DISTINGUISHED THETA REPRESENTATIONS FOR CERTAIN COVERING GROUPS

For Brylinski-Deligne covering groups of an arbitrary split reductive group, we consider theta representations attached to certain exceptional genuine characters. The goal of the paper is to study the dimension of the space of Whittaker functionals of a theta representation. In particular, we investigate when the dimension is exactly one, in which case the theta representation is called distinguished. For this purpose, we first give effective lower and upper bounds for the dimension of Whittaker functionals for general theta representations. Consequently, the dimension in many cases can be reduced to simple combinatorial computations, e.g., the Kazhdan-Patterson covering groups of the general linear groups, or covering groups whose complex dual groups (a la Finkelberg, Lysenko, McNamara and Reich) are of adjoint type. In the second part of the paper, we consider coverings of certain semisimple simply connected groups and give necessary and sufficient conditions for the theta representation to be distinguished. There are subtleties arising from the relation between the rank and the degree of the covering group. However, in each case we will determine the exceptional character whose associated theta representation is distinguished.