Galois order realization of noncommutative type $D$ Kleinian singularities

TL;DR

This paper demonstrates that noncommutative type D Kleinian singularities can be represented as principal Galois orders, providing explicit generators and structural insights into their modules.

Contribution

It establishes a realization of noncommutative type D Kleinian singularities as principal Galois orders and computes explicit generators and module structures.

Findings

Realization of type D Kleinian singularities as Galois orders

Explicit generators for associated flag orders

Structure constants for Harish-Chandra modules

Abstract

Galois orders, introduced by Futorny and Ovsienko, is a class of noncommutative algebras that includes generalized Weyl algebras, the enveloping algebra of the general linear Lie algebra and many others. We prove that the noncommutative Kleinian singularities of type $D$ can be realized as principal Galois orders. Our starting point is an embedding theorem due to Boddington. We also compute explicit generators for the corresponding (Morita equivalent) flag order, as a subalgebra of the nil-Hecke algebra of type $A_1^{(1)}$. Lastly, we compute structure constants for Harish-Chandra modules of local distributions and give a visual description of their structure from which subquotients are easily obtained.