On the generalized commuting varieties of a reductive Lie algebra

TL;DR

This paper proves that the normalizations of generalized commuting varieties in reductive Lie algebras are Gorenstein with rational singularities, and their canonical modules are free of rank 1, extending known results to a broader class.

Contribution

It establishes that the normalizations of generalized commuting varieties are Gorenstein with rational singularities and have canonical modules that are free of rank 1, generalizing previous results.

Findings

Normalizations are Gorenstein with rational singularities

Canonical modules are free of rank 1

Results apply to the usual commuting variety as a special case

Abstract

The generalized commuting and isospectral commuting varieties of a reductive Lie algebra have been introduced in a preceding article. In this note, it is proved that their normalizations are Gorenstein with rational singularities. Moreover, their canonical modules are free of rank 1. In particular, the usual commuting variety is Gorenstein with rational singularities and its canonical module is free of rank 1.