Cocharge and skewing formulas for $\Delta$-Springer modules and the Delta Conjecture

Abstract

We prove that $\omega \Delta'_{e_{k}}e_n|_{t=0}$, the symmetric function in the Delta Conjecture at $t=0$, is a skewing operator applied to a Hall-Littlewood polynomial, and generalize this formula to the Frobenius series of all $\Delta$-Springer modules. We use this to give an explicit Schur expansion in terms of the Lascoux-Sch\"utzenberger cocharge statistic on a new combinatorial object that we call a \textit{battery-powered tableau}. Our proof is geometric, and shows that the $\Delta$-Springer varieties of Levinson, Woo, and the second author are generalized Springer fibers coming from the partial resolutions of the nilpotent cone due to Borho and MacPherson. We also give alternative combinatorial proofs of our Schur expansion for several special cases, and give conjectural skewing formulas for the $t$ and $t^2$ coefficients of $\omega \Delta'_{e_{k}}e_n$.