Igusa zeta functions and flag Hilbert--Poincaré series

We define a class of multivariate rational functions associated with hyperplane arrangements called flag Hilbert--Poincaré series, and we show these rational functions are closely related to Igusa local zeta functions and other generating functions. We report on a general self-reciprocity result and explore other connections within algebraic combinatorics via Hilbert series of Stanley--Reisner rings. This is joint work with Christopher Voll.