Let G be a semisimple group over an algebraically closed field. Steinberg has associated to a Coxeter element G of minimal length G a subvariety G of G isomorphic to an affine space of dimension G which meets the regular unipotent class G in exactly one point. In this paper this is generalized to the case where G is replaced by any elliptic element in the Weyl group of minimal length G in its conjugacy class, G is replaced by a subvariety G of G isomorphic to an affine space of dimension G , and G is replaced by a unipotent class G of codimension G in such a way that the intersection of G and G is finite.

Let W a be an affine Weyl group and : W a W 0 be the natural projection to the corresponding finite Weyl group. We say that w W a has finite Coxeter part if (w) is conjugate to a Coxeter element of W 0. The elements with finite Coxeter part are a union of conjugacy classes of W a. We show that for each conjugacy class O of W a with finite Coxeter part there exists a unique maximal proper parabolic subgroup W J of W a, such that the set of minimal length elements in O is exactly the set of Coxeter elements in W J. Similar results hold for twisted conjugacy classes.

We prove that the DeligneLusztig variety associated to minimal length elements in any <i></i>-conjugacy class of the Weyl group is affine, which was conjectured by Orlik and Rapoport in [S. Orlik, M. Rapoport, DeligneLusztig varieties and period domains over finite fields, arXiv: 0705.1646].

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 .

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.)