On the classification of primitive ideals for complex classical Lie algebras, IV

TL;DR

This paper completes the classification of primitive ideals for complex classical Lie algebras by extending previous results to type D, introducing a new tau-invariant, and parametrizing ideals via domino tableaux.

Contribution

It introduces a new definition of the tau-invariant for type D and completes the classification of primitive ideals for all classical types.

Findings

Primitive ideals with trivial infinitesimal character are characterized by tau-invariants.

Primitive ideals are parametrized by standard domino tableaux.

The generalized tau-invariant is defined using operators related to type D_4 subsystems.

Abstract

This paper is the fourth and last in the series "On the classification of primitive ideals for complex classical Lie algebras", extending earlier results in other classical types to type D. The generalized tau-invariant used in earlier work must now be defined in a different way, using a family of operators attached to a quadruple of simple roots spanning a subsystem of type D_4. each taking one or two values. Using these operators we show that primitive ideals of trivial infinitesimal character are characterized by their generalized tau-invariants and are parametrized by standard domino tableaux of the appropriate special shape.