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On 1-dimensional representations of finite W-algebras associated to simple Lie algebras of exceptional type

Published online by Cambridge University Press:  01 August 2010

Simon M. Goodwin
School of Mathematics, University of Birmingham, Birmingham, B15 2TT, United Kingdom (email:
Gerhard Röhrle
Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany (email:
Glenn Ubly
School of Mathematics, University of Southampton, Southampton, SO17 1BJ, United Kingdom (email:


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We consider the finite W-algebra U(𝔤,e) associated to a nilpotent element e∈𝔤 in a simple complex Lie algebra 𝔤 of exceptional type. Using presentations obtained through an algorithm based on the PBW-theorem for U(𝔤,e), we verify a conjecture of Premet, that U(𝔤,e) always has a 1-dimensional representation when 𝔤 is of type G2, F4, E6 or E7. Thanks to a theorem of Premet, this allows one to deduce the existence of minimal dimension representations of reduced enveloping algebras of modular Lie algebras of the above types. In addition, a theorem of Losev allows us to deduce that there exists a completely prime primitive ideal in U(𝔤) whose associated variety is the coadjoint orbit corresponding to e.

Research Article
Copyright © London Mathematical Society 2010


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