{"project":[{"grant_number":"P35847","_id":"34b2c9cb-11ca-11ed-8bc3-a50ba74ca4a3","name":"Geometry of the tip of the global nilpotent cone"}],"intvolume":"       121","abstract":[{"lang":"eng","text":"Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing the latter with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig’s q-weight multiplicity polynomials i.e., certain affine Kazhdan–Lusztig polynomials."}],"publication_status":"published","external_id":{"pmid":["39259592"]},"article_processing_charge":"Yes (in subscription journal)","has_accepted_license":"1","_id":"18108","ddc":["510"],"volume":121,"file":[{"file_name":"2024_PNAS_Hausel.pdf","access_level":"open_access","date_updated":"2024-09-23T11:22:56Z","file_size":3764695,"file_id":"18127","relation":"main_file","creator":"dernst","date_created":"2024-09-23T11:22:56Z","checksum":"df80c873633c6734d2e324841e69db58","success":1,"content_type":"application/pdf"}],"title":"Commutative avatars of representations of semisimple Lie groups","oa":1,"department":[{"_id":"TaHa"}],"scopus_import":"1","month":"09","quality_controlled":"1","tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"doi":"10.1073/pnas.2319341121","publication_identifier":{"eissn":["1091-6490"]},"publication":"Proceedings of the National Academy of Sciences of the United States of America","year":"2024","date_updated":"2024-09-23T11:25:37Z","citation":{"mla":"Hausel, Tamás. “Commutative Avatars of Representations of Semisimple Lie Groups.” <i>Proceedings of the National Academy of Sciences of the United States of America</i>, vol. 121, no. 38, e2319341121, National Academy of Sciences, 2024, doi:<a href=\"https://doi.org/10.1073/pnas.2319341121\">10.1073/pnas.2319341121</a>.","apa":"Hausel, T. (2024). Commutative avatars of representations of semisimple Lie groups. <i>Proceedings of the National Academy of Sciences of the United States of America</i>. National Academy of Sciences. <a href=\"https://doi.org/10.1073/pnas.2319341121\">https://doi.org/10.1073/pnas.2319341121</a>","chicago":"Hausel, Tamás. “Commutative Avatars of Representations of Semisimple Lie Groups.” <i>Proceedings of the National Academy of Sciences of the United States of America</i>. National Academy of Sciences, 2024. <a href=\"https://doi.org/10.1073/pnas.2319341121\">https://doi.org/10.1073/pnas.2319341121</a>.","ama":"Hausel T. Commutative avatars of representations of semisimple Lie groups. <i>Proceedings of the National Academy of Sciences of the United States of America</i>. 2024;121(38). doi:<a href=\"https://doi.org/10.1073/pnas.2319341121\">10.1073/pnas.2319341121</a>","short":"T. Hausel, Proceedings of the National Academy of Sciences of the United States of America 121 (2024).","ieee":"T. Hausel, “Commutative avatars of representations of semisimple Lie groups,” <i>Proceedings of the National Academy of Sciences of the United States of America</i>, vol. 121, no. 38. National Academy of Sciences, 2024.","ista":"Hausel T. 2024. Commutative avatars of representations of semisimple Lie groups. Proceedings of the National Academy of Sciences of the United States of America. 121(38), e2319341121."},"oa_version":"Published Version","corr_author":"1","article_number":"e2319341121","publisher":"National Academy of Sciences","type":"journal_article","file_date_updated":"2024-09-23T11:22:56Z","pmid":1,"acknowledgement":"We thank Nigel Hitchin for discussions and the joint projects this paper has grown out from. We thank Vladyslav Zveryk for collaboration on Theorem 2.3 and on the corresponding Magma code which implements big algebras. We thank Hiraku Nakajima for discussions and pointing out Theorem 3.1.2, a result generalizing our original observation in the= = 0 case. Special thanks go to Leonid Rybnikov for patiently explaining his works, in particular crucial to Theorem 2.1. We thank Michel Brion, Michael Finkelberg, Oscar García-Prada, Jakub Löwit, Joel Kamnitzer, Friedrich Knop, Michael McBreen, Anton Mellit, Takuro Mochizuki, Shon Ngô, Kamil Rychlewicz, Shiyu Shen, Leslie Spencer, Balázs Szendr ˝ oi, András Szenes, and Oksana\r\nYakimova for comments and discussions. Kamil Rychlewicz and Daniel Bedats helped with the Mathematica files for the figures, and we used the SM_isospin Tikz package of Izaak Neutelings for drawing the baryon multiplets. We thank the referees for many useful comments. We acknowledge funding from FWF grant “Geometry of the tip of the global nilpotent cone” no. P 35847.","author":[{"last_name":"Hausel","id":"4A0666D8-F248-11E8-B48F-1D18A9856A87","first_name":"Tamás","full_name":"Hausel, Tamás","orcid":"0000-0002-9582-2634"}],"license":"https://creativecommons.org/licenses/by/4.0/","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2024-09-22T22:01:41Z","related_material":{"link":[{"relation":"press_release","url":"https://ista.ac.at/en/news/big-algebras-a-dictionary-of-abstract-math/"}]},"day":"17","status":"public","language":[{"iso":"eng"}],"issue":"38","article_type":"original","date_published":"2024-09-17T00:00:00Z"}