Geometric Langlands duality relates a representation of a simple Lie group G to the cohomology of a certain moduli space associated with the dual group G. In this correspondence, a principal SL2 subgroup of G makes an unexpected appearance. Why this happens can be explained using gauge theory, as we will see in this article, with the help of the equations of Nahm and Bogomolny. (Based on a lect...

We determine essentially completely the theta correspondence arising from the dual pair PGL3 × G2 ⊂ E6 over a p-adic field. Our first result determines the theta lift of any non-supercuspidal representation of PGL3 and shows that the lifting respects Langlands functoriality. Our second result shows that the theta lift θ(π) of a (non-self-dual) supercuspidal representation π of PGL3 is an irredu...

Let G be the group of points of a split reductive group over a finite extension of Qp. In this paper, we compute the dimensions of certain classes of locally analytic G-representations. This includes principal series representations and certain representations coming from homogeneous line bundles on p-adic symmetric spaces. As an application, we compute the dimensions of the unitary GL2pQpq-rep...

In this mostly expository article we give a survey of some of the generalizations of Serre’s conjecture and results towards them that have been obtained in recent years. We also discuss recent progress towards a mod p local Langlands correspondence for p-adic fields and its connections with Serre’s conjecture. A theorem describing the structure of some mod p Hecke algebras for GLn is proved.

Let F be a non-Archimedean local field with finite residual field of characteristic p. In this article we calculate the ε-factor of pairs for GLl(F ) ×GLl′(F ) where l and l are distinct primes including the case l = p. For this calculation, we use the local Langlands correspondence and non-Galois base change lift. This method leads to the explicit conjecture of the ε-factor of the representati...

1.1. Let X be a smooth, complete, geometrically connected curve over Fq. Denote by F the field of rational functions on X , by A the ring of adeles of F , and by Gal(F/F ) the Galois group of F . The present paper may be considered as a step towards understanding the geometric Langlands correspondence between n–dimensional `–adic representations of Gal(F/F ) and automorphic forms on the group G...

In the late 1960’s, Robert Langlands introduced a number of ideas to the theory of automorphic forms and formulated a number of conjectures which gave the theory a new focus. I was a colleague of his at this time, and a good deal of my professional energy since then has been directed to problems posed by him. Thus it was not entirely inappropriate that when I was invited to this conference, Miy...

In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive p-adic groups, analogous to Deligne–Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig’s program. Precisely, let X be the Deligne–Lusztig (ind-pro-)scheme associated to a division algebra D over a non-Archimedean local field K of positive characteris...

Let OK = FqJπK be the ring of power series in one variable over a finite field, and let K be its fraction field. We introduce the notion of a “formal K-vector space”; this is a certain kind of K-vector space object in the category of formal schemes. This concept runs parallel to the established notion of a formal OK-module, but in many ways formal K-vector spaces are much simpler objects. Our m...