Linear Factorization of Conical Polynomials over Certain Nonassociative Algebras

Conical polynomials are defined as certain polynomials in quadratic elements of the universal enveloping algebra of a semisimple symmetric Lie algebra over a field of characteristic zero. These polynomials were used in an earlier paper to describe the conical vectors in certain induced modules. Here it is shown that when the base field is extended to a certain type of nonassociative algebra, the conical polynomials can be factored "linearly". One such nonassociative algebra is discussed in detail—an (alternative) composition algebra intimately related to the structure of the Lie algebra and studied earlier by B. Kostant in the context of real semisimple Lie algebras. The linear factorization leads in a later paper to an extension of the earlier work on conical vectors in induced modules.