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 $\mathfrak{g}_2 \supset \mathfrak{sl}_3$.