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.