Theta correspondence for almost unramified representations of unitary groups

TL;DR

This paper introduces and studies almost unramified representations of quasi-split unitary groups over local fields, focusing on their behavior under the local theta correspondence, highlighting their minimal ramification with root number -1.

Contribution

It defines the concept of almost unramified representations for unitary groups and analyzes their properties under the local theta correspondence.

Findings

Almost unramified representations have root number -1.

These representations are minimally ramified among those with root number -1.

Behavior under theta correspondence is characterized for these representations.

Abstract

In this note, we introduce the notion of almost unramified representations of quasi-split unitary groups of even ranks with respect to an unramified quadratic extension of local fields, and study their behavior under the local theta correspondence. These representations are in some sense the least ramified representations that have root number $-1$.