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Let $\phi$ be an $L^2$-normalized Hecke--Maa{\ss} cusp form for$\mathrm{PGL}_n(\mathbb{Z}[i])$ on the locally symmetric space$X:=\mathrm{PGL}_n(\mathbb{Z}[i])\backslash \mathrm{PGL}_n(\mathbb{C}) /\mathrm{PU}_n$. If $\Omega$ is a compact subset of $X$, then we prove the bound$\|\phi|_{\Omega}\|_{\infty}\ll_{\Omega} \lambda_{\phi}^{n(n-1)/4-\delta}$ forsome $\delta>0$ depending only on $n$, where $\lambda_{\phi}$ is the Laplaceeigenvalue of $\phi$.

The sup-norm problem for automorphic cusp forms of $\mathrm{PGL}(n,\mathbb{Z}[i])$