Beyond the spherical sup-norm problem

TL;DR

This paper extends the sup-norm problem to non-spherical Maass forms on SL_2(C), providing new analytic tools, solving for specific cases, and achieving sub-Weyl bounds with applications to counting matrices near manifolds.

Contribution

It introduces a new framework for the sup-norm problem for non-spherical Maass forms with large K-type, including analytic theory and optimal counting results.

Findings

Achieved sub-Weyl bounds for certain Maass forms.

Developed uniform localization estimates for spherical functions.

Solved counting problems for matrices near manifolds.

Abstract

We open a new perspective on the sup-norm problem and propose a version for non-spherical Maass 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=SL_2(C) with K=SU_2(C). Our results cover the case of vector-valued Maass forms as well as all the individual scalar-valued Maass 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.