Scattered representations of $SL(n, \mathbb{C})$

TL;DR

This paper characterizes specific parameters and types of scattered representations of the special linear group over complex numbers, contributing to the understanding of their unitary dual with non-zero Dirac cohomology.

Contribution

It provides a detailed description of Zhelobenko parameters and spin-lowest $K$-types for scattered representations of $SL(n, C)$, verifying related conjectures.

Findings

Explicit description of Zhelobenko parameters.

Identification of spin-lowest $K$-types.

Verification of conjectures for $SL(n, C)$.

Abstract

Let $G$ be $SL(n, \mathbb{C})$. This paper aims to describe the Zhelobenko parameters and the spin-lowest $K$-types of the scattered representations of $G$, which lie at the heart of $\hat{G}^d$ - the set of all the equivalence classes of irreducible unitary representations of $G$ with non-vanishing Dirac cohomology. As a consequence, we will verify a couple of conjectures of the first-named author for $G$.