On the nonexistence of automorphic eigenfunctions of exponential growth on SL(3,Z)\SL(3,R)/SO(3,R)

Abstract

It is well-known that there are automorphic eigenfunctions on SL(2,Z)\SL(2,R)/SO(2,R) -- such as the classical $j$-function -- that have exponential growth and have exponentially growing Fourier coefficients (e.g., negative powers of $q=e^{2\pi i z}$, or an I-Bessel function). We show that this phenomenon does not occur on the quotient SL(3,Z)\SL(3,R)/SO(3,R) and eigenvalues in general position (a removable technical assumption). More precisely, if such an automorphic eigenfunction has at most exponential growth, it cannot have non-decaying Whittaker functions in its Fourier expansion. This confirms part of a conjecture of Miatello and Wallach, who assert all automorphic eigenfunctions on this quotient (among other rank $\ge$ 2 examples) always have moderate growth. We additionally confirm their conjecture under certain natural hypotheses, such as the absolute convergence of the eigenfunction's Fourier expansion.