Twisting of Paramodular Vectors

Let $F$ be a non-archimedean local field of characteristic zero
and let $(\pi,V)$ be an irreducible, admissible representation of
$\Gsp(4,F)$ with trivial central character. For a nontrivial quadratic
character $\chi$ of $F$, we define a nonzero twisting map from $\pi$ to
$\chi\otimes\pi$ which is a paramodular level-raising operator. I will
explain the construction of this map, compare it to the analogous map for
$\GL(2)$, and show that it is zero if $\pi$ is unramified of
Saito-Kurokawa type. This is joint work with Brooks Roberts.