Local newforms for generic representations of unramified ${\mathrm{U}}_{2n+1}$ and Rankin-Selberg integrals

Recently Atobe-Oi-Yasuda established the newform theory for irreducible tempered generic representations of unramified ${\mathrm{U}}_{2n+1}$ over non-archimedean local fields. In this paper we extend their result to every irreducible generic representations and compute the dimensions of the spaces of oldforms. We also compute the Rankin-Selberg integrals attached to newforms and oldforms under a natural assumption on the $\gamma$-factors defined by these integrals.