X (and hence a polynomial function on the fibers) is an order d symbol. The order d symbol map sends Vd(X) into Rd(T* X) where Vd(X) is the space of differential operators of order at most d and Rd(T* X) is the space of order d symbols. We say that ¢ E Rd(T* X) quantizes into D if D E Vd(X) has order d and the principal symbol of D is ¢. Unless X is smooth and affine, there is no guarantee that a given symbol will quantize.
There are natural non-compact analogues to the flag varieties. These homogeneous spaces of K arise in the following way. Suppose that K sits inside a larger reductive complex algebraic group G as (the identity component of) a spherical subgroup. Let g = t EEl P be the corresponding complex Cartan decomposition where t = Lie K and g = Lie G. Then the K-orbits in p are quasi-affine homogeneous spaces X of K (and so noncompact except if X is a point). The choice G = K x K gives rise to all the adjoint orbits of K.
Advances in Geometry by Alexander Astashkevich (auth.), Jean-Luc Brylinski, Ranee Brylinski, Victor Nistor, Boris Tsygan, Ping Xu (eds.)