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

Let A be an abelian category and B be the Happel-Reiten-Smalo tilt of A with respect to a torsion pair. We give necessary and sufficient conditions for the existence of a derived equivalence between A and B, which is compatible with the inclusion of B into the derived category of A. As applications, we obtain new derived equivalences related to splitting torsion pairs, TTF-triples and two-term silting subcategories, respectively. We prove that for the realization functor of any bounded t-structure, its denseness implies its fully-faithfulness.

In this paper, we give explicit descriptions of the centers and cocenters of 0-Hecke algebras associated to finite Coxeter groups.

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 find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra g^, from g-module homomorphisms. When g=sl_2, these results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to g^, and we identify obstructions to the construction arising from the possible non-existence of series solutions having a prescribed form.