Transverse properties of parabolic subgroups of Garside groups

Let $G$ be a Garside group endowed with the generating set $\mathcal{S}$ of non-trivial simple elements, and let $H$ be a parabolic subgroup of $G$. We determine a transversal $T$ of $H$ in $G$ such that each $\theta \in T$ is of minimal length in its right-coset, $H\theta$, for the word length with respect to $\mathcal{S}$. We show that there exists a regular language $L$ on $\mathcal{S}\cup {\mathcal{S}}^{-1}$ and a bijection $\mathrm{e}\mathrm{v}:L\to T$ satisfying $\mathrm{l}\mathrm{g}(U)={\mathrm{l}\mathrm{g}}_{\mathcal{S}}(\mathrm{e}\mathrm{v}(U))$ for all $U\in L$. From this we deduce that the coset growth series of $H$ in $G$ is rational. Finally, we show that $G$ has fellow projections on $H$ but does not have bounded projections on $H$.