Lifting of Characters for Nonlinear Simply Laced Groups

TL;DR

This paper extends the Langlands character lifting theory from linear groups to nonlinear two-fold covers of real simply laced groups, providing explicit computation methods for stable virtual characters.

Contribution

It introduces a new lifting operation for stable virtual characters from real groups to their nonlinear two-fold covers in the simply laced case, expanding the Langlands program.

Findings

Defined the lifting operation $ ext{Lift}_G^{ ilde{G}}$ for characters.

Proved the operation maps stable virtual characters to genuine characters.

Provided explicit formulas for computing the lift of stable sums of standard modules.

Abstract

One aspect of the Langlands program for linear groups is lifting of characters, which relates virtual representations on a group $G$ with those on an endoscopic group for $G$. The goal of this paper is to extend this theory to nonlinear two-fold covers of real groups in the simply laced case. Suppose $\tG$ is a two-fold cover of a real reductive group $G$. The main result is that there is an operation, denoted $\Lift_G^{\tG}$, taking a stable virtual character of $G$ to 0 or a virtual genuine character of $\tG$, and $\Lift_G^{\tG}(\Theta_\pi)$ may be explicitly computed if $\pi$ is a stable sum of standard modules.