Products of longitudinal pseudodifferential operators on flag varieties

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), whichmay be described as ‘essential orthogonality of subrepresentations’. © 2009 Elsevier Inc. All rights reserved.