The multiplicity of weights in nonprimitive pairs of weights

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.