Soit G un groupe réductif défini sur un corps algébriquement clos de caractéristique positive. Nous montrons que le foncteur contraction par Frobenius de la catégorie des G-modules est adjoint à droite de celui de tensorisation deux fois par le module de Steinberg du tordu de Frobenius du G-module de départ. Il s’ensuit en particulier que le foncteur de contraction par Frobenius préserve le caractère injectif et l’existence de bonne filtration mais pas la semi-simplicité. (For a reductive group G defined over an algebraically closed field of positive characteristic, we show that the Frobenius contraction functor of $Gs-modules is right adjoint to the Frobenius twist of the modules tensored with the Steinberg module twice. It follows that the Frobenius contraction functor preserves injectivity, good filtrations, but not semi-simplicity.)

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.

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.

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).

Let <i>H</i> denote a spherical subgroup within a semisimple algebraic group <i>G</i>. In this paper we study the closures of the finitely many <i>H</i>-orbits in the flag variety of <i>G</i>. Using the language of Frobenius splitting we provide a criterion for these closures to have nice geometric and cohomological properties. We then show how the criterion applies to the spherical subgroups of minimal rank studied by N. Ressayre. Finally, we also provide applications of the criterion to orbit closures which are not multiplicity-free in the sense defined by M. Brion.