Langlands correspondence and L-functions of symmetric and outside squares

Let p be a prime number, F a finite extension of Qp and ψ a non trivial additive character of F. The Langlands correspondence is a bijection σ π (σ) between Φ-semisimple degree n representations of the Weil-Deligne group of F, up to isomorphism, and smooth irreducible representations of, up to isomorphism. For some representations r of the dual group of, local-global methods attach factors L(π, r, s) and ε (π, r, s, ψ) to any smooth irreducible representation π of. Conjecturally we have L(π (σ), r, s) = L (r ○ σ, s), and similarly for the ε-factors, when σ is a degree n representation of the Weil-Deligne group of F. © The Author 2009. Published by Oxford University Press. All rights reserved.