Pure weights

Summary

A highest weight $\mu$ for $G_E$ is a family $(\mu_\tau)$ indexed by the set $S_\infty$ of infinite embeddings of the base field $E$. Each $\mu_\tau$ is a dominant weight of the diagonal torus of $\mathrm{GL}_3$, written as a triple $(m^+_\tau, m^-_\tau, v_\tau)$.

The weight $\mu$ is pure when $v_\tau$ is the same for all $\tau$ in $S_\infty$, and $m^+_\tau = m^-_\tau$ for every $\tau$.

The source paper uses this condition when it studies divisibility of automorphic periods for the base change of $\mathrm{GL}_3$ to a real quadratic field.