Computing Parabolically Induced Embeddings of Semisimple Complex Lie Algebras in Weyl Algebras

An arbitrary proper parabolic subalgebra p of a simple complex Lie algebra g induces an embedding g W n , and more generally an embedding g W n End V , where W n is the Weyl algebra in n variables, n is the dimension of the nilradical of p, and V is an arbitrary p-module.We give an elementary proof of this known fact, report on a computer program computing the embeddings, and tabulate exceptional Lie algebra embeddings G 2 W 5 , F 4 W 15 , E 6 W 16 , E 7 W 27 , E 8 W 57 arising in this fashion.