On the local Bump-Friedberg L-function

Let $F$ be a $p$-adic field. If $\pi$ be an irreducible representation of $GL(n,F)$, Bump and Friedberg associated to $\pi$ an Euler fator $L(\pi,BF,s_1,s_2)$ in \cite{BF}, that should be equal to $L(\phi(\pi),s_1)L(\phi(\pi),\Lambda^2,s_2)$, where $\phi(\pi)$ is the Langlands' parameter of $\pi$. The main result of this paper is to show that this equality is true when $(s_1,s_2)=(s+1/2,2s)$, for $s$ in $\C$. To prove this, we classify in terms of distinguished discrete series, generic representations of $GL(n,F)$ which are $\chi_\alpha$-distinguished by the Levi subgroup $GL([(n+1)/2],F) \times GL([n/2],F)$, for $\chi_\alpha(g_1,g_2)=\alpha(det(g_1)/det(g_2))$, where $\alpha$ is a character of $F^*$ of real part between -1/2 and 1/2. We then adapt the technique of \cite{CP} to reduce the proof of the equality to the case of discrete series. The equality for discrete series is a consequence of the relation between linear periods and Shalika periods for discrete series, and the main result of \cite{KR}.