{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,8,26]],"date-time":"2022-08-26T03:40:42Z","timestamp":1661485242111},"reference-count":15,"publisher":"Walter de Gruyter GmbH","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,11,1]]},"abstract":"Abstract<\/jats:title>\n We consider smooth representations of the unit group \n \n \n \n G<\/m:mi>\n =<\/m:mo>\n \n A<\/m:mi>\n \u00d7<\/m:mo>\n <\/m:msup>\n <\/m:mrow>\n <\/m:math>\n \n G=\\mathcal{A}^{\\times}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of a finite-dimensional split basic algebra \ud835\udc9c over a non-Archimedean local field.\nIn particular, we prove a version of Gutkin\u2019s conjecture, namely, we prove that every irreducible smooth representation of \ud835\udc3a is compactly induced by a one-dimensional representation of the unit group of some subalgebra of \ud835\udc9c.\nWe also discuss admissibility and unitarisability of smooth representations of \ud835\udc3a.<\/jats:p>","DOI":"10.1515\/jgth-2019-0084","type":"journal-article","created":{"date-parts":[[2021,5,26]],"date-time":"2021-05-26T05:33:29Z","timestamp":1622007209000},"page":"1069-1097","source":"Crossref","is-referenced-by-count":0,"title":["Smooth representations of unit groups of split basic algebras over non-Archimedean local fields"],"prefix":"10.1515","volume":"24","author":[{"given":"Carlos A.\u2009M.","family":"Andr\u00e9","sequence":"first","affiliation":[{"name":"Centro de An\u00e1lise Funcional, Estruturas Lineares e Aplica\u00e7\u00f5es (Grupo de Estruturas Lineares e Combinat\u00f3rias) , Departamento de Matem\u00e1tica , Faculdade de Ci\u00eancias da Universidade de Lisboa , Campo Grande, Edif\u00edcio C6, Piso 2, 1749-016 Lisboa , Portugal"}]},{"given":"Jo\u00e3o","family":"Dias","sequence":"additional","affiliation":[{"name":"Centro de An\u00e1lise Funcional, Estruturas Lineares e Aplica\u00e7\u00f5es (Grupo de Estruturas Lineares e Combinat\u00f3rias) , Departamento de Matem\u00e1tica , Faculdade de Ci\u00eancias da Universidade de Lisboa , Campo Grande, Edif\u00edcio C6, Piso 2, 1749-016 Lisboa , Portugal"}]}],"member":"374","published-online":{"date-parts":[[2021,5,26]]},"reference":[{"key":"2022082603131013788_j_jgth-2019-0084_ref_001","doi-asserted-by":"crossref","unstructured":"I.\u2009N. Bern\u0161te\u012dn and A.\u2009V. Zelevinski\u012d,\nRepresentations of the group \n \n \n \n \n GL\n \u2062\n \n (\n n\n ,\n F\n )\n \n \n ,\n \n \n \n \\mathrm{GL}(n,F),\n \n where \ud835\udc39 is a local non-Archimedean field,\nUspehi Mat. Nauk 31 (1976), no. 3\u2009(189), 5\u201370.","DOI":"10.1070\/RM1976v031n03ABEH001532"},{"key":"2022082603131013788_j_jgth-2019-0084_ref_002","doi-asserted-by":"crossref","unstructured":"A. Borel,\nLinear Algebraic Groups, 2nd ed.,\nGrad. Texts Math. 126,\nSpringer, New York, 1991.","DOI":"10.1007\/978-1-4612-0941-6"},{"key":"2022082603131013788_j_jgth-2019-0084_ref_003","doi-asserted-by":"crossref","unstructured":"M. Boyarchenko,\nRepresentations of unipotent groups over local fields and Gutkin\u2019s conjecture,\nMath. Res. Lett. 18 (2011), no. 3, 539\u2013557.","DOI":"10.4310\/MRL.2011.v18.n3.a14"},{"key":"2022082603131013788_j_jgth-2019-0084_ref_004","doi-asserted-by":"crossref","unstructured":"C.\u2009J. Bushnell and G. Henniart,\nThe Local Langlands Conjecture for \n \n \n \n \n GL\n \u2062\n \n (\n 2\n )\n \n \n \n \n \\mathrm{GL}(2)\n \n \n ,\nGrundlehren Math. Wiss. 335,\nSpringer, Berlin, 2006.","DOI":"10.1007\/3-540-31511-X"},{"key":"2022082603131013788_j_jgth-2019-0084_ref_005","doi-asserted-by":"crossref","unstructured":"P. Cartier,\nRepresentations of \ud835\udc5d-adic groups: A survey,\nAutomorphic Forms, Representations and \ud835\udc3f-Functions,\nProc. Sympos. Pure Math. 33,\nAmerican Mathematical Society, Providence (1979), 111\u2013155.","DOI":"10.1090\/pspum\/033.1\/546593"},{"key":"2022082603131013788_j_jgth-2019-0084_ref_006","doi-asserted-by":"crossref","unstructured":"A. Deitmar and S. Echterhoff,\nPrinciples of Harmonic Analysis, 2nd ed.,\nUniversitext,\nSpringer, Cham, 2014.","DOI":"10.1007\/978-3-319-05792-7"},{"key":"2022082603131013788_j_jgth-2019-0084_ref_007","doi-asserted-by":"crossref","unstructured":"G.\u2009B. Folland,\nA Course in Abstract Harmonic Analysis, 2nd ed.,\nTextb. Math.,\nCRC Press, Boca Raton, 2016.","DOI":"10.1201\/b19172"},{"key":"2022082603131013788_j_jgth-2019-0084_ref_008","doi-asserted-by":"crossref","unstructured":"E.\u2009A. Gutkin,\nRepresentations of algebraic unipotent groups over a selfdual field,\nFunct. Anal. Appl. 7 (1974), 322\u2013323.","DOI":"10.1007\/BF01075739"},{"key":"2022082603131013788_j_jgth-2019-0084_ref_009","doi-asserted-by":"crossref","unstructured":"Z. Halasi,\nOn the characters and commutators of finite algebra groups,\nJ. Algebra 275 (2004), no. 2, 481\u2013487.","DOI":"10.1016\/j.jalgebra.2004.01.021"},{"key":"2022082603131013788_j_jgth-2019-0084_ref_010","doi-asserted-by":"crossref","unstructured":"Z. Halasi,\nOn the characters of the unit group of DN-algebras,\nJ. Algebra 302 (2006), no. 2, 678\u2013685.","DOI":"10.1016\/j.jalgebra.2006.02.036"},{"key":"2022082603131013788_j_jgth-2019-0084_ref_011","doi-asserted-by":"crossref","unstructured":"I.\u2009M. Isaacs,\nCharacters of groups associated with finite algebras,\nJ. Algebra 177 (1995), no. 3, 708\u2013730.","DOI":"10.1006\/jabr.1995.1325"},{"key":"2022082603131013788_j_jgth-2019-0084_ref_012","unstructured":"H. Jacquet,\nSur les repr\u00e9sentations des groupes r\u00e9ductifs \ud835\udc5d-adiques,\nC. R. Acad. Sci. Paris S\u00e9r. A-B 280 (1975), 1271\u20131272."},{"key":"2022082603131013788_j_jgth-2019-0084_ref_013","unstructured":"H. Kl\u00fcver,\nThe unitary character group of abelian unipotent groups,\nM\u00fcnster J. Math. 1 (2008), 181\u2013219."},{"key":"2022082603131013788_j_jgth-2019-0084_ref_014","doi-asserted-by":"crossref","unstructured":"F. Rodier,\nD\u00e9composition spectrale des repr\u00e9sentations lisses,\nNon-Commutative Harmonic Analysis,\nLecture Notes in Math. 587,\nSpringer, Berlin (1977), 177\u2013195.","DOI":"10.1007\/BFb0087921"},{"key":"2022082603131013788_j_jgth-2019-0084_ref_015","doi-asserted-by":"crossref","unstructured":"B. Szegedy,\nOn the characters of the group of upper-triangular matrices,\nJ. Algebra 186 (1996), no. 1, 113\u2013119.","DOI":"10.1006\/jabr.1996.0365"}],"container-title":["Journal of Group Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/jgth-2019-0084\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/jgth-2019-0084\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,26]],"date-time":"2022-08-26T03:13:33Z","timestamp":1661483613000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/jgth-2019-0084\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,26]]},"references-count":15,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,5,26]]},"published-print":{"date-parts":[[2021,11,1]]}},"alternative-id":["10.1515\/jgth-2019-0084"],"URL":"https:\/\/doi.org\/10.1515\/jgth-2019-0084","relation":{},"ISSN":["1433-5883","1435-4446"],"issn-type":[{"value":"1433-5883","type":"print"},{"value":"1435-4446","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,26]]}}}