Parabolic-equivariant modules over polynomial rings in infinitely many variables

We study the category of $\mathbf{P}$-equivariant modules over the infinite variable polynomial ring, where $\mathbf{P}$ denotes the subgroup of the infinite general linear group $\mathbf{G}\mathbf{L}({\mathbf{C}}^{\mathrm{\infty }})$ consisting of elements fixing a flag in ${\mathbf{C}}^{\mathrm{\infty }}$ with each graded piece infinite-dimensional. We decompose the category into simpler pieces that can be described combinatorially, and prove a number of finiteness results, such as finite generation of local cohomology and rationality of Hilbert series. Furthermore, we show that this category is equivalent to the category of representations of a particular combinatorial category generalizing $\mathbf{F}\mathbf{I}$.