Abstract

Let F be a p-adic field. The local Langlands correspondence for GL(n,F) relates irreducible degree n representations of the absolute Galois group of F to cuspidal representations of GL(n,F). For n=1 it is given by class field theory, and for n>1 it is characterized by the preservation of fine invariants called “epsilon factors for pairs”, obtained from the tensor product of two representations on the Galois side, and by Rankin-Selberg convolutions on the GL(n) side. But there are other invariants defined on both sides, and naturally they should correspond via the Langlands correspondence too.

After a general introduction to the topic, we shall look at the local factors which correspond on the Galois side to taking the exterior and symmetric square of a representation, and are obtained on the GL(n) side by a method of Langlands-Shahidi.

We shall indicate a global-local proof of their preservation by the Langlands correspondence, which uses the Galois representations attached to regular algebraic cuspidal automorphic representations of GL(n) over (totally real) number fields.