Let <i>G</i> be a semisimple adjoint group over <b>C</b> and <i></i> be the De ConciniProcesi completion of <i>G</i>. In this paper, we define a Lagrangian subvariety of the cotangent bundle of <i></i> such that the singular support of any character sheaf on <i></i> is contained in .

Let G be a semisimple group over an algebraically closed field. Steinberg has associated to a Coxeter element G of minimal length G a subvariety G of G isomorphic to an affine space of dimension G which meets the regular unipotent class G in exactly one point. In this paper this is generalized to the case where G is replaced by any elliptic element in the Weyl group of minimal length G in its conjugacy class, G is replaced by a subvariety G of G isomorphic to an affine space of dimension G , and G is replaced by a unipotent class G of codimension G in such a way that the intersection of G and G is finite.

We give a new description of the set {m Adm}(\mu) of admissible alcoves as an intersection of certain``obtuse cones''of alcoves, and we show this description may be given by imposing conditions vertexwise. We use this to prove the vertexwise admissibility conjecture of Pappas-Rapoport-Smithling. The same idea gives simple proofs of two ingredients used in the proof of the Kottwitz-Rapoport conjecture on existence of crystals with additional structure.

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).

In this paper, we discuss some partitions of affine flag varieties. These partitions include as special cases the partition of affine flag variety into affine Deligne-Lusztig varieties and the affine analogue of the partition of flag varieties into $\cb_w (b) $ introduced by Lusztig in\cite {L1} as part of the definition of character sheaves.