Beyond the spherical sup-norm problem (Q2107143)

In the paper under the review, the authors aim to open a new perspective on the sup-norm problem and propose a detailed investigation of the first nontrivial case \(\mathrm{SL}_2(\mathbb{C})\), i.e., for an arithmetic quotient \(G/K\) where \(G= \mathrm{SL}_2(\mathbb{C})\). In general, they are interested in the case when the maximal compact subgroup \(K\) of \(G\) is non-abelian and the dimension of the \(K\)-type is large. The first main result of the paper covers the case of vector-valued forms, while the second main result deals with the individual basis elements. Along the way, bounds for the spherical trace functions are obtained, together with the new Paley-Wiener theorem for \(K\)-finite Schwartz class functions on \(\mathrm{SL}_2(\mathbb{C})\).