Parabolic-equivariant modules over polynomial rings in infinitely many variables

We study the category of P-equivariant modules over the infinite variable polynomial ring, where P denotes the subgroup of the infinite general linear group GL (C^) consisting of elements fixing a flag in C^ 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 FI.