Deletion formulas for equivariant Kazhdan-Lusztig polynomials of matroids

We study equivariant Kazhdan--Lusztig (KL) and Z-polynomials of matroids. We formulate an equivariant generalization of a result by Braden and Vysogorets that relates the equivariant KL and Z-polynomials of a matroid with those of a single-element deletion. We also discuss the failure of equivariant -positivity for the Z-polynomial. As an application of our main result, we obtain a formula for the equivariant KL polynomial of the graphic matroid gotten by gluing two cycles. Furthermore, we compute the equivariant KL polynomials of all matroids of corank~2 via valuations. This provides an application of the machinery of Elias, Miyata, Proudfoot, and Vecchi to corank 2 matroids, and it extends results of Ferroni and Schr\"oter.