  Lewis, Reiner, and Stanton conjectured a Hilbert seriesfor a space of
invariants under an action of finite general linear groups using
$(q,t)$-binomial coefficients. This work gives an analog in positive
characteristic of theorems relating various Catalan numbers to the
representation theory of rational Cherednik algebras. They consider a finite
general linear group as a reflection group acting on the quotient of a
polynomial ring by iterated powers of the irrelevant ideal under the Frobenius
map. We prove a variant of their conjecture in the local case, when the group
acting fixes a reflecting hyperplane.
