Local newforms for generic representations of unramified even unitary groups I: Even conductor case

TL;DR

This work extends local newform theory to quasi-split even unitary groups $ ext{U}_{2n}$ by defining compact subgroups $K_{2m}^W$ at even levels and analyzing the Fourier–Jacobi module $ extpi_ extpsi$ for irreducible tempered representations with central-character constraints. The authors combine local Gan–Gross–Prasad periods, Rankin–Selberg integrals, and the theta correspondence to (i) characterize when $ extpi_ extpsi^{K_{2m}^W}$ vanishes or is 1-dimensional at critical levels, (ii) prove existence of local newforms via theta lifts and lattice models, and (iii) establish a multiplicity-one type framework for the Fourier–Jacobi data and its compatibility with theta lifts. The results culminate in a precise relation between the conductor $c( extphi_ pi)$ of the L-parameter and the smallest level at which Fourier–Jacobi fixed vectors appear, providing a robust foundation for higher-level generalizations and connections to hermitian modular forms. Overall, the paper advances local newform theory for unitary groups and highlights deep interactions among GGP periods, Rankin–Selberg theory, and the theta correspondence with potential applications to automorphic forms and beyond.

Abstract

In this paper, we define compact open subgroups of quasi-split even unitary groups for each even non-negative integers, and establish the theory of local newforms for irreducible tempered generic representations with a certain condition on the central characters. To do this, we use the local Gan-Gross-Prasad conjecture, the local Rankin-Selberg integrals, and the local theta correspondence.

Theorems & Definitions (30)

• Theorem 1.1: Theorem \ref{['even']}
• Remark 1.2
• Theorem 2.1
• Theorem 2.2
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['even']} (1)
• Proposition 3.3