Quantization of nilpotent coadjoint $GL_N$-orbit closures in positive characteristics

TL;DR

This paper classifies filtered Hamiltonian quantizations of nilpotent coadjoint orbit closures for GL_N over fields of positive characteristic, introducing a new construction method from primitive quotients of the enveloping algebra.

Contribution

It provides a classification of quantizations for nilpotent orbit closures in positive characteristic and introduces a novel construction technique from primitive quotients.

Findings

Classified all filtered Hamiltonian quantizations for GL_N in positive characteristic.

Developed a new method to construct quantizations from primitive quotients of the enveloping algebra.

Connected quantizations to stabilizers of Frobenius twisted p-characters.

Abstract

Let $G$ be a reductive group over an algebraically closed field of positive characteristic $p$, good for the root system of $G$. The closures of $G$-orbits in the Hilbert nullcone of the coadjoint representation are conical affine Poisson varieties, generically of full rank, known as {\em nilpotent coadjoint orbits}. In this paper, we classify the filtered Hamiltonian quantizations of these orbit closures for $G = GL_N$ and any $p > 0$. Our main new technique is a construction of quantizations from certain primitive quotients of the enveloping algebra, inducing them from the stabiliser in $G$ of the Frobenius twisted $p$-character.