Let K/F be a quadratic extension of p-adic fields. The Bernstein-Zelevinsky’s classification asserts that generic representations are parabolically induced from quasi-squareintegrable representations. We show, following a method developed by Cogdell and PiatetskiShapiro, that expected equality of the Rankin-Selberg type Asai L-function of a generic representation of GL(n, K), and the Asai L-fun...

Let p and q be two distinct prime ideals of Fq[T ]. We use the Eisenstein ideal of the Hecke algebra of the Drinfeld modular curveX0(pq) to compare the rational torsion subgroup of the Jacobian J0(pq) with its subgroup generated by the cuspidal divisors, and to produce explicit examples of Jacquet-Langlands isogenies. Our results are stronger than what is currently known about the analogues of ...

Let G = Sp(2n,C) be a complex symplectic group. In the companion paper [math.AG/0601154], we introduced a (G × C × C)-variety N, which we call the exotic nilpotent cone. In this paper, we realize the Hecke algebra H of type C n with unequal parameters via equivariant algebraic K-theory in terms of the geometry of N. This enables us to establish a Deligne-Langlands type classification of “non-cr...

‎We illustrate a method of computing Langlands-Shahidi local coefficients via the theory of types and covers‎. ‎The purpose of this paper is to illustrate a method of computing the Langlands-Shahidi local coefficients using the theory of types and covers.

This text gives an introduction to the Langlands correspondence for function fields and in particular to some recent works in this subject. We begin with a short historical account (all notions used below are recalled in the text). The Langlands correspondence [Lan70] is a conjecture of utmost importance, concerning global fields, i.e. number fields and function fields. Many excellent surveys a...

In the late 1960s Robert Langlands launched what has become known as the Langlands Program with the ambitious goal of relating deep questions in Number Theory to Harmonic Analysis [L]. In particular, Langlands conjectured that Galois representations and motives can be described in terms of the more tangible data of automorphic representations. A striking application of this general principle is...

I present several applications of the Dirac inequality to determination isolated unitary representations and associated “spectral gaps” in case unramified principal series. The method works particularly well order attach irreducible large nilpotent orbits (e.g., regular, subregular) Langlands dual complex Lie algebra. results could be viewed as a p-adic analogue Salamanca-Riba’s classification ...

26 NOTICES OF THE AMS VOLUME 47, NUMBER 1 H armonic analysis can be interpreted broadly as a general principle that relates geometric objects and spectral objects. The two kinds of objects are sometimes related by explicit formulas, and sometimes simply by parallel theories. This principle runs throughout much of mathematics. The rather impressionistic table at the top of the opposite page prov...

Part I. The origins of the Langlands Program 9 1. The Langlands correspondence over number fields 9 1.1. Galois group 9 1.2. Abelian class field theory 10 1.3. Frobenius automorphisms 13 1.4. Rigidifying ACFT 14 1.5. Non-abelian generalization? 15 1.6. Automorphic representations of GL2(AQ) and modular forms 18 1.7. Elliptic curves and Galois representations 22 2. From number fields to function...

The local Langlands correspondence is a conjectural connection between representations of groups G(k) for connected reductive groups G over a p-adic field k and certain homomorphisms (Langlands parameters) from the Galois (or WeilDeligne group) of k into a complex Lie group Gwhich is dual, in a certain sense, to G and which encodes the splitting structure of G over k. More introductory remarks ...