Narrow Results

In our earlier work on a new functor from $E_6$-Mod to $E_7$-Mod, we found a one-parameter ($c$) family of inhomogeneous first-order differential operator representations of the simple Lie algebra of type $E_7$ in $27$ variables. Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector $\overrightarrow{a}\in {ℂ}^{27}\backslash \{\overrightarrow{0}\}$, we prove that the space forms an irreducible $E_7$-module for any $c\in ℂ$ if $\overrightarrow{a}$ is not on an explicitly given projective algebraic variety. Certain equivalent combinatorial properties of the basic oscillator representation of $E_6$ over its 27-dimensional module play key roles in our proof. Our result can also be used to study free bosonic field irreducible representations of the corresponding affine Kac–Moody algebra.