A Gelfand–Tsetlin type basis for the algebra $\mathfrak g_2$

A construction of irreducible finite-dimensional representations of the Lie algebra $\mathfrak{g}_2$ is given. A space of a representation is constructed as a space of polynomial solutions of some system of partial differential equations of hypregeometric type which is closely related to Gelfand–Kapranov–Zelevinsky systems. This relations allows to construct a base in a representation. An orthogonalization of this base with respect to an invariant scalar product is a Gelfand–Tsetlin type base for the chain os subalgebras $\mathfrak{g}_2 \supset \mathfrak{sl}_3$.