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.

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.

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.

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 study the nonnegative part \overline {G_ {&gt; 0}} of the De Concini-Procesi compactification of a semisimple algebraic group \overline {G_ {&gt; 0}} , as defined by Lusztig. Using positivity properties of the canonical basis and parametrization of flag varieties, we will give an explicit description of \overline {G_ {&gt; 0}} . This answers the question of Lusztig in Total positivity and canonical bases, Algebraic groups and Lie groups (ed. GI Lehrer), Cambridge Univ. Press, 1997, pp. 281-295. We will also prove that \overline {G_ {&gt; 0}} has a cell decomposition which was conjectured by Lusztig.