Lattice Points and Rational $q$-Catalan Numbers

For each pair of coprime integers $a$ and $b$ we have a rational $q$-Catalan number $\operatorname{Cat}(a,b)_q=\binom{a+b}{a}_q/[a+b]_q$. It is known that this is a polynomial in $q$ with nonnegative integer coefficients, but the nature of these coefficients is still mysterious. Our current understanding is based on the rational shuffle conjecture that was conjectured by Bergeron, Garsia, Leven and Xin in 2014 and proved by Mellit in 2016, based on earlier work with Carlsson. This theorem realizes $\operatorname{Cat}(a,b)_q$ as the generating function for the statistic "area $-\ \mathrm{dinv}+\frac{(a-1)(b-1)}{2}$" defined on rational Dyck paths. However, this statistic is difficult to work with and leaves some phenomena unexplained. For example, it does not prove the conjecture that the difference $\operatorname{Cat}(a,c)_q-\operatorname{Cat}(a,b)_q$ has nonnegative coefficients whenever $\gcd(a,b)=\gcd(a,c)=1$ and $b<c$. The current paper proposes to look at lattice points instead of Dyck paths. Our idea is to fix $a$ and express everything in terms of the weight lattice $\mathrm{L}$ and root lattice $\mathrm{R}$ of type $A_{a-1}$. Based on ideas of Paul Johnson, we conjecture the existence of certain "Johnson statistics" $J:\mathrm{R}\to\mathbb{Z}$ and we prove this conjecture for $a\le 20$. We show that these statistics satisfy many remarkable properties including a $q$-analogue of Brion's theorem for simplices.