1-dimensional representations and parabolic induction for W-algebras

TL;DR

This paper investigates 1-dimensional representations of W-algebras derived from semisimple Lie algebras, providing criteria for their existence, explicit computations, and a new parabolic induction functor that preserves module dimensions.

Contribution

It introduces a criterion for 1-dimensional modules based on highest weight theory and develops a dimension-preserving parabolic induction functor for W-algebras.

Findings

Criteria for 1-dimensional modules under certain conditions.

Explicit computation in type E8 case.

A dimension-preserving parabolic induction functor.

Abstract

A W-algebra is an associative algebra constructed from a semisimple Lie algebra and its nilpotent element. This paper concentrates on the study of 1-dimensional representations of these algebras. Under some conditions on a nilpotent element (satisfied by all rigid elements) we obtain a criterium for a finite dimensional module to have dimension 1. It is stated in terms of the Brundan-Goodwin-Kleshchev highest weight theory. This criterium allows to compute highest weights for certain completely prime primitive ideals in universal enveloping algebras. We make an explicit computation in a special case in type $E_8$. Our second principal result is a version of a parabolic induction for W-algebras. In this case, the parabolic induction is an exact functor between the categories of finite dimensional modules for two different W-algebras. The most important feature of the functor is that it preserves dimensions. In particular, it preserves one-dimensional representations. A closely related result was obtained previously by Premet. We also establish some other properties of the parabolic induction functor.