arXiv subjects
Publications and source records attributed to G. Lusztig.
In a previous paper we have defined a second basis of the Grothendieck group of a split reductive group over a finite field. In this paper we extend this to the case of nonsplit special orthogonal groups.
The set of strata of a reductive group can be viewed as an enlargement of the set of unipotent classes. In this paper the notion of distinguished unipotent class is extended to this larger set. The strata of a Weyl group are introduced and studied.
In a recent paper I defined a new basis for the Grothendieck group of unipotent representations of an almost simple Chevalley group over a finite field. The definition for classical types was different from that for exceptional types. In this paper we reconcile the two definitions.
The theory of total positivity for reductive groups is here extended to the case of symmetric spaces.
Let w be an element of the Weyl group of a reductive group G defined and split over a finite field. We consider the variety of triples (g,B,B') where g is a unipotent element of G and B, B' are Borel subgroups of G such that B contains g and B',gB'g^{-1} are in relative position w. We show that the number of rational points of this variety can be expressed in terms of a trace on the Iwahori-Hecke algebra. We also show that this variety is smooth, irreducible, if w is elliptic, of minimal length in its conjugacy class.
The special representations of a Weyl group can be regarded as the vertices of a graph with an involution i such that any edge e has the following property: either e or i(e) joins two vertices whose a-functions differ by 1.
We point out similarities between two partitions of a flag manifold with pieces indexed by Weyl group elements. One partition is defined using the action of a Frobenius map, the other partition is defined using conjugation by a regular semisimple element.
Consider a nilpotent element e in a simple complex Lie algebra. The Springer fibre corresponding to e admits a discretization (discrete analogue) introduced by the author in 1999. In this paper we propose a conjectural description of that discretization which is more amenable to computation. We also propose a conjectural PBW analogue of that discretization. This version differs substantially from the previous one.
In a 1983 paper the author has established a (decategorified) Satake equivalence for affine Hecke algebras. In this paper we give new proofs for some results of that paper, one based on the theory of J-rings and one based on the known character formula for rational representations of a reductive group in positive, large, characteristic. We also give an extension of that character formula to disconnected groups.
Let G be a connected reductive group over a finite field F_q. A map from irreducible representations of G(F_q) to semisimple classes in the dual group was defined by a cohomological method by Deligne and the author in 1976. Here we show that when q is large enough this map has an alternative (non-cohomological) definition.
Let G be a simple reductive group over the complex numbers. Let W be the Weyl group of G. We propose a description of the Springer representations of W associated to various unipotent classes of G by a purely algebraic method involving the families of various reflection subgroups of W and which is suitable to computer calculations.
We show that various properties of unipotent elements in a reductive group over the complex numbers can be recovered purely in terms of the affine Weyl group of the dual group.
Let $D$ be a connected component of a possibly disconnected reductive group $G$ over an algebraic closed field. We define a partition of $D$ into finitely many Strata each of which is a union of $G^0$-conjugacy classes of fixed dimension. In the case where $D=G^0$ this recovers a known partition.
Let W be a Weyl group. We introduce the notion of positive conjugacy class in W. This generalizes the notion of regular elliptic conjugacy class in the sense of Springer.
In this paper we discuss some of Springer's work on unipotent elements in a reductive groups and on representations of Weyl groups. Among the topics considered are Springer's bijection from the unipotent variety to the nilpotent variety, Springer's example of nonrationality of certain representations of the Hecke algebras associated to a Weyl group and the Springer representation of the Weyl group associated to a unipotent element.
We state four open problems on Iwahori-Hecke algebras. The first one states a relation between some algebras appearing in Solleveld's work and some explicit Hecke algebras appearing in the study of unipotent representations. The second one is a boundedness conjecture. The third one is a conjecture of well-definedness of an asymptotic Hecke algebra. The fourth one concerns positive conjugacy classes in a Weyl group.
Let G(F_q) be the group of rational points of a simple algebraic group defined and split over a finite field F_q. In this paper we define a new basis for the Grothendieck group of unipotent representations of G(F_q).
We discuss the non-negative monoid attached to a Cartan matrix following my 1994 paper.