Poly-analytic Functions and Representation Theory

Abstract

We propose the Lie-algebraic interpretation of poly-analytic functions in L2(C, dμ), with the Gaussian measure dμ, based on a flag structure formed by the representation spaces of the sl(2)-algebra realized by differential operators in z and z̄. Following the pattern of the onedimensional 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 sl(d+ 1), we consider also the family of algebras sl(m1 + 1) ⊗ . . .⊗ sl(mn + 1) for tuples m = (m1,m2, . . . ,mn) of positive integers whose sum is equal to d.