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

Abstract

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 $\theta :{\mathcal{O}}_q(P_J)\to {\mathcal{O}}_q(L_J)\otimes {\mathcal{O}}_q(P_J)$ of an induced coaction and the corresponding subalgebra ${\mathcal{O}}_q(P_J{)}^{\mathrm{co}\theta }\subseteq {\mathcal{O}}_q(P_J)$ of coinvariants. It was shown that the smash product algebra ${\mathcal{O}}_q(L_J)\#{\mathcal{O}}_q(P_J{)}^{\mathrm{co}\theta }$ is isomorphic to ${\mathcal{O}}_q(P_J)$. In view of this, ${\mathcal{O}}_q(P_J{)}^{\mathrm{co}\theta }$ -- 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 $\mathfrak{sl}(n)$. An explicit presentation in terms of generators and relations is found for these quantum Schubert cells.

19 pages, AMS Latex