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

TL;DR

The paper establishes that non-zero spaces of newforms for all irreducible generic supercuspidal representations imply the same for all irreducible generic representations of ${ m SO}_{2n+1}$ over p-adic fields.

Contribution

It proves a reduction principle linking the existence of newforms in supercuspidal and all generic representations of ${ m SO}_{2n+1}$.

Findings

Non-zero newform spaces in supercuspidal representations imply non-zero in all generic representations.

The result applies to the representation theory of ${ m SO}_{2n+1}$ over p-adic fields.

Provides a reduction step in understanding newforms for classical groups.

Abstract

We prove that if the space of newforms is non-zero for every irreducible generic supercuspidal representation of ${\rm SO}_{2n+1}$ then it is also non-zero for all irreducible generic representations of ${\rm SO}_{2n+1}$.