2000/02/15 by Ranee Brylinski, Brylinski, Ranee
Mathematics · #17B35 #17C20 #22E46 #43A85 #53D55 #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Symplectic Geometry (math.SG) #math.QA #math.RT #math.SG #msc:17B35 #msc:17C20 #msc:22E46 #msc:43A85 #msc:53D55
paper · pdf · doi:10.48550/arxiv.math/0002117
37 pages; Latex; see http://www.math.psu.edu/rkb for more papers
arxiv created 2000/02/15 · openalex publication_date 2000/02/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct by geometric methods a noncommutative model E of the algebra of regular functions on the universal (2-fold) cover M of certain nilpotent coadjoint orbits O for a complex simple Lie algebra g. Here O is the dense orbit in the cotangent bundle of the generalized flag variety X associated to a complexified Cartan decomposition g=(p+)+k+(p-) where p+- are Jordan algebras by the TKK construction. We obtain E as the algebra of g-finite differential operators on a smooth Lagrangian subvariety in M where g is given by differential operators twisted according to a critical parameter. After Fourier transform, E is a quadratic extension of the algebra of twisted differential operators for a (formal) tensor power of the canonical bundle. Not only is E a Dixmier algebra for M, in the sense of the orbit method, but also E has a lot of additional structure,including an anti-automorphism, a supertrace, and a non-degenerate supersymmetric bilinear pairing. We show that E is the specialization at t=1 of a graded (non-local) equivariant star product with parity.