Pure weights

Definition

Let $E$ be a quadratic field that is real. Let $S_\infty$ be the set of its infinite places, that is, its embeddings into the real numbers. Let $G_E$ be the group $\mathrm{GL}_3$ over $E$, and let $\mu$ be a highest weight for $G_E$.

Such a weight is a family $\mu=(\mu_\tau)_{\tau \in S_\infty}$. Each $\mu_\tau=(m^+_\tau,m^-_\tau,v_\tau)$ is a dominant weight of the diagonal torus of $\mathrm{GL}_3$.

The weight $\mu$ is called pure when both clauses hold:

1. The entry $v_\tau$ is the same for every $\tau \in S_\infty$.
2. The equality $m^+_\tau=m^-_\tau$ holds for every $\tau \in S_\infty$.

The pure weights are the highest weights for $G_E$ that satisfy both clauses.