On automorphisms of quantum Schubert cells

Automorphisms of the quantum Schubert cell algebras ${\mathcal U}_q^\pm[w]$ of De Concini, Kac, Procesi and Lusztig and their restrictions to some key invariant subalgebras are studied. We develop some general rigidity results and apply them to completely determine the automorphism group in several cases. We focus primarily on those cases when the underlying Lie algebra $\mathfrak{g}$ is finite dimensional and simple with rank $r > 1$, and $w$ is a parabolic element of the Weyl group, say $w = w_o^Jw_o$, for some nonempty subset $J$ of simple roots. Here, ${\mathcal U}_q^\pm[w]$ is a deformation of the universal enveloping algebra of the nilradical of a parabolic subalgebra of $\mathfrak{g}$. In this setting we conjecture that, with the exception of two specific low rank cases, the automorphism group of ${\mathcal U}_q^{\pm}[w]$ is the semidirect product of an algebraic torus of rank $r$ with the group of Dynkin diagram symmetries that preserve $J$. This conjecture is a more general form of the Launois-Lenagan and Andruskiewitsch-Dumas conjectures regarding the automorphism groups of the algebras of quantum matrices and the algebras ${\mathcal U}_q^+(\mathfrak{g})$, respectively. We completely determine the automorphism group in several instances, including all cases when $\mathfrak{g}$ is of type $F_4$ or $G_2$, as well as those cases when the quantum Schubert cell algebras are the algebras of quantum symmetric matrices.