Models of representations for classical series of Lie algebras

By a model of representations of a Lie algebra we mean a representation which is a direct sum of all irreducible finite-dimensional representations taken with multiplicity $1$. An explicit construction of a model of representations for all classical series of simple Lie algebras is given. This construction is generic for all classical series of Lie algebras. The space of the model is constructed as the space of polynomial solutions of a system of partial differential equations, where the equations are constructed form relations between minors of matrices taken from the corresponding Lie group. This system admits a simplification very close to the GKZ system, which is satisfied by $A$-hypergeometric functions.