  We establish the existence and uniqueness of twisted exterior and symmetric
square $\gamma$-factors in positive characteristic by studying the Siegel Levi
case of generalized spinor groups. The corresponding theory in characteristic
zero is due to Shahidi. In addition, in characteristic $p$ we prove that these
twisted local factors are compatible with the local Langlands correspondence.
As a consequence, still in characteristic $p$, we obtain a proof of the
stability property of $\gamma$-factors under twists by highly ramified
characters. Next we use the results on the compatibility of the
Langlands-Shahidi local coefficients with the Deligne-Kazhdan theory over close
local fields to show that the twisted symmetric and exterior square
$\gamma$-factors, $L$-functions and $\varepsilon$-factors are preserved over
close local fields. Furthermore, we obtain a formula for Plancherel measures in
terms of local factors and we also show that they also preserved over close
local fields.
