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

Abstract

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

30 pages, fix typos