Differential Operators on Grassmann Varieties

Following Weyl’s account in The Classical Groups we develop an analogue of the (first and second) Fundamental Theorems of Invariant Theory for rings of differential operators: when V is a k-dimensional complex vector space with the standard SLkC action, we give a presentation of the ring of invariant differential operators D(C[V n])SLkC and a description of the ring of differential operators on the G.I.T. quotient, D(C[V n]SLkC), which is the ring of differential operators on the (affine cone over the) Grassmann variety of k-planes in n-dimensional space. We also compute the Hilbert series of the associated graded rings GrD(C[V ])k and Gr(D(C[V n]SLkC)). This computation shows that earlier claims that the kernel of the map from D(C[V n])SLkC to D(C[V n]SLkC) is generated by the Casimir operator are incorrect. Something can be gleaned from these earlier incorrect computations though: the kernel meets the universal enveloping algebra of slkC precisely in the central elements of U(slkC).