Thu, Nov 16, 3:30pm Matt Szczesny (Boston Univ.) The Langlands program: from arithmetic to conformal field theory This will be a very elementary attempt to trace the sequence of analogies that lead from the Langlands program for global fields to the geometric formulation given by A. Beilinson and V. Drinfeld. I will attempt to explain the formulation of the latter in the language of conformal f...

Arthur's conjectures predict the existence of some very interesting unitary representations occurring in spaces automorphic forms. We prove unitarity ``Langlands element'' (i.e., representation specified by Arthur) all unipotent Arthur packets for split real exceptional groups. The proof uses Eisenstein series, Langlands' constant term formula and square integrability criterion, analytic proper...

One of the fundamental problems of local harmonic analysis is to classify the set Π(G) of equivalence classes of irreducible representations of G(F ). The problem separates naturally into two parts. The first is to establish the local Langlands correspondence. This conjecture of Langlands asserts that Π(G) is a disjoint union of finite subsets Πφ, indexed by (equivalence classes of) Langlands p...

Let ∧ : GLn(C) −→ GLN (C), where N = n(n−1) 2 , be the map given by the exterior square. Then Langlands’ functoriality predicts that there is a map from cuspidal representations of GLn to automorphic representations of GLN , which satisfies certain canonical properties. To explain, let F be a number field, and let A be its ring of adeles. Let π = ⊗ v πv be a cuspidal (automorphic) representatio...

If ρ is a selfdual representation of a group G on a vector space V over C, we will say that ρ is orthogonal, resp. symplectic, if G leaves a nondegenerate symmetric, resp. alternating, bilinear form B : V ×V → C invariant. If ρ is irreducible, exactly one of these possibilities will occur, and we may define a sign c(ρ) ∈ {±1}, taken to be +1, resp. −1, in the orthogonal, resp. symplectic, case....

The local Langlands correspondence for GL(n) of a non-Archimedean local field F parametrizes irreducible admissible representations of GL(n, F ) in terms of representations of the Weil-Deligne group WDF of F . The correspondence, whose existence for p-adic fields was proved in joint work of the author with R. Taylor, and then more simply by G. Henniart, is characterized by its preservation of s...

We present a gauge-theoretic interpretation of the “analytic” version geometric Langlands program, in which Hitchin Hamiltonians and Hecke operators are viewed as concrete acting on Hilbert space quantum states. The gauge theory ingredients required to understand this construction—such electric–magnetic duality between Wilson ’t Hooft line four-dimensional theory—are same ones that enter unders...