Gamma factors and root numbers of pairs for the Galois and the linear model

Abstract: Using harmonic analysis on Harish-Chandra Schwartz spaces of various spherical spaces, we extend a relative local converse theorem of Youngbin Ok for the Galois model of p-adic GLn, from the class of cuspidal representations to that of square integrable representations, which is its optimal form. We also prove a variant of this result for linear models by the same method. The above statements are luckily non empty as we verify triviality results for gamma and epsilon factors of pairs of distinguished representations at the central value s=1/2. Along the way, we offer a new proof of conjectures of D. Prasad and D. Ramakrishnan on local components of symplectic cuspidal automorphic representations, and root numbers of pairs of symplectic representations.