Restrictions, L-parameters, and local coefficients for genuine representations

2021/02/17 by Gao, Fan, Shahidi, Freydoon, Szpruch, Dani
#FOS: Mathematics #Number Theory (math.NT) #Primary 11F70 #Representation Theory (math.RT) #Secondary 22E50

paper · doi:10.48550/arxiv.2102.08859

Abstract

We consider the restriction and induction of representations between a covering group and 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 arbitrary 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 an unramified L-packet on how the parametrization of elements inside such a 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 the symplectic similitudes groups are discussed in detail in the last part of the paper.

Related

• L-groups and parameters for covering groups
• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
• Restrictions of representations of classical groups: examples
• Irreducible admissible mod-p representations of metaplectic groups
• Doubling Constructions and Tensor Product L-Functions: coverings of the symplectic group