{"data":{"id":"10.13016/m23d0q","type":"dois","attributes":{"doi":"10.13016/m23d0q","prefix":"10.13016","suffix":"m23d0q","identifiers":[],"alternateIdentifiers":[],"creators":[{"name":"Ralston, Jacob","nameType":"Personal","givenName":"Jacob","familyName":"Ralston","affiliation":[],"nameIdentifiers":[]}],"titles":[{"title":"The Relative Lie Algebra Cohomology of the Weil Representation"}],"publisher":"Digital Repository at the University of Maryland","container":{},"publicationYear":2015,"subjects":[{"subject":"Mathematics","subjectScheme":"pqcontrolled"},{"subject":"FOS: Mathematics","schemeUri":"http://www.oecd.org/science/inno/38235147.pdf","subjectScheme":"Fields of Science and Technology (FOS)"},{"subjectScheme":"pquncontrolled"}],"contributors":[],"dates":[{"date":"2015","dateType":"Issued"}],"language":"en","types":{"ris":"THES","bibtex":"phdthesis","citeproc":"thesis","schemaOrg":"Thesis","resourceType":"Dissertation","resourceTypeGeneral":"Collection"},"relatedIdentifiers":[],"relatedItems":[],"sizes":[],"formats":[],"version":null,"rightsList":[],"descriptions":[{"description":"We study the relative Lie algebra cohomology of $\\mathfrak{so}(p,q)$ with values in the Weil representation $\\varpi$ of the dual pair $\\mathrm{Sp}(2k, \\R) \\times \\OO(p,q)$.  Using the Fock model defined in Chapter \\ref{Introchapter}, we filter this complex and construct the associated spectral sequence. We then prove that the resulting spectral sequence converges to the relative Lie algebra cohomology and has  $E_0$ term, the associated graded complex, isomorphic to a Koszul complex, see Section \\ref{defofkoszulsection}.  It is immediate that the construction of the spectral sequence of Chapter \\ref{spectralchapter} can be applied to any reductive subalgebra $\\mathfrak{g} \\subset \\mathfrak{sp}(2k(p+q), \\R)$.  By the Weil representation of $\\OO(p,q)$, we mean the twist of the Weil representation of the two-fold cover $\\widetilde{\\OO(p,q)}$ by a suitable character. We do this to make the center of $\\widetilde{\\OO(p,q)}$ act trivially.  Otherwise, all relative Lie algebra cohomology groups would vanish, see Proposition \\ref{genuineprop}.\n\nIn case the symplectic group is large relative to the orthogonal group ($k \\geq pq$), the $E_0$ term is isomorphic to a Koszul complex defined by a regular sequence, see \\ref{defofkoszulsection}.  Thus, the cohomology vanishes except in top degree.  This result is obtained without calculating the space of cochains and hence without using any representation theory.  On the other hand, in case $k \u0026lt; p$, we know the Koszul complex is not that of a regular sequence from the existence of the class $\\varphi_{kq}$ of Kudla and Millson, see \\cite{KM2}, a nonzero element of the relative Lie algebra cohomology of degree $kq$.\n\nFor the case of $\\SO_0(p,1)$ we compute the cohomology groups in these remaining cases, namely $k \u0026lt; p$.  We do this by first computing a basis for the relative Lie algebra cochains and then splitting the complex into a sum of two complexes, each of whose $E_0$ term is then isomorphic to a Koszul complex defined by a regular sequence.\n\nThis thesis is adapted from the paper, \\cite{BMR}, this author wrote with his advisor John Millson and Nicolas Bergeron of the University of Paris.","descriptionType":"Abstract"}],"geoLocations":[],"fundingReferences":[],"xml":"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","url":"http://hdl.handle.net/1903/16665","contentUrl":null,"metadataVersion":0,"schemaVersion":null,"source":null,"isActive":true,"state":"findable","reason":null,"viewCount":0,"viewsOverTime":[],"downloadCount":0,"downloadsOverTime":[],"referenceCount":0,"citationCount":0,"citationsOverTime":[],"partCount":0,"partOfCount":0,"versionCount":0,"versionOfCount":0,"created":"2015-06-26T05:44:09.000Z","registered":"2015-06-26T05:44:10.000Z","published":"2015","updated":"2020-08-19T19:59:37.000Z"},"relationships":{"client":{"data":{"id":"umd.lib","type":"clients"}},"provider":{"data":{"id":"umd","type":"providers"}},"media":{"data":{"id":"10.13016/m23d0q","type":"media"}},"references":{"data":[]},"citations":{"data"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