Fock projections on vector-valued $L^p$-spaces with matrix weights

In this paper, we characterize the $d\times d$ matrix weights $W$ on ${\mathbb{C}}^n$ such that the Fock projection $P_{\alpha }$ is bounded on the vector-valued spaces $L_{\alpha ,W}^p({\mathbb{C}}^n;{\mathbb{C}}^d)$ induced by $W$ and the Gaussian measures. It is proved that for $1\le p\le \mathrm{\infty }$, the Fock projection $P_{\alpha }$ is bounded on $L_{\alpha ,W}^p({\mathbb{C}}^n;{\mathbb{C}}^d)$ if and only if $W$ satisfies a restricted ${\mathcal{A}}_p$-condition. Our result is new even in the scalar setting at the endpoint $p=\mathrm{\infty }$.