CAP representations of \(G_{2}\) and the spin \(L\)-function of \(\mathrm{PGSp}_6\) (Q839903)

Let \(F\) be a number field and let \(\mathbb A\) be its ring of adeles. Let \(G\) be a reductive algebraic group over \(F\). Piatetski-Shapiro introduced the concept of cuspidal almost parabolic (CAP) automorphic representations. They should not exist for \(G=\mathrm{GL}_n\) by the Ramanujan conjecture. Roughly speaking, CAP representations describe how the Ramanujan conjecture fails for other groups (for example \(\mathrm{Sp}_4\)). Their conjectural description is given by \textit{J. Arthur} [Astérisque 171--172, 13--71 (1989; Zbl 0728.22014)]. The construction of CAP representations is usually done by using global theta lifts. The paper under review is a part of the authors' project of determining CAP representations for the split group of type \(G_2\). The others are \textit{W. T. Gan} [Duke J. Math. 130, No.~2, 297--320 (2005; Zbl 1087.11034)], \textit{W. T. Gan, N. Gurevich} and \textit{D. H. Jiang} [Invent. Math. 149, No.~2, 225--265 (2002; Zbl 1036.11020)], and \textit{W. T. Gan} and \textit{N. Gurevich} [Am. J. Math. 128, No.~5, 1105--1185 (2006; Zbl 1109.22013)]. The introduction of the paper under review explains very well the strategy and the results obtained so far and in the present paper.