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

TL;DR

This paper investigates the restriction and induction of genuine representations between covering groups and their derived subgroups, analyzing L-parameters, multiplicity formulas, and the structure of L-packets, with a focus on symplectic similitudes groups.

Contribution

It introduces new results on restriction of genuine principal series, a metaplectic tensor product construction, and multiplicity formulas in the context of covering groups and L-parameters.

Findings

Restriction of genuine principal series is explicitly analyzed.

A multiplicity formula for restriction to the derived subgroup is established.

Parametrization and genericity within unramified L-packets are studied.

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.