  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 $\mathbf{GL}(\mathbf{C}^\infty)$ consisting of
elements fixing a flag in $\mathbf{C}^\infty$ 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
$\mathbf{FI}$.
