{"paper":{"title":"Remarks on the abelian ideals of a Borel subalgebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Chao-Ping Dong","submitted_at":"2013-08-21T01:30:49Z","abstract_excerpt":"Let $\\frb$ be a fixed Borel subalgebra of a finite-dimensional complex simple Lie algebra $\\frg$. The Shi bijection associates to every ad-nilpotent ideal $\\fri$ of $\\frb$ a region $V_{\\fri}$. In this paper, we show that $\\fri$ is abelian if and only if $V_{\\fri}\\cap 2A$ is empty, if and only if the volume of $V_{\\fri}\\cap 2A$ equals to that of $A$, where $A$ is the fundamental alcove of the affine Weyl group. For certain flag of abelian ideals, we record an ascending property of their associated regions. We also determine the maximal eigenvalue $m_{r-1}$ of the Casimir operator on $\\wedge^{r-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4464","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}