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It is well-known that there are automorphic eigenfunctions onSL(2,Z)\SL(2,R)/SO(2,R) -- such as the classical $j$-function -- that haveexponential 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 thatthis phenomenon does not occur on the quotient SL(3,Z)\SL(3,R)/SO(3,R) andeigenvalues in general position (a removable technical assumption).  More precisely, if such an automorphic eigenfunction has at most exponentialgrowth, it cannot have non-decaying Whittaker functions in its Fourierexpansion. This confirms part of a conjecture of Miatello and Wallach, whoassert all automorphic eigenfunctions on this quotient (among other rank $\ge$2 examples) always have moderate growth. We additionally confirm theirconjecture under certain natural hypotheses, such as the absolute convergenceof the eigenfunction's Fourier expansion.

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