Invariant Prime Ideals in Quantizations of Nilpotent Lie Algebras

De Concini, Kac and Procesi defined a family of subalgebras U + of a quantized universal enveloping algebra Uq(g), associated to the elements of the corresponding Weyl group W . They are deformations of the universal enveloping algebras U(n+ ∩ Adw(n−)) where n± are the nilradicals of a pair of dual Borel subalgebras. Based on results of Gorelik and Joseph and an interpretation of U + as quantized algebras of functions on Schubert cells, we construct explicitly the H invariant prime ideals of each U + and show that the corresponding poset is isomorphic to W, where H is the group of group-like elements of Uq(g). Moreover, for each H-prime of U w + we construct a generating set in terms of Demazure modules related to fundamental representations. Using results of Ramanathan and Kempf we prove similar theorems for vanishing ideals of closures of torus orbits of symplectic leaves of related Poisson structures on Schubert cells in flag varieties.