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

Abstract

Let $\varphi$ be an $L^2$-normalized Hecke--Maaß 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 $\Vert \varphi {|}_{\Omega }{\Vert }_{\infty }{\ll }_{\Omega }{\lambda }_{\varphi }^{n(n-1)/4-\delta }$ for some $\delta >0$ depending only on $n$, where ${\lambda }_{\varphi }$ is the Laplace eigenvalue of $\varphi$.