Computation of Gelfand–Kirillov dimension for B-type structures

Let (Formula presented.) and (Formula presented.) denote the quantized algebras of regular functions on the Lie groups (Formula presented.) and (Formula presented.), respectively. In this article, we prove that the Gelfand–Kirillov dimension of a simple unitarizable (Formula presented.) -module (Formula presented.) is the same as the length of the associated Weyl word w. We further show that the same result holds for the (Formula presented.) -module (Formula presented.), which is obtained from (Formula presented.) by restricting the algebra action to the subalgebra (Formula presented.) of (Formula presented.). Moreover, we consider the quantized algebras of regular functions on certain homogeneous spaces of (Formula presented.) and (Formula presented.), and show that their Gelfand–Kirillov dimension coincides with the dimension of the corresponding homogeneous space as a real differentiable manifold.