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.)

Let G be a connected semi-simple algebraic group of adjoint type over an algebraically closed field and let ar G be the wonderful compactification of G. For a fixed pair (B,B  ) of opposite Borel subgroups of G, we look at intersections of Lusztigs G-stable pieces and the B  B-orbits in ar G , as well as intersections of BB-orbits and B  B  -orbits in ar G . We give explicit conditions for such intersections to be nonempty, and in each case, we show that every nonempty intersection is smooth and irreducible, that the closure of the intersection is equal to the intersection of the closures, and that the nonempty intersections form a strongly admissible partition of .

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.

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).

In this paper, we discuss some partitions of affine flag varieties. These partitions include as special cases the partition of affine flag variety into affine Deligne-Lusztig varieties and the affine analogue of the partition of flag varieties into $\cb_w (b) $ introduced by Lusztig in\cite {L1} as part of the definition of character sheaves.