We solve the sup-norm problem for non-spherical Maass forms on an arithmetic quotient of G=SL_2(C) with maximal compact K=SU_2(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.
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