Kostant's problem for Whittaker modules

TL;DR

This paper investigates Kostant's problem for Whittaker modules over Lie algebras and superalgebras, providing conditions for positive solutions and reduction techniques to simpler modules.

Contribution

It offers a sufficient condition for positive answers to Kostant's problem for standard Whittaker modules and reduces the problem for superalgebras to their even parts.

Findings

Provides a criterion for positive Kostant's problem for standard Whittaker modules.

Reduces the problem for simple Whittaker modules to highest weight modules.

Develops reduction methods for Lie superalgebras to their even parts.

Abstract

We study the classical problem of Kostant for Whittaker modules over Lie algebras and Lie superalgebras. We give a sufficient condition for a positive answer to Kostant's problem for the standard Whittaker modules over reductive Lie algebras. Under the same condition, the positivity of the answer for simple Whittaker modules is reduced to that for simple highest weight modules. We develop several reduction results to reduce the Kostant's problem for standard and simple Whittaker modules over a type I Lie superalgebra to that for the corresponding Whittaker modules over the even part of this Lie superalgebra.