The non-spherical sup-norm problem for $\mathrm{G}\mathrm{L}(n)$ and generalized spherical functions

We solve the sup-norm problem for minimal weight vectors in an arbitrary cuspidal representation $\pi$ of $\mathrm{G}\mathrm{L}(n,\mathbb{Z})\mathrm{\backslash }\mathrm{G}\mathrm{L}(n,\mathbb{R})$ with a uniform power saving bound in terms of the archimedean data of $\pi$, 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.