In this talk, we will discuss smooth random dynamical systems and group actions on surfaces. Random dynamical systems, especially understanding stationary measures, can play an important role for understanding a group action. For instance, when a group action on torus is given by toral automorphisms, using random dynamics, Benoist-Quint classified all orbit closures under a mild assumption. We will study group actions on surfaces by diffeomorphisms, using random dynamics. We will discuss absolute continuity of stationary measures, classification of orbit closure, and exact dimensionality of stationary measures. This talk will be mostly about the ongoing joint work with Aaron Brown, Davi Obata, and Yuping Ruan.

Kazhdan and Lusztig identified the affine Hecke algebra with equivariant K-theory of Steinberg variety of the dual group. Bezrukavnikov categorified this identification, and in particular, gave a geometric description of the Kazhdan–Lusztig canonical basis. It is given by classes of simple perverse modules over the non-commutative resolution. We propose an analogous construction for a different situation: resolution of an affine Schubert variety in type A. We construct the non-commutative resolution and describe the corresponding basis in terms of the quantum loop group action. We emphasize a relation to categorical Howe duality and K-theoretic Satake.

Based on a work in progress, joint with E. Bodish and V. Krylov.