Beyond the spherical sup-norm problem

We solve the sup-norm problem for non-spherical Maaß forms on an arithmetic quotient of G = SL2(C) with maximal compact K = SU2(C) when the dimension of the associated K-type gets large. Our results cover the case of vector-valued Maaß forms as well as all the individual scalar-valued Maaß forms of the Wigner basis. They establish the first subconvex bounds for the sup-norm problem in the K-aspect in a non-abelian situation and yield sub-Weyl exponents in some cases. On the way, we develop theory of independent interest for the group G, including localization estimates for generalized spherical functions of high K-type and a Paley–Wiener theorem for the corresponding spherical transform acting on the space of rapidly decreasing functions.