Primitive ideals in quantum Schubert cells: Dimension of the strata

Abstract

Abstract. The aim of this paper is to study the representation theory of quantum Schubert cells. Let 𝔤 $\mathfrak {g}$ be a simple complex Lie algebra. To each element w of the Weyl group W of 𝔤 $\mathfrak {g}$ , De Concini, Kac and Procesi have attached a subalgebra U q [ w ] $U_q[w]$ of the quantised enveloping algebra U q ( 𝔤 ) $U_q(\mathfrak {g})$ . 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 U q [ w ] $U_q[w]$ . 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 U q [ w ] $U_q[w]$ admits a stratification indexed by those elements v ∈ W $v \in W$ with v ≤ w $v \le w$ 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 v ≤ w $v \le w$ .