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Journal of Lie Theory 31 (2021), No. 4, 1113--1140
Copyright Heldermann Verlag 2021

Noncubic Dirac Operators for Finite-Dimensional Modules

Spyridon Afentoulidis-Almpanis
Faculty of Mathematics, University of Lorraine, Metz, France

We study the decomposition into irreducibles of the kernel of noncubic Dirac operators attached to finite-dimensional modules. We compare this decomposition with features of Kostant's cubic Dirac operator. In particular, we show that the kernel of noncubic Dirac operators need not contain full isotypic components. The cases of classical and exceptional complex Lie algebras are studied in details. As a by-product, we deduce some information on the kernel of noncubic geometric Dirac operators acting on sections over compact manifolds studied by Slebarski.

Keywords: Complex semisimple Lie algebras, highest weight representations, Dirac operators, Dirac cohomology, Weyl inequalities.

MSC: 17B45; 20G05.

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