Let G be a connected almost simple algebraic group with a Dynkin automorphism . Let G be the connected almost simple algebraic group associated with G and . We prove that the dimension of the tensor invariant space of G is equal to the trace of on the corresponding tensor invariant space of G. We prove that if G has the saturation property then so does G . As a consequence, we show that the spin group Spin (2 n+ 1) has saturation factor 2, which strengthens the results of BelkaleKumar [1] and Sam [28] in the case of type B n.

The unipotent variety of a reductive algebraic group G plays an important role in the representation theory. In this paper, we will consider the closure U of the unipotent variety in the De ConciniProcesi compactification G of a connected simple algebraic group G. We will prove that U-U is a union of some G-stable pieces introduced by Lusztig in [Moscow Math. J 4 (2004) 869896]. This was first conjectured by Lusztig. We will also give an explicit description of U. It turns out that similar results hold for the closure of any Steinberg fiber in G.

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 characterize the indecomposable injective objects in the category of finitely presented representations of an interval finite quiver.

We give a generalization of the classical tilting theorem of Brenner and Butler. We show that for a 2-term silting complex P in the bounded homotopy category K^b(proj A) of finitely generated projective modules of a finite dimensional algebra A, the algebra B=End_{K^b(proj A)}(P) admits a 2-term silting complex Q with the following properties: (i) The endomorphism algebra of Q in K^b(proj B) is a factor algebra of A, and (ii) there are induced torsion pairs in mod A and mod B, such that we obtain natural equivalences induced by Hom- and Ext-functors. Moreover, we show how the Auslander–Reiten theory of mod B can be described in terms of the Auslander–Reiten theory of mod A.