arXiv subjects
Publications and source records attributed to G. Lusztig.
We study the new basis of the (complexified) Grothendieck group of unipotent representations of a split reductive group over a finite field. For exceptional types we use a definition of the new basis which differs from the earlier one.
We define the new basis of the Grothendieck group of unipotent representations in an exceptional Weyl group, making more precise and correcting results in previous papers.
We define the totally positive part of the coordinate algebra associated to a Cartan datum of simply laced type. We show its relation to the canonical basis of the corresponding plus part of the enveloping algebra.
Let $V$ be an $F_2$-vector space with a given nondegenerate symplectic form and a circular basis. We define a partition of $V$ into subsets each of which has cardinal equal to a product of Catalan numbers. This can be viewed as a partition of the set of unipotent representations of a finite symplectic group corresponding to a certain two-sided cell.
This is essentially the content of a talk at the workshop on Character Sheaves on Loop Groups at the University of Minnesota (June 8,2026). It describes the early days of the theory and also some recent developments.
This document contains a description of several of my papers, including remarks on history and connection with subsequent work. It also contains some new results and conjectures.
We define a partition of a reductive group into finitely many subsets, refining the partition of the group into strata. We state some conjectural properties of these subsets (called substrata) and verify them in some examples.
A Weyl group W is a union of strata (certain subsets which are unions of conjugacy classes) which are the nonempty fibres of a map from W to the set of irreducible representations of W. We give an explicit description of strata in terms of a certain modified centralizer group. We also define superstrata of a connected reductive group and state a conjecture about them.
Let W be a Weyl group. We define a class of irreducible representations of W that we call antispecial. They are in bijection with the constructible representations of W. We define an oriented graph structure on the set of antispecial representations or equivalently on the set of constructible representations of W. We describe explicitly the antispecial representations in each case. Section 2 has been modified.
Let W be a Weyl group. We can define the notion of positivity of a W-module in terms of the corresponding module over the asymptotic Iwahori-Hecke algebra. We state a conjecture which says that certain explicit W-modules are positive and we prove it in the case where W is of classical type.
Consider a Chevalley group over a finite field $F_q$ such that the longest element of the Weyl group is central. We construct an involution $ξ\mapstoξ^!$ of the set of unipotent representations of this group such that the degree polynomial of a unipotent representation $ξ$ is obtained up to sign from the degree polynomial of $ξ^!$ by changing $q$ to $-q$.
Consider a Chevalley group over a finite field F_q such that the longest element in the Weyl group is central. In this paper we study the effect of changing q to -q in the polynomials which give the character values of unipotent representations of our group at semisimple elements.
In this (partly expository) paper we show, using ideas from the theory of total positivity, how a number of properties of a semisimple group over the complex numbers can be presented purely in terms of the Weyl group. We also describe some new connections of the theory of canonical bases with total positivity.
This represents a talk given at the International Conference for Basic Science, July 2025. We review the theory of canonical bases of quantum groups and its relation with the theory of total positivity.
The history of the canonical basis and crystal basis of a quantized enveloping algebra and its representations is presented
We define a subset of the set of special representations of a Weyl group. This subset contains at most one representation.
Let W be a Weyl group. In my 1984 book a group was attached to any special representation of W using the theory of Springer representations. In this paper we give a new definition of this group which is purely algebraic (no use of geometry is made).
We show that certain Iwahori-Hecke algebras with unequal parameters can be realized in the framework of parabolic character sheaves.