Gorenstein algebras and toric bundles

We study commutative algebras with Gorenstein duality, i.e. algebras $A$ equipped with a non-degenerate bilinear pairing such that $\langle ac,b\rangle=\langle a,bc\rangle$ for any $a,b,c\in A$. If an algebra $A$ is Artinian, such pairing exists if and only if $A$ is Gorenstein. We give a description of algebras with Gorenstein duality as a quotients of the ring of differential operators by the annihilator of an explicit polynomial (or more generally formal polynomial series). This provides a calculation of Macaulay's inverse systems for (not necessarily Artinian) algebras with Gorenstein duality. Our description generalizes previously know description of graded algebras with Gorenstein duality generated in degree 1. Our main motivation comes from the study of even degree cohomology rings. In particular, we apply our main result to compute the ring of cohomology classes of even degree of toric bundles and the ring of conditions of horospherical homogeneous spaces.