A Star Product for Complex Grasmann Manifolds

We explicitly construct a star product for the complex Grassmann manifolds using the method of phase space reduction. Functions on $\mathbb{C}^{(p+q)\cdot p~*}$, the space of $(p+q)\times p$ matrices of rank p, invariant under the right action of $Gl(p,\mathbb{C})$ can be regarded as functions on the Grassmann manifold $G_{p,q}(\mathbb{C})$, but do not form a subalgebra whereas functions only invariant under the unitary subgroup $U(p)\subset Gl(p,\mathbb{C})$ do. The idea is to construct a projection from $U(p)$- onto $Gl(p,\mathbb{C})$-invariant functions, whose kernel is an ideal. This projection can be used to define a star-algebra on $G_{p,q}(\mathbb{C})$ onto which this projection acts as an algebra-epimorphism.