$\Delta$-Springer varieties and Hall-Littlewood polynomials

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 $t=0$.