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

TL;DR

This paper extends the newform theory for irreducible generic representations of unramified odd unitary groups, computes oldform dimensions, and analyzes Rankin-Selberg integrals for these forms.

Contribution

It generalizes existing results to all irreducible generic representations and provides explicit computations of integrals and dimensions.

Findings

Dimensions of oldform spaces are computed.

Rankin-Selberg integrals for newforms and oldforms are explicitly calculated.

The results apply under a natural assumption on gamma-factors.

Abstract

Recently Atobe-Oi-Yasuda established the newform theory for irreducible tempered generic representations of unramified ${\rm U}_{2n+1}$ 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.