We reduce some key calculations of compositions of morphisms between Soergel bimodules (" Soergel calculus") to calculations in the nil Hecke ring (" Schubert calculus"). This formula has several applications in modular representation theory.

The cocenter of an affine Hecke algebra plays an important role in the study of representations of the affine Hecke algebra and the geometry of affine DeligneLusztig varieties (see for example, He and Nie in Compos Math 150(11):19031927, 2014; He in Ann Math 179:367404, 2014; Ciubotaru and He in Cocenter and representations of affine Hecke algebras, 2014). In this paper, we give a BernsteinLusztig type presentation of the cocenter. We also obtain a comparison theorem between the class polynomials of the affine Hecke algebra and those of its parabolic subalgebras, which is an algebraic analog of the HodgeNewton decomposition theorem for affine DeligneLusztig varieties. As a consequence, we present a new proof of the emptiness pattern of affine DeligneLusztig varieties (Grtz et al. in Compos Math 146(5):13391382, 2010; Grtz et al. in Ann Sci cole Norm Sup, 2012).

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.

There are two changes in comparison to the original version: Part (1). The union must be split up according to Jb/(Jb Ker (G)). This is also important for the description of the closure relations between strata: An analogous correction needs to be made in Prop. 7.2. 2. See Section 3 below.

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.