Cyclotomic expansion of exceptional spectral measures

TL;DR

This paper derives explicit formulas for the circular spectral measures of the exceptional Lie groups E7 and E8, revealing their structure as combinations of measures supported on roots of unity with densities from degree 3 polynomials.

Contribution

It provides explicit formulas for the spectral measures of E7 and E8, advancing understanding of ADE circular measures and their polynomial densities.

Findings

Spectral measures for E7 and E8 are explicitly formulated.

These measures are linear combinations supported on roots of unity.

Densities are given by specific degree 3 polynomials.

Abstract

We find explicit formulae for the circular spectral measures of $E_7,E_8$. This leads to a number of general observations regarding the ADE circular measures: these are linear combinations of measures supported by the roots of unity, with real density given by certain degree 3 polynomials.