Cubic metaplectic forms on GL(3) (Q1074626)

This interesting paper concerns automorphic forms on the metaplectic cover of degree n of GL(r, \({\mathbb{R}})\). When \(n=r=2\), the usual theta function is an example which can be viewed as a residue of an Eisenstein series formed from summing \(J(r,z)^{-1} | cz+d|^{-2s}\) over \(\Gamma\) (2) modulo the upper triangular subgroup. Patterson has investigated the case \(r=2\) and \(n=3\). The authors of the paper under review believe that the case \(n=r=3\) is closer to the classical situation. They proceed to support this view by studying the Fourier expansion of a certain maximal parabolic Eisenstein series for GL(3) which is formed with the cubic theta function for GL(2). The Fourier coefficients turn out to be L-functions and when the appropriate residue is taken the Whittaker function simplifies as in the case \(n=2\). The authors note connections with results of Waldspurger provided there is an analogue of the Shimura correspondence for GL(3).