Quantized nilradicals of parabolic subalgebras of $\mathfrak{s}\mathfrak{l}(n)$ and algebras of coinvariants

Let $P_J$ be the standard parabolic subgroup of $S{L}_n$ obtained by deleting a subset $J$ of negative simple roots, and let $P_J=L_J{U}_J$ be the standard Levi decomposition. Following work of the first author, we study the quantum analogue $\theta: {\mathcal O}_q(P_J) \to{\mathcal O}_q(L_J) \otimes {\mathcal O}_q(P_J)$ofaninducedcoactionandthecorrespondingsubalgebra${\mathcal O}_q(P_J)^{\operatorname{co} \theta} \subseteq {\mathcal O}_q(P_J)$ of coinvariants. It was shown that the smash product algebra ${\mathcal O}_q(L_J)# {\mathcal O}_q(P_J)^{\operatorname{co} \theta}$isisomorphicto${\mathcal O}_q(P_J)$\mathrm{.}Inviewofthis,${\mathcal O}_q(P_J)^{\operatorname{co} \theta}$ – while it is not a Hopf algebra – can be viewed as a quantum analogue of the coordinate ring $\mathcal{O}(U_J)$. In this paper we prove that when $q\in \mathbb{K}$ is nonzero and not a root of unity, ${\mathcal{O}}_q(P_J{)}^{\mathrm{co}\theta }$ is isomorphic to a quantum Schubert cell algebra ${\mathcal{U}}_q^{+}[w]$ associated to a parabolic element $w$ in the Weyl group of $\mathfrak{s}\mathfrak{l}(n)$. An explicit presentation in terms of generators and relations is found for these quantum Schubert cells.