Local Langlands correspondence and exterior and symmetric square root numbers for $GL(n)$

Let $\rho$ be an $n$-dimensional continuous complex representation of $W'_F$, the Weil-Deligne group of a local field $F$. Denote by $\pi(\rho)$ the irreducible admissible representation of $GL_n(F)$ attached to $\rho$ by the local Langlands correspondence. Let $\Lambda^2$ be the exterior square representation of $GL_n(\Bbb C)$. Let $L(s,\Lambda^2\rho)$ and $\varepsilon(s,\Lambda^2\rho,\psi_F)$ be the Artin $L$-function and root number defined by $\Lambda^2\rho$. Denote by $L(s,\pi(\rho),\Lambda^2)$ and $\varepsilon(s,\pi(\rho),\Lambda^2,\psi_F)$ the local factors attached to $\pi(\rho)$ and $\Lambda^2$ using the Langlands-Shahidi method. In this talk we will prove $\varepsilon(s,\Lambda^2\rho,\psi_F)=\varepsilon(s,\pi(\rho),\Lambda^2,\psi_F)$ as well as $L(s,\Lambda^2\rho)=L(s,\pi(\rho),\Lambda^2)$. While the case of $L$-function is already proved by Henniart, the equality of root numbers is much more subtle. The proof uses a deformation argument, complemented by an asymptotics of certain Bessel functions derived from certain generalized Shalika germ expansions of Jacquet and Ye. This is a joint work with J. Cogdell and T.-L. Tsai.