Primitive ideals in quantum Schubert cells: Dimension of the strata

The aim of this paper is to study the representation theory of quantum Schubert cells. Let be a simple complex Lie algebra. To each element w of the Weyl group W of , De Concini, Kac and Procesi have attached a subalgebra Uq[w] of the quantised enveloping algebra Uq(). Recently, Yakimov showed that these algebras can be interpreted as the (quantum) Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of Uq[w]. More precisely, it follows from the Stratification Theorem of Goodearl and Letzter, and from recent works of Mériaux–Cauchon and Yakimov, that the primitive spectrum of Uq[w] admits a stratification indexed by those elements vW with vw in the Bruhat order. Moreover each stratum is homeomorphic to the spectrum of maximal ideals of a torus. The main result of this paper gives an explicit formula for the dimension of the stratum associated to a pair.