Computation of Gelfand-Kirillov dimension for B-type structures

Abstract

Let O(Spinq1/2(2n+1)) and O(SOq(2n+1)) be the quantized algebras of regular functions on the Lie groups Spin(2n+1) and SO(2n+1), respectively. In this article, we prove that the Gelfand-Kirillov dimension of a simple unitarizable O(Spinq1/2(2n+1))-module VSpint,w is the same as the length of the Weyl word w. We show that the same result holds for the O(SOq(2n+1))-module Vt,w, which is obtained from VSpint,w by restricting the algebra action to the subalgebra O(SOq(2n+1)) of O(Spinq1/2(2n+1)). Moreover, we consider the quantized algebras of regular functions on certain homogeneous spaces of SO(2n+1) and Spin(2n+1) and show that its Gelfand-Kirillov dimension is equal to the dimension of the homogeneous space as a real differentiable manifold.