Local newforms for generic representations of $p$-adic ${\rm SO}_{2n+1}$: Uniqueness

TL;DR

This work resolves the uniqueness aspect of Gross’s local newform conjecture for generic representations of ${\rm SO}_{2n+1}(F)$ by showing the newform space $\pi^{K_{n,m}}$ is at most $1$-dimensional and exhibits the expected arithmetic behavior when nonzero, conditional on existence. The authors leverage a triad of tools—explicit double coset decompositions, local Rankin–Selberg integrals for ${\rm SO}_{2n+1}\times GL_r$, and the Gross–Prasad local period uniqueness—to reduce to tempered representations and construct maps $\Xi_{r,m}$ that detect newforms via symmetric polynomials attached to $L$-parameter data. A tempered-case analysis, aided by Gross–Prasad uniqueness and Rankin–Selberg theory, yields injectivity of these maps and the desired dimension and pairing properties, which then extend to all generic representations through Langlands classification. The results provide the missing half of the local newform picture for ${\rm SO}_{2n+1}$, laying groundwork for the existence part of the conjecture and informing the structure of oldforms and related $L$-packets in this setting.

Abstract

The conjectural theory of local newofmrs for the split $p$-adic group ${\rm SO}_{2n+1}$, proposed by Gross, predicts that the space of local newforms in a generic representation is one-dimensional. In this note, we prove that this space is at most one-dimensional and verify its expected arithmetic properties, conditional on existence. These results play an important role in our proof of the existence part of the newform conjecture.

Theorems & Definitions (31)

• Conjecture
• Remark 1.1
• Theorem A
• Remark 1.2
• Lemma 3.1
• proof
• Remark 3.2
• Corollary 3.3
• proof
• Lemma 4.1