Parabolic-equivariant modules over polynomial rings in infinitely many variables

Abstract We study the category of $$\textbf{P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> -equivariant modules over the infinite variable polynomial ring, where $$\textbf{P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> denotes the subgroup of the infinite general linear group $$\textbf{GL}(\textbf{C}^\infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>GL</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> consisting of elements fixing a flag in $$\textbf{C}^\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> 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 $$\textbf{FI}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>FI</mml:mi> </mml:math> .