Distinguished theta representations for certain covering groups

TL;DR

This paper investigates when theta representations of Brylinski-Deligne covering groups are distinguished by having a one-dimensional space of Whittaker functionals, providing bounds and explicit conditions for such cases.

Contribution

It offers effective bounds for Whittaker functional dimensions and characterizes distinguished theta representations for certain covering groups, including explicit conditions.

Findings

Bounds for Whittaker functional dimensions are established.

Explicit criteria for distinguished theta representations are provided.

Conditions depend on the group's rank, degree, and the associated exceptional characters.

Abstract

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 determine when a theta representation has exactly a one-dimensional space of Whittaker functionals, in which case it is called distinguished. For this purpose, we first give effective lower and upper bounds for the dimension of Whittaker functionals for general theta representations. As a consequence, the dimension in many cases can be reduced to simple combinatorial computations, e.g., the Kazhdan-Patterson covering groups, or covering groups whose complex dual group is of adjoint type. In the second part of the paper, we consider covering groups of certain simply-connected groups and give necessary and sufficient condition 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 such that its associated theta representation is distinguished.