  We study multipliers associated to the Hermite operator $H=-\Delta + |x|^2$
on modulation spaces $M^{p,q}(\mathbb R^d)$. We prove that the operator $m(H)$
is bounded on $M^{p,q}(\mathbb R^d)$ under standard conditions on $m,$ for
suitable choice of $p$ and $q$. As an application, we point out that the
solutions to the free wave and Schr\"odinger equations associated to $H$ with
initial data in a modulation space will remain in the same modulation space for
all times. We also point out that Riesz transforms associated to $H$ are
bounded on some modulation spaces.
