensions. The same six-dimensional theory also ...

This is an attempt to formulate a geometric Langlands conjecture for G = GSpin2n+1 and GSp2n, n ≥ 1. Namely, let Ǧ be the Langlands dual group (over Q̄l), let X be a smooth projective connected curve. Given a Ǧ-local system E on X , assuming some irreducibility type conditions on VE for some representations V of Ǧ, we propose a conjectural construction of a distinguished E-Hecke automorphic shea...

The local Langlands conjectures imply that to every generic supercuspidal irreducible representation of G2 over a p-adic field, one can associate a generic supercuspidal irreducible representation of either PGSp6 orPGL3. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of G2 and other representations of PGSp6 and PGL3 when p 6= 2. This ...

In the context of the local Langlands philosopy for R, Adams, Barbasch and Vogan describe a bijection between the simple Harish-Chandra modules for a real reductive group G(R) and the space of “complete geometric parameters”—a space of equivariant local systems on a variety on which the Langlands-dual of G(R) acts. By a conjecture of Soergel, this bijection can be enhanced to an equivalence of ...

This is the fourth in a series of papers (the earlier ones are [Casselman:1989], [Casselman:1993] and [Casselman:1999]) intended to present eventually a new way of proving, among other things, the well known results of Chapter 7 of [Langlands:1977] on the completeness of the spectrum arising from cusp forms and Eisenstein series. That Langlands’ results have lasted for nearly 40 years without m...

The local Langlands conjecture for GL(n) over a p-adic field, n < p.