2016/11/05 by Dijols, Sarah, Prasad, Dipendra
#FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1611.01621
In analogy with the study of representations of GL2n(F) distinguished by Sp2n(F), where F is a local field, in this paper we study representations of U2n(F) distinguished by Sp2n(F). (Only quasi-split unitary groups are considered in this paper since they are the only ones which contain Sp2n(F).) We prove that there are no cuspidal representations of U2n(F) distinguished by Sp2n(F) for F a non-archimedean local field. We also prove the corresponding global theorem that there are no cuspidal representations of U2n(\mathbb Ak) with nonzero period integral on Sp2n(k) \backslash Sp2n(\mathbb Ak) for k any number field or a function field. We completely classify representations of quasi-split unitary group in four variables over local and global fields with nontrivial symplectic periods using methods of theta correspondence. We propose a conjectural answer for the classification of all representations of a quasi-split unitary group distinguished by Sp2n(F).