Introduction to the Alexandru Conjecture

TL;DR

This paper proves that a Verma module can be transformed into another Verma module via a self-equivalence, introducing a new notion of standard object in Harish-Chandra modules that aligns with traditional concepts in many cases.

Contribution

It establishes the affirmative answer to the Alexandru Conjecture and proposes a new perspective on standard objects in Harish-Chandra modules.

Findings

Verma modules can be transformed into each other by self-equivalences.

Introduces a notion of standard object in Harish-Chandra modules.

The new notion coincides with the classical one in many cases.

Abstract

Is a Verma module transformed into another Verma module by a selfequivalence? The answer is affirmative and the proof suggests a notion of standard object in the category of Harish-Chandra modules that coincides often, but not always, with the usual one.