Abstract: This is from joint work in progress with Mina Aganagic, Jinghang Miao, Spencer Tamagni and Peng Zhou. McKay correspondence for simply laced Lie algebras gives two different realizations of ADE root systems via the ADE singularities. These singularties are algebraic Poisson varieties. Starting from the singularity, one can consider its semi-universal symplectic deformation and consider the monodromy action (Picard-Fuchs theory) on the middle-dimensional homology of smooth fibers. This can be categorified to a braid group action on the Fukaya category. On the other hand, one can also consider its symplectic resolution, which is also its minimal resolution. There is also a braid group action on the category of coherent sheaves on the resolution, generated by spherical twists around the exceptional divisors. In type A, we show that these two pictures are mirror to each other, summarizing the work of Seidel, Smith and Thomas. When the Lie algebra is gl(m|n), we give an analogous result. The corresponding singularity will be the threefold singularity xy = z^mw^n. Note that in this case there are many minimal resolutions, related to each other by local Atiyah flops. Correspondingly, gl(m|n) has many inequivalent Dynkin diagrams, organized into the Weyl groupoid. We’ll category this groupoid action to coherent sheaves on the minimal resolutions and describe their mirrors.