A Gelfand–Tsetlin type basis for the algebra 𝔤₂

The paper presents a construction of finite-dimensional irreducible representations of the Lie algebra g 2 \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 relationship allows for the construction of a basis in the representation. An orthogonalization of the constructed basis with respect to an invariant scalar product turns out to be a Gelfand–Tsetlin-type basis for the chain of subalgebras g 2 ⊃ s l 3 \mathfrak {g}_2 \supset \mathfrak {sl}_3 .