The Gelfand-Tsetlin type base for the algebra ${\mathfrak{g}}_2$

The paper presents a construction of finite-dimensional irreducible representations of the Lie algebra ${\mathfrak{g}}_2$. The representation space is constructed as the space of solutions to a certain system of partial differential equations of hypergeometric type, which is closely related to the Gelfand-Kapranov-Zelevinsky systems. This connection allows for the construction of a basis in the representation. The orthogonalization of the constructed basis with respect to the invariant scalar product turns out to be a Gelfand-Tsetlin-type basis for the chain of subalgebras ${\mathfrak{g}}_2\supset {\mathfrak{s}\mathfrak{l}}_3$.