Transfer of ideals and quantization of small nilpotent orbits

TL;DR

This paper introduces a transfer map between ideals in universal enveloping algebras of dual Lie algebra pairs, linking representation annihilators and quantizations of nilpotent orbits, with applications to classical Lie algebras.

Contribution

It defines a transfer map motivated by Capelli identities, proving it bounds annihilators of theta lifts and respects quantizations in the stable range, with explicit orbit quantizations and differential operator rings.

Findings

Transfer map bounds annihilators of theta lifts.

In stable range, transfer respects nilpotent orbit quantizations.

Explicit descriptions of quantizations for small nilpotent orbits.

Abstract

We introduce and study a transfer map between ideals of the universal enveloping algebras of two members of a reductive dual pair of Lie algebras. Its definition is motivated by the approach to the real Howe duality through the theory of Capelli identities. We prove that this map provides a lower bound on the annihilators of theta lifts of representations with a fixed annihilator ideal. We also show that in the algebraic stable range, transfer respects the class of quantizations of nilpotent orbit closures. As an application, we explicitly describe quantizations of small nilpotent orbits of general linear and orthogonal Lie algebras and give presentations of certain rings of algebraic differential operators. We consider two algebraic versions of Howe duality and reformulate our results in terms of noncommutative Capelli identities.