{"paper":{"title":"Geometric and algebraic parameterizations for Dirac cohomology of simple modules in $\\mathcal{O}^\\mathfrak{p}$ and their applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Ho-Man Cheung","submitted_at":"2019-07-22T01:11:41Z","abstract_excerpt":"In this paper, we show that the Dirac cohomology $H_{D}(L(\\lambda))$ of a simple highest weight module $L(\\lambda)$ in $\\mathcal{O}^\\mathfrak{p}$ can be parameterized by a specific set of weights: a subset $\\mathcal{W}_I(\\lambda)$ of the orbit of the Weyl group $W$ acting on $\\lambda+\\rho$. As an application, we show that any simple module in $\\mathcal{O}^\\mathfrak{p}$ is determined up to isomorphism by its Dirac cohomology. We describe four parameterizations of $H_D(L(\\lambda))$ when $\\lambda$ is regular. Two of these parameterizations are geometric in terms of a partial ordering on the dual "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.09069","kind":"arxiv","version":1},"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"}