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

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 iz), 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 >= 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.