On unitary representations of disconnected real reductive groups

TL;DR

This paper develops a method to construct the unitary dual of disconnected real reductive groups using automorphisms of the identity component and intertwining operators, extending known results for the connected case.

Contribution

It introduces a new construction of the unitary dual for disconnected groups based on automorphisms and intertwining operators, expanding the understanding of their representation theory.

Findings

Constructed the unitary dual $\\hat{G}$ from known $\\hat{G_0}$.

Analyzed properties of intertwining operators $S$ for outer automorphisms.

Investigated automorphisms of $so(4,4)$ and their relation to group automorphisms.

Abstract

Let $G$ be the real reductive group and let $G_0$ be the identity component. Let us assume that the unitary dual $\hat{G_0}$ is known. In this paper (in Section 5) the unitary dual $\hat{G}$ is constructed. Automorphisms of $G_0$ generated by elements of $G$ are the main ingredient of the construction. If the automorphism is outer, one has to consider the corresponding intertwining operators $S$. Operators $S$ and their properties are analyzed in Section 4. Automorphisms of $g_0$ are closely related to automorphisms of $G_0$. They are investigated in Section 3. Automorphisms of so(4,4)$ are analyzed in Subsection 3.1.