  Let ${\cal S}^{lin}:=\{a(t)(-2\II \partial_t-\partial_r^2+V(t,r) | a\in
C^{\infty}(\R/2\pi\Z),
  V\in C^{\infty}(\R/2\pi\Z\times\R)\}$ be the space of Schr\"odinger operators
in $(1+1)$-dimensions with periodic time-dependent potential. The action on
${\cal S}^{lin}$ of a large infinite-dimensional reparametrization group $SV$
with Lie algebra $\sv$ \cite{RogUnt06,Unt08}, called the Schr\"odinger-Virasoro
group and containing the Virasoro group, is proved to be Hamiltonian for a
certain Poisson structure on ${\cal S}^{lin}$. More precisely, the
infinitesimal action of $\sv$ appears to be part of a coadjoint action of a Lie
algebra of pseudo-differential symbols, $\g$, of which $\sv$ is a quotient,
while the Poisson structure is inherited from the corresponding
Kirillov-Kostant-Souriau form.
