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...

Abstract This survey reports on some of the recent developments in area Galois representations and automorphic forms, with a particular focus author Thorne’s work symmetric power functoriality for modular forms.

— By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type ...

In this paper, we take some initial steps towards illuminating the (hypothetical) p-adic local Langlands functoriality principle relating Galois representations of a p-adic field L and admissible unitary Banach space representations of G(L) when G is a split reductive group over L. 2000 Mathematics Subject Classification: 11F80, 11S37, 22E50

In this paper we give an overview over the basic features of the continuous representation theory of p-adic Lie groups as it has emerged during the last five years. The main motivation for developing such a theory is a possible extension of the local Langlands program to p-adic Galois representations. This is still very much in its infancy. But in the last section we will describe a first appro...

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...

We consider distinction of representations in the context p $p$ -adic Galois symmetric spaces. provide new sufficient conditions for parabolically induced terms similar on inducing data and deduce a characterization from cuspidal. explicate results further classical groups give several applications, particular, concerning preservation via Langlands functoriality. relate our with conjecture Dipe...

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...