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 $$SL(2,{\mathbb {Z}})\backslash SL(2,{\mathbb {R}})/SO(2,{\mathbb {R}})$$—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 $$SL(3,{\mathbb {Z}})\backslash SL(3,{\mathbb {R}})/SO(3,{\mathbb {R}})$$ 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.