Products of longitudinal pseudodifferential operators on flag varieties

Abstract

Associated to each set S of simple roots of SL ( n , C ) is an equivariant fibration X → X S of the complete flag variety X of C n . To each such fibration we associate an algebra J S of operators on L 2 ( X ) , or more generally on L 2 -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 → X S and X → X T , 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’.