Computation of Gelfand-Kirillov dimension for $B$-type structures

Let $\mathcal{O}(Spi{n}_{q^{1\mathrm{/}2}}(2n+1))$ and $\mathcal{O}(S{O}_q(2n+1))$ 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 $\mathcal{O}(Spi{n}_{q^{1\mathrm{/}2}}(2n+1))$-module $V_{t,w}^{Spin}$ is the same as the length of the Weyl word $w$. We show that the same result holds for the $\mathcal{O}(S{O}_q(2n+1))$-module $V_{t,w}$, which is obtained from $V_{t,w}^{Spin}$ by restricting the algebra action to the subalgebra $\mathcal{O}(S{O}_q(2n+1))$ of $\mathcal{O}(Spi{n}_{q^{1\mathrm{/}2}}(2n+1))$. 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.