Δ–Springer varieties and Hall–Littlewood polynomials

Abstract

The $\Delta $-Springer varieties are a generalization of Springer fibers introduced by Levinson, Woo and the author that have connections to the Delta Conjecture from algebraic combinatorics. We prove a positive Hall–Littlewood expansion formula for the graded Frobenius characteristic of the cohomology ring of a $\Delta $-Springer variety. We do this by interpreting the Frobenius characteristic in terms of counting points over a finite field $\mathbb {F}_q$ and partitioning the $\Delta $-Springer variety into copies of Springer fibers crossed with affine spaces. As a special case, our proof method gives a geometric meaning to a formula of Haglund, Rhoades and Shimozono for the Hall–Littlewood expansion of the symmetric function in the Delta Conjecture at $t=0$.