Abstract

This chapter studies explicit representations of the simple Lie algebra of type \(E_6\). First we prove the cubic \(E_6\)-generalization of the classical theorem on harmonic polynomials. Then we study the functor from the module category of \(D_5\) to the module category of \(E_6\). Finally, we give a family of inhomogeneous oscillator representations of the simple Lie algebra of type \(E_6\) on a space of exponential-polynomial functions and prove that their irreducibility is related to an explicit given algebraic variety.

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