The multiplicity of weights in nonprimitive pairs of weights

TL;DR

This paper classifies certain dominant weights in classical Lie algebras where specific weight spaces are one-dimensional, revealing examples of nilmanifolds with unresolved geodesic density questions.

Contribution

It provides a detailed listing of nonprimitive pairs of weights with one-dimensional weight spaces across classical Lie algebras, highlighting cases with open geometric problems.

Findings

Identifies weights with one-dimensional weight spaces in classical Lie algebras.

Provides examples of nilmanifolds with unresolved geodesic density questions.

Classifies nonprimitive pairs of weights for each classical Lie algebra type.

Abstract

For each type of classical Lie algebra, we list the dominant highest weights $\zeta$ for which $(\zeta;\mu_i)$ is not a primitive pair and the weight space $V_{\mu_i}$ has dimension one where $\mu_i$ are the highest long and short roots in each case. These dimension one weight spaces lead to examples of nilmanifolds for which we cannot prove or disprove the density of closed geodesics.