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  "body": "\n## Summary\n\nA 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 $(m^+_\\tau, m^-_\\tau, v_\\tau)$.\n\nThe weight $\\mu$ is **pure** when $v_\\tau$ is the same for all $\\tau$ in $S_\\infty$, and $m^+_\\tau = m^-_\\tau$ for every $\\tau$.\n\nThe 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.\n",
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      "notes": "The source is arXiv:2609.26477v1 (2026-09), by Tristan Ricoul. A paper also restates older results, so this is not the first statement until a person or a later pass confirms it."
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  "name": "Pure weights",
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