We will use the combinatorics of the G-stable pieces to describe the closure relation of the partition of partial flag varieties in [L3, section 4].

In this paper, we give a uniform construction of irreducible genuine characters of the Pin cover W of a Weyl group W, and put them into the context of theory of Springer representations. In the process, we provide a direct connection between Springer theory, via Green polynomials, the irreducible representations of W, and an extended Dirac operator for graded Hecke algebras. We also introduce a q-elliptic pairing for W with respect to the reflection representation V. These constructions are of independent interest. The q-elliptic pairing is a generalization of the elliptic pairing of W introduced by Reeder, and it is also related to S. Kato's notion of (graded) Kostka systems for the semidirect product A W= C [W] S (V).

Suppose G is a reductive algebraic group, T is a Cartan subgroup of G, N= Norm (T), and W= N/T is the Weyl group. If w W has order d, it is natural to ask about the orders lifts of w to N. It is straightforward to see that the minimal order of a lift of w has order d or 2d, but it can be a subtle question which holds. We first consider the question of when W itself lifts to a subgroup of N (in which case every element of W lifts to an element of N of the same order). We then consider two natural classes of elements: regular and elliptic. In the latter case all lifts of w are conjugate, and therefore have the same order. We also consider the twisted case.

We give a definition of character sheaves on the group compactification which is equivalent to Lusztig's definition in [G. Lusztig, Parabolic character sheaves, II, Mosc. Math. J. 4 (4) (2004) 869896]. We also prove some properties of the character sheaves on the group compactification.

Soit G un groupe algbrique complexe quasi-simple et G/P une varit de drapeaux partiels. La projection sur G/P des varits de Richardson (de la varit des drapeaux complets) forment une stratification de G/P. Nous montrons que les relations dadhrence des varits de Richardson projetes correspondent celles dun certain sous-ensemble de varits de Schubert sur la varit de drapeaux affine de G. Nous comparons aussi les classes de cohomologie quivariante et de K-thorie de ces deux stratifications. Notre travail gnralise celui de Knutson, Lam et Speyer pour la grassmannienne de type A.