The tale of Kostant's problem

TL;DR

This survey explores the history and recent developments of Kostant's problem, which concerns the surjectivity of the universal enveloping algebra onto certain endomorphism algebras for modules over semi-simple Lie algebras.

Contribution

It provides a comprehensive overview of both classical and new results related to Kostant's problem, highlighting progress and open questions.

Findings

Summary of historical context and key results

Introduction of new partial solutions or conjectures

Identification of modules where surjectivity holds or fails

Abstract

This is a survey paper presenting the history and both old and new results related to Kostant's problem. This problem asks for which modules over a semi-simple finite dimensional complex Lie algebra, the universal enveloping algebra surjects onto the algebra of adjointly locally finite linear endomorphism.