Vol. 290, No. 2, 2017

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Distinguished theta representations for certain covering groups

Fan Gao

Vol. 290 (2017), No. 2, 333–379

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 (à 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.

To Professor Freydoon Shahidi on his 70th birthday

Brylinski–Deligne covering groups, theta representations, Whittaker functionals, distinguished characters, dual groups
Mathematical Subject Classification 2010
Primary: 11F70
Secondary: 22E50
Received: 14 June 2016
Revised: 21 March 2017
Accepted: 21 March 2017
Published: 25 July 2017
Fan Gao
Department of Mathematics
Purdue University
150 N. University Street
West Lafayette, IN 47907
United States