Spherical Shalika models on $\mathrm{PGU}_{2,2}$ and the theta correspondence for $(\mathrm{PGSp}_4,\mathrm{PGU}_{2,2})$

TL;DR

The paper develops a local theory of Shalika models for generic unramified representations of ${\rm PGU}_{2,2}$ over non-archimedean fields, proving multiplicity one of Shalika functionals via the local theta correspondence with ${\rm PGSp}_4$ and showing that every such representation is a small theta lift from ${\rm PGSp}_4^+$. It then proves a Casselman–Shalika type formula for spherical Shalika functionals on ${\rm PGU}_{2,2}$, expressing their values in terms of dual-group data for ${\rm PGSp}_4$, with split and inert cases handled through explicit Weyl-group calculations and an exceptional isomorphism to ${\rm PGSO}_{4,2}$. The Casselman–Shalika formula is applied to local zeta integrals representing the degree-5 standard $L$-function of ${\rm PGSp}_4$ twisted by the quadratic character $\chi_{E/F}$, establishing Eulerian local factors and connecting Shalika models to functorial liftings from ${\rm PGSp}_4$. These results reinforce the link between Shalika models on ${\rm PGU}_{2,2}$ and the theta correspondence, and provide tools for explicit L-function calculations via Rankin–Selberg type integrals. Overall, the work advances local Langlands phenomena for unitary groups by giving explicit multiplicity one results and a practical Casselman–Shalika formula with direct L-function consequences.

Abstract

We study Shalika models for generic unramified representations of $\mathrm{PGU}_{2,2}$ over non-archimedean local fields of characteristic zero. We show that they are unique up to constant by means of the theta correspondence for $(\mathrm{PGSp}_4,\mathrm{PGU}_{2,2})$. We then prove a Casselman-Shalika formula which relates the values of spherical Shalika functionals on ${\rm PGU}_{2,2}$ to the values of finite dimensional complex representations of the dual group of $\mathrm{PGSp}_4$.

Theorems & Definitions (88)

• Theorem 1.1: Theorem \ref{['MainSummarizingS3']}
• Theorem 1.2: Theorem \ref{['FinalMackey']}
• Theorem 1.3: Theorem \ref{['CasselmanShalikaformula']}
• Theorem 1.4
• Remark 2.1
• Lemma 2.2
• Definition 3.1
• Proposition 3.2: JacquetRallis
• Proposition 3.3
• Lemma 3.4