A New Functor from $D_5$-Mod to $E_6$-Mod

We find a new representation of the simple Lie algebra of type $E_6$ on the polynomial algebra in 16 variables, which gives a fractional representation of the corresponding Lie group on 16-dimensional space. Using this representation and Shen's idea of mixed product, we construct a functor from $D_5$-{\bf Mod} to $E_6$-{\bf Mod}. A condition for the functor to map a finite-dimensional irreducible $D_5$-module to an infinite-dimensional irreducible $E_6$-module is obtained. Our general frame also gives a direct polynomial extension from irreducible $D_5$-modules to irreducible $E_6$-modules. The obtained infinite-dimensional irreducible $E_6$-modules are $({\cal G},K)$-modules in terms of Lie group representations. The results could be used in studying the quantum field theory with $E_6$ symmetry and symmetry of partial differential equations.