Higher rank (𝑞,𝑡)-Catalan polynomials, affine Springer fibers, and a finite rational shuffle theorem

Abstract

We introduce the higher rank ( q , t ) (q,t) -Catalan polynomials and prove they equal truncations of the Hikita polynomial to a finite number of variables. Using affine compositions and a certain standardization map, we define a d i n v \mathtt {dinv} statistic on rank r r semistandard ( m , n ) (m,n) -parking functions and prove c o d i n v \mathtt {codinv} counts the dimension of an affine space in an affine paving of a parabolic affine Springer fiber. Combining these results, we give a finite analogue of the Rational Shuffle Theorem in the context of double affine Hecke algebras. Lastly, we also give a Bizley-type formula for the higher rank Catalan numbers in the non-coprime case.