Scattered representations of complex classical Lie groups

TL;DR

This paper investigates scattered representations of certain complex classical Lie groups, detailing their parameters, counting them, and analyzing their properties, while also disproving a 2015 conjecture about the structure of the unitary dual.

Contribution

It provides a detailed description and enumeration of scattered representations for complex classical Lie groups and refutes a previous conjecture on the construction of the unitary dual.

Findings

Described Zhelobenko parameters of scattered representations

Counted the number of such representations

Disproved the 2015 conjecture on unitary dual construction

Abstract

This paper studies scattered representations of $G = SO(2n+1, \mathbb{C})$, $Sp(2n, \mathbb{C})$ and $SO(2n, \mathbb{C})$, which lies in the `core' of the unitary spectrum $G$ with nonzero Dirac cohomology. We describe the Zhelobenko parameters of these representations, count their cardinality, and determine their spin-lowest $K$-types. We also disprove a conjecture raised in 2015 asserting that the unitary dual can be obtained via parabolic induction from irreducible unitary representations with non-zero Dirac cohomology.