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).

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.

We study a class of perverse sheaves on some spherical varieties which include the strata of the De ConciniProcesi completion of a symmetric variety. This is a generalization of the theory of (parabolic) character sheaves.