We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface S with marked points and non-empty boundary, which generalizes Brüstle–Zhang’s result for the puncture free case. As an application, we show that the intersection of the shifts in the 3-Calabi–Yau derived category D(Γ_S) associated to the surface and the corresponding Seidel–Thomas braid group of D(Γ_S) is empty, unless S is a polygon with at most one puncture (i.e. of type A or D).

We give a block decomposition of the dg category of character sheaves on a simple and simply-connected complex reductive group G, similar to the one in generalized Springer correspondence. As a corollary, we identify the category of character sheaves on G as the category of quasi-coherent sheaves on an explicitly defined derived stack G^.

We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's <i>G</i>-stable pieces and the generalization of <i>G</i>-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in [M. Geck, S. Kim, G. Pfeiffer, Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. Algebra 229 (2) (2000) 570600].