Some results in the theory of genuine representations of the metaplectic double cover of GSp2n(F) over p-adic fields

TL;DR

This paper investigates the structure of genuine irreducible representations of the metaplectic double cover of GSp(2n) over p-adic fields, establishing induction relations, uniqueness of Whittaker functionals, and criteria for reducibility, with distinctions based on the parity of n.

Contribution

It demonstrates that for odd n, all genuine irreducible representations are induced from a related subgroup, simplifying their classification, and provides new irreducibility criteria for parabolic induction.

Findings

All genuine irreducible representations of G(n) for odd n are induced from a subgroup related to G`(n).

Uniqueness of certain Whittaker functionals and Rodier-type heredity are established for odd n.

Reductibility points on the unitary axis are identified for even n, with explicit classification for G(2).

Abstract

Let F be a p-adic field and let G(n) and G`(n) be the metaplectic double covers of the general symplectic group and symplectic group attached to a 2n dimensional symplectic space over F. We show here that if n is odd then all the genuine irreducible representations of G(n) are induced from a normal subgroup of finite index closely related to G`(n). Thus, we reduce, in this case, the theory of genuine admissible representations of G(n) to the better understood corresponding theory of G`(n). For odd n we also prove the uniqueness of certain Whittaker functionals along with Rodier type of Heredity. Our results apply also to all parabolic subgroups of G(n) if n is odd and to some of the parabolic subgroups of G(n) if n is even. We prove some irreducibility criteria for parabolic induction on G(n) for both even and odd n. As a corollary we show, among other results, that while for odd n, all genuine principal series representations of G(n) induced from unitary representations are irreducible, there exist reducibility points on the unitary axis if n is even. We also list all the reducible genuine principal series representations of G(2) provided that the F is not 2-adic.