Lift of the trivial representation to a nonlinear double cover

TL;DR

This paper characterizes the genuine irreducible representations of a nonlinear double cover of a semisimple group obtained via Kazhdan-Patterson lifting, specifically for simply laced split groups, building on prior work on small representations.

Contribution

It identifies the set of representations arising from Kazhdan-Patterson lifting as a specific set of small representations for simply laced split groups.

Findings

The set of representations from Kazhdan-Patterson lifting matches the small representations with infinitesimal character ρ/2.

The result applies to simply laced split groups, providing a precise classification.

Connects the theory of small representations with explicit lifting constructions.

Abstract

Let $\widetilde G$ be the nonlinear double cover of the real points of a connected, simply connected, semisimple complex group. In [Ts], we introduce a set of genuine small representations of $\widetilde G$ with infinitesimal character $\lambda$, denoted $\prod _\lambda ^s (\widetilde G)$. In this paper, we show that $\prod _{\rho/2} ^s (\widetilde G)$ is precisely the set of genuine irreducible representations arising from the Kazhdan-Patterson lifting of the trivial representation, when $\widetilde G$ is simply laced and split.