Poly-analytic Functions and Representation Theory

Abstract

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