The Degenerate Principal Series Representations of Exceptional Groups of Type $E_8$ over $p$-adic Fields

TL;DR

This paper investigates the reducibility and substructure of degenerate principal series representations of the exceptional group $E_8$ over p-adic fields, providing detailed calculations of their maximal semi-simple subrepresentations and quotients.

Contribution

It offers new explicit calculations of the subrepresentations and quotients of degenerate principal series for $E_8$, a complex and less understood exceptional group.

Findings

Determined reducibility conditions for these representations.

Calculated maximal semi-simple subrepresentations.

Identified quotient structures for almost all cases.

Abstract

In this paper, we study the reducibility of degenerate principal series of the simple, simply-connected exceptional group of type $E_8$. Furthermore, we calculate the maximal semi-simple subrepresentation and quotient of these representations for almost all cases.