Poly-analytic Functions and Representation Theory

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