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Process."],"published-print":{"date-parts":[[2024,3]]},"abstract":"<jats:p> In this paper, we consider the Plancherel formula for Fourier integral operator on the space [Formula: see text], where [Formula: see text] is the quaternion skew field. By using the metaplectic representations of a noncompact classical dual pair [Formula: see text] and combining with the Plancherel formula of Poisson transform, we obtain the direct integral decomposition of all irreducible unitary representations of Fourier integral operator. <\/jats:p>","DOI":"10.1142\/s021969132350042x","type":"journal-article","created":{"date-parts":[[2023,8,12]],"date-time":"2023-08-12T04:11:17Z","timestamp":1691813477000},"source":"Crossref","is-referenced-by-count":0,"title":["The Plancherel formula of Fourier integral operators: The case of Sp(1,1)"],"prefix":"10.1142","volume":"22","author":[{"given":"Xiaoke","family":"Jia","sequence":"first","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2051-5397","authenticated-orcid":false,"given":"Xingya","family":"Fan","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2023,8,31]]},"reference":[{"key":"S021969132350042XBIB001","series-title":"Graduate Texts in Mathematics","volume-title":"Representations of Compact Lie Groups","volume":"98","author":"Br\u00f6cker T.","year":"1995"},{"key":"S021969132350042XBIB002","doi-asserted-by":"publisher","DOI":"10.1515\/9781400882427"},{"key":"S021969132350042XBIB003","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1007\/BF01418850","volume":"19","author":"Gelbart S.","year":"1973","journal-title":"Invent. 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