{"ID":22955972,"CreatedAt":"2026-09-17T02:12:05.498442134Z","UpdatedAt":"2026-09-17T02:12:05.498442134Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.19121","arxiv_id":"2609.19121","title":"The non-spherical sup-norm problem for $\\mathrm{GL}(n)$ and generalized spherical functions","abstract":"We solve the sup-norm problem for minimal weight vectors in an arbitrary cuspidal representation $π$ of $\\mathrm{GL}(n,\\mathbb{Z})\\backslash\\mathrm{GL}(n,\\mathbb{R})$ with a uniform power saving bound in terms of the archimedean data of $π$, including its spectral parameters and the dimension of its minimal $K$-type. As a key ingredient, we establish new uniform decay bounds for generalized spherical functions.","short_abstract":"We solve the sup-norm problem for minimal weight vectors in an arbitrary cuspidal representation $π$ of $\\mathrm{GL}(n,\\mathbb{Z})\\backslash\\mathrm{GL}(n,\\mathbb{R})$ with a uniform power saving bound in terms of the archimedean data of $π$, including its spectral parameters and the dimension of its minimal $K$-type. A...","url_abs":"https://arxiv.org/abs/2609.19121","url_pdf":"https://arxiv.org/pdf/2609.19121v1","authors":"[\"Valentin Blomer\",\"Gergely Harcos\",\"Péter Maga\",\"Djordje Milićević\"]","published":"2026-09-16T17:43:40Z","proceeding":"math.NT","tasks":"[\"math.NT\",\"math.RT\",\"math.SP\"]","methods":"[]","has_code":false}