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

The conjectural theory of local newofmrs for the split $p$-adic group ${\mathrm{S}\mathrm{O}}_{2n+1}$, 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.