Covariant deformations on polarized orbits of a Lie group
Let \({\mathcal O}\) be a coadjoint orbit in the dual of the Lie algebra of a real Lie group, with its natural structure of a symplectic manifold. The aim is to construct a star-product on \({\mathcal O}\): \[ u*v = C_0 (u,v) + tC_1 (u,v) + t^2 C_2 (u,v) + \cdots \] where \(u,v\) are smooth functions on \({\mathcal O},t\) is the deformation parameter, \(C_0(u,v) = uv\) is the ordinary product and \(C_1 (u,v) = \{u,v\}\) is the Poisson bracket. Star-products are known to exist on any symplectic manifold, but here it is natural to ask for a covariant star-product, i.e. \(u*v - v*u = 2t \{u,v\}\) whenever \(u(\xi) = \langle \xi,X \rangle\) and \(v(\xi) = \langle \xi, Y \rangle \) are coordinate functions associated to elements \(X,Y\) in the Lie algebra. -- The main result is the existence of such a covariant star-product on the orbit \({\mathcal O}\), as soon as it admits a (real or complex) polarization. Its construction is obtained by means of suitably chosen coordinates on a Kähler manifold naturally associated to this geometric setting.
scientific article; zbMATH DE number 700988