Poly-analytic Functions and Representation Theory

We propose the Lie-algebraic interpretation of poly-analytic functions in $$L_2({{\mathbb {C}}},d\mu )$$ , with Gaussian measure $$d\mu $$ based on a flag structure formed by representation spaces $$\mathfrak {sl}(2)$$ -algebra realized differential operators z and $${\bar{z}}$$ . Following pattern one-dimensional situation, we define poly-Fock d complex variables way, as invariant for action generators certain Lie algebra. In addition to basic case algebra {sl}(d+1)$$ consider also family algebras {sl}(m_1+1) \otimes \ldots \mathfrak {sl}(m_n+1)$$ tuples $$\mathbf {m} = (m_1,m_2,\ldots ,m_n)$$ positive integers whose sum is equal d.