Products of longitudinal pseudodifferential operators on flag varieties |
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Authors: | Robert Yuncken |
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Institution: | University of Victoria, Department of Mathematics and Statistics, PO Box 3060 STN CSC, Victoria, BC, Canada |
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Abstract: | Associated to each set S of simple roots of SL(n,C) is an equivariant fibration X→XS of the complete flag variety X of Cn. To each such fibration we associate an algebra JS of operators on L2(X), or more generally on L2-sections of vector bundles over X. This ideal contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the fibres. Together, they form a lattice of operator ideals whose common intersection is the compact operators. Thus, for instance, the product of negative order pseudodifferential operators along the fibres of two such fibrations, X→XS and X→XT, is a compact operator if S∪T is the full set of simple roots. The construction of the ideals uses noncommutative harmonic analysis, and hinges upon a representation theoretic property of subgroups of SU(n), which may be described as ‘essential orthogonality of subrepresentations’. |
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Keywords: | Semisimple Lie groups Pseudodifferential operators Noncommutative harmonic analysis Operator algebras |
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