Some Genuine Small Representations of a Nonlinear Double Cover

TL;DR

This paper studies special small genuine irreducible representations of the nonlinear double cover of a simply connected, semisimple complex Lie group, characterizing their properties and establishing explicit constructions and correspondences.

Contribution

It introduces a new class of small genuine representations with specific properties and constructs them explicitly for split groups, establishing a correspondence with central characters and real forms.

Findings

Explicit construction of small genuine representations for split groups

Characterization of these representations by infinitesimal character and associated variety

Establishment of a bijective correspondence with central characters and real forms

Abstract

Let G be the real points of a simply connected, semisimple, simply laced complex Lie group, and let \tilde{G} be the nonlinear double cover of G. We discuss a set of small genuine irreducible representations of \tilde{G} which can be characterized by the following properties: (a) the infinitesimal character is \rho/2; (b) they have maximal tau-invariant; (c) they have a particular associated variety O. When G is split, we construct them explicitly. Furthermore, in many cases, there is a one-to-one correspondence between these small representations and the pairs (genuine central characters of \tilde{G}, real forms of O) via the map \pi mapped to (central character of \pi, real associated variety of \pi).