2015/02/25 by Libor Křižka, Křižka, Libor, Petr Somberg +1 · 1 citation
Mathematics · #22E47 #33C45 #53A30 #58J70 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1502.07095
openalex publication_date 2015/02/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In the present article, we combine some techniques in the harmonic analysis together with the geometric approach given by modules over sheaves of rings of twisted differential operators (D-modules), and reformulate the composition series and branching problems for objects in the Bernstein-Gelfand-Gelfand parabolic category O^\mathfrakp geometrically realized on certain orbits in the generalized flag manifolds. The general framework is then applied to the scalar generalized Verma modules supported on the closed Schubert cell of the generalized flag manifold G/P for G=\rm SL(n+2,ℂ) and P the Heisenberg parabolic subgroup, and the algebraic analysis gives a complete classification of \mathfrakg'r-singular vectors for all \mathfrakg'r=\mathfraksl(n-r+2,ℂ) ⊂ \mathfrakg=\mathfraksl(n+2,ℂ), n-r > 2. A consequence of our results is that we classify \rm SL(n-r+2,ℂ)-covariant differential operators acting on homogeneous line bundles over the complexification of the odd dimensional CR-sphere S2n+1 and valued in homogeneous vector bundles over the complexification of the CR-subspheres S2(n-r)+1.