Quantized nilradicals of parabolic subalgebras of and algebras of coinvariants

Abstract

Let PJ be the standard parabolic subgroup of SLn associated to a subset J of simple roots, and let be the standard Levi decomposition. We let and denote the quantized algebras of regular functions on PJ and LJ , respectively. Following work of the first author, we study the quantum analogue of an induced coaction and the corresponding subalgebra of coinvariants. It was previously shown that the smash product algebra is isomorphic to In view of this, – while it is not a Hopf algebra – can be viewed as a quantum analogue of the coordinate ring In this article we prove that when is nonzero and not a root of unity, is isomorphic to a quantum Schubert cell algebra associated to a certain parabolic element w in the Weyl group of In this setting, the quantum Schubert cell algebra is a q-deformation of the universal enveloping algebra of the nilradical of a parabolic subalgebra of We also compute explicit commutation relations among the Lusztig root vectors for these particular quantized nilradicals and we give explicit algebra isomorphisms from to