Quantization of Double Covers of Nilpotent Coadjoint Orbits I: Noncommutative Models

TL;DR

This paper constructs a noncommutative algebraic model for the universal double cover of certain nilpotent coadjoint orbits using geometric and differential operator methods, revealing rich algebraic structures.

Contribution

It introduces a novel geometric construction of a noncommutative algebra for nilpotent orbit covers, with detailed structural properties and a connection to star products.

Findings

E is a Dixmier algebra for the orbit cover

E has an anti-automorphism, supertrace, and bilinear pairing

E is a specialization of a graded equivariant star product

Abstract

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.