  Let $F$ be a $p$--adic field, i.e., a finite extension of $\mathbb Q_p$ for
some prime $p$. The local Langlands correspondence attaches to each continuous
$n$--dimensional $\Phi$-semisimple representation $\rho$ of $W'_F$, the
Weil--Deligne group for $\bar F/F$, an irreducible admissible representation
$\pi(\rho)$ of $GL_n(F)$ such that, among other things, the local $L$- and
$\varepsilon$-factors of pairs are preserved. This correspondence should be
robust and preserve various parallel operations on the arithmetic and analytic
sides, such as taking the exterior square or symmetric square. In this paper,
we show that this is the case for the local arithmetic and analytic symmetric
square and exterior square $\varepsilon$--factors, that is, that
$\varepsilon(s,\Lambda^2\rho,\psi)=\varepsilon(s,\pi(\rho),\Lambda^2,\psi)$ and
$\varepsilon(s,Sym^2\rho,\psi)=\varepsilon(s,\pi(\rho),Sym^2,\psi)$. The
agreement of the $L$-functions also follows by our methods, but this was
already known by Henniart. The proof is a robust deformation argument, combined
with local/global techniques, which reduces the problem to the stability of the
analytic $\gamma$-factor $\gamma(s,\pi,\Lambda^2,\psi)$ under highly ramified
twists when $\pi$ is supercuspidal. This last step is achieved by relating the
$\gamma$-factor to a Mellin transform of a partial Bessel function attached to
the representation and then analyzing the asymptotics of the partial Bessel
function, inspired in part by the theory of Shalika germs for Bessel integrals.
The stability for every irreducible admissible representation $\pi$ then
follows from those of the corresponding arithmetic $\gamma$--factors as a
corollary.
