The sup-norm problem for automorphic cusp forms of 𝑃𝐺𝐿(𝑛,ℤ[𝕚])

Abstract

Let ϕ \phi be an L 2 L^2 -normalized Hecke–Maaß cusp form on the locally symmetric space X ≔ P G L n ( Z [ i ] ) ∖ P G L n ( C ) / P U n X≔\mathrm {PGL}_n(\mathbb {Z}[i])\backslash \mathrm {PGL}_n(\mathbb {C}) / \mathrm {PU}_n . If Ω \Omega is a compact subset of X X , then we prove the bound ‖ ϕ | Ω ‖ ∞ ≪ Ω λ ϕ n ( n − 1 ) / 4 − δ \|\phi |_{\Omega }\|_{\infty }\ll _{\Omega } \lambda _{\phi }^{n(n-1)/4-\delta } for some δ > 0 \delta >0 depending only on n n , where λ ϕ \lambda _{\phi } is the Laplace eigenvalue of ϕ \phi .