Some genuine small representations of a nonlinear double cover

Abstract

Let G G be the real points of a simply connected, semisimple, simply laced complex Lie group, and let G ~ \widetilde {G} be the nonlinear double cover of G G . We discuss a set of small genuine irreducible representations of G ~ \widetilde {G} which can be characterized by the following properties: (a) the infinitesimal character is ρ / 2 \rho /2 ; (b) they have maximal τ \tau -invariant; (c) they have a particular associated variety O \mathcal {O} . When G 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 G ~ \widetilde {G} , real forms of O \mathcal {O} ) via the map π ~ ↦ ( χ π ~ , A V ( π ~ ) ) \widetilde {\pi } \mapsto (\chi _{\widetilde {\pi }}, AV(\widetilde {\pi }) ) .