  We study $L^p$-$L^q$ estimate for the spectral projection operator
$\Pi_\lambda$ associated to the Hermite operator $H=|x|^2-\Delta$ in $\mathbb
R^d$. Here $\Pi_\lambda$ denotes the projection to the subspace spanned by the
Hermite functions which are the eigenfunctions of $H$ with eigenvalue
$\lambda$. Such estimates were previously available only for $q=p'$,
equivalently with $p=2$ or $q=2$ (by $TT^*$ argument) except for the estimates
which are straightforward consequences of interpolation between those
estimates. As shown in the works of Karadzhov, Thangavelu, and Koch and Tataru,
the local and global estimates for $\Pi_\lambda$ are of different nature.
Especially, $\Pi_\lambda$ exhibits complicated behaviors near the set
$\sqrt\lambda\mathbb S^{d-1}$. Compared with the spectral projection operator
associated to the Laplacian, $L^p$-$L^q$ estimate for $\Pi_\lambda$ is not so
well understood up to now for general $p,q$. In this paper we consider
$L^p$--$L^q$ estimate for $\Pi_\lambda$ in a general framework including the
local and global estimates with $1\le p\le 2\le q\le \infty$ and undertake the
work of characterizing the sharp bounds on $\Pi_\lambda$. We establish various
new sharp estimates in extended ranges of $p,q$. First of all, we provide a
complete characterization of the local estimate for $\Pi_\lambda$ which was
first considered by Thangavelu. Secondly, for $d\ge5$, we prove the endpoint
$L^2$--$L^{2(d+3)/(d+1)}$ estimate for $\Pi_\lambda$ which has been left open
since the work of Koch and Tataru. Thirdly, we extend the range of $p,q$ for
which the operator $\Pi_\lambda$ is uniformly bounded from $L^p$ to $L^q$.
