On the automorphic theta representation for simply laced groups
This paper is devoted to the investigation of certain automorphic representations of certain split, semisimple algebraic groups over a number field \(k\). The groups in question are to be of type \(D_m\) or \(E_m\). It was first shown by \textit{D. Kazhdan} and \textit{G. Savin} [Isr. Math. Conf. Proc. 2, 209-223 (1990; Zbl 0737.22008)] that there exists a ``very small'' representation of these groups and that this representation is a quotient of a principal series representation. The purpose of this paper is mainly to show that analogous representations exist globally in the cases listed and that they can be obtained as ``residues of Eisenstein series''. These remarkable representations are analogues of the classical theta representations. The authors consider the Fourier coefficients with respect to the unipotent radical of a certain parabolic subgroup (defined ad hoc) and show that these are often zero and have stronger invariance properties than one would expect. Furthermore the authors can show that there is a multiplicity one theorem in the case of \(D_m\). Finally, they give some indications as to how the emerging theory of these representations would lead to novel and intriguing analogues of the theta correspondences which arise between representations of elements of dual reductive pairs in symplectic groups, or their double covers.
scientific article; zbMATH DE number 1062357
scientific article