Subconvexity for sup-norms of cusp forms on \(\mathrm{PGL}(n)\)
The main theorem of the paper may be summarized as follows. Let \(G = \mathrm{PGL}_n(\mathbb{R})\), \(K = \mathrm{PO}_n(\mathbb{R})\) and let \(F\) be a Hecke Maass cusp form for \(\Gamma_0(N) \subset G\), with spectral parameter \(\mu\) and \(\mu^* = \mathrm{Re}(\mu)\). For any \(\varepsilon > 0\) and any compact subset \(\Omega\) of the Riemannian symmetric space \(G/K\), there is an estimate \[ \| F \|_{L^\infty(\Omega)} \ll_{\Omega, \varepsilon} N^\varepsilon D(\mu^*)^{\frac{1}{2} - \delta}, \] for some effectively computable \(\delta = \delta_n > 0\). Here \(D(\lambda) := \prod_\alpha (1 + |\langle \alpha, \lambda \rangle|)^{m(\alpha)}\) for \(\alpha\) ranging over the positive roots and \(m(\alpha)\) standing for the dimension of \(\mathfrak{g}_\alpha\). In particular, we have \[ \| F \|_{L^\infty(\Omega)} \ll_{\Omega, \varepsilon} N^\varepsilon \lambda_F^{\frac{n(n-1)}{4} \cdot (\frac{1}{2} - \delta)}, \] where \(\lambda_F\) denotes the Laplacian eigenvalue of \(F\). This leads to a subconvexity bound for sup-norms of cusp forms, which is in clear analogy with the well-know problem for critical \(L\)-values. It seems to be the first subconvexity result of this kind on groups of arbitrary rank. The main tool is a comparison of spectral and geometric expansions of the automorphic kernel, i.e., an amplified trace formula. In the geometric side, the key point is a counting problem of integral matrices.
scientific article; zbMATH DE number 6618476
scientific article