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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.

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.

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.

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 study of affine Deligne-Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne-Lusztig varieties. The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned.

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.

In this paper, we give explicit descriptions of the centers and cocenters of 0-Hecke algebras associated to finite Coxeter groups.

We study the intermediate extension of the character sheaves on an adjoint group to the semi-stable locus of its wonderful compactification. We show that the intermediate extension can be described by a direct image construction. As a consequence, we show that the ordinary restriction of a character sheaf on the compactification to a semi-stable stratum is a shift of semisimple perverse sheaf and is closely related to Lusztig's restriction functor (from a character sheaf on a reductive group to a direct sum of character sheaves on a Levi subgroup). We also provide a (conjectural) formula for the boundary values inside the semi-stable locus of an irreducible character of a finite group of Lie type, which gives a partial answer to a question of Springer (2006) [21]. This formula holds for Steinberg character and characters coming from generic character sheaves. In the end, we verify Lusztig's conjecture Lusztig (2004) [16

In this paper, we study the relation between the cocenter \overline {{ilde {\mathcal H}}} and the representations of an affine pro-\overline {{ilde {\mathcal H}}} Hecke algebra \overline {{ilde {\mathcal H}}} . As a consequence, we obtain a new criterion on supersingular representations: a (virtual) representation of \overline {{ilde {\mathcal H}}} is supersingular if and only if its character vanishes on the non-supersingular part of the cocenter \overline {{ilde {\mathcal H}}} .

Let X denote an equivariant embedding of a connected reductive group X over an algebraically closed field X . Let X denote a Borel subgroup of X and let X denote a X -orbit closure in X . When the characteristic of X is positive and X is projective we prove that X is globally X -regular. As a consequence, X is normal and Cohen-Macaulay for arbitrary X and arbitrary characteristics. Moreover, in characteristic zero it follows that X has rational singularities. This extends earlier results by the second author and M. Brion.