Faculty Publication: Associate Professor of Mathematics Djordje Mili膰evi膰
Beyond the Spherical Sup-norm Problem
Authors: Blomer, Valentin; Harcos, Gergely; Maga, Peter; Mili膰evi膰, Djordje
Source: Journal de Math茅matiques Pures et Appliqu茅es, Volume: 168, Pages: 1-64, DOI: 10.1016/j.matpur.2022.09.009, Dec. 2022
Type of Publication: Article
Abstract: We open a new perspective on the sup-norm problem and propose a version for non-spherical Maa脽 forms when the maximal compact K is non-abelian and the dimension of the K -type gets large. We solve this problem for an arithmetic quotient of G = SL2(C) with K = SU2(C). Our results cover the case of vector-valued Maa脽 forms as well as all the individual scalar-valued Maa脽 forms of the Wigner basis, reaching sub-Weyl exponents in some cases. On the way, we develop analytic theory of independent interest, including uniform strong 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. The new analytic properties of the generalized spherical functions lead to novel counting problems of matrices close to various manifolds that we solve optimally.