RESTRICTIONS, L-PARAMETERS, AND LOCAL COEFFICIENTS FOR GENUINE REPRESENTATIONS

We consider the restriction and induction of representations between a covering group and subgroups closely related to its derived subgroup, both on the representation-theoretic side and the L-parameter side. In particular, restriction of a genuine principal series is analyzed in detail. We also discuss a metaplectic tensor product construction for covers of the symplectic similitudes groups, and remark on the generality of such a construction for other groups. Furthermore, working with an irreducible constituent of a unitary unramified principal series, we prove a multiplicity formula for its restriction to the derived subgroup in terms of three associated R groups. Later in the paper, we study how the parametrization of elements inside an unramified L-packet varies along with different choices of hyperspecial maximal compact subgroups and their splittings. We also investigate the genericity of elements inside such an L-packet with respect to varying Whittaker datum. Pertaining to the above two problems, covers of symplectic similitudes groups are discussed in detail in the last part of the paper.