Theta Correspondence for U(1,1) and U(2)

TL;DR

This paper explicitly parametrizes certain supercuspidal representations of U(1,1) and U(2) using induction data linked to quadratic extensions, and determines their theta correspondence via the lattice model of the Weil representation.

Contribution

It provides a new explicit parametrization of supercuspidal representations and clarifies their theta correspondence for the dual pair U(1,1) and U(2).

Findings

Parametrization depends on traceless elements of negative valuation.

Lattice model of Weil representation used to identify involved traceless elements.

Clarifies the theta correspondence for the dual pair U(1,1) and U(2).

Abstract

In this paper, we parametrize certain irreducible supercuspidal representations of U(1,1) and U(2) via explicit induction data. The parametrization depends on traceless elements of negative valuation in a quadratic extension of base field. We use the lattice model of the Weil representation to determine which traceless elements are involved in the theta correspondence for reductive dual pair U(1,1) and U(2).