Lusztig Theory Research Articles

Combinatorial descriptions of the crystal structure on certain PBW bases

Lusztig's theory of PBW bases gives a way to realize the crystal B(∞) for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations, one for each reduced expression for the longest element of the Weyl group. There is an algorithm to calculate the actions of the crystal operators, but it can be quite complicated. For ADE types, we give conditions on the reduced expression which ensure that the corresponding crystal operators are given by simple combinatorial bracketing rules. We then give at least one reduced expression satisfying our conditions in every type except E8, and discuss the resulting combinatorics. Finally, we describe the relationship with more standard tableaux combinatorics in types A and D.

An asymptotic cell category for cyclic groups

In his theory of unipotent characters of finite groups of Lie type, Lusztig constructed modular categories from two-sided cells in Weyl groups. Broué, Malle and Michel have extended parts of Lusztig's theory to complex reflection groups. This includes generalizations of the corresponding fusion algebras, although the presence of negative structure constants prevents them from arising from modular categories. We give here the first construction of braided pivotal monoidal categories associated with non-real reflection groups (later reinterpreted by Lacabanne as super modular categories). They are associated with cyclic groups, and their fusion algebras are those constructed by Malle.

On the values of unipotent characters of finite Chevalley groups of type E6 in characteristic 3

Let G be a finite Chevalley group. We are concerned with computing the values of the unipotent characters of G by making use of Lusztig's theory of character sheaves. In this framework, one has to find the transformation between several bases for the class functions on G. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. We shall determine these scalars in the specific case where G is one of the groups E6(q), E62(q), and q is a power of the bad prime p=3 for E6, by exploiting known facts about the representation theory of the Hecke algebra associated with G.

Complements of Coxeter group quotients

We consider the complement $$W\setminus W^J$$W\WJ of any quotient $$W^J$$WJ of a Coxeter system $$(W,S)$$(W,S) and we investigate its algebraic, combinatorial and geometric properties, emphasizing its connection with parabolic Kazhdan---Lusztig theory. In particular, we define two families of polynomials which are the analogues, for the poset $$W\setminus W^J$$W\WJ, of the parabolic Kazhdan---Lusztig and $$R$$R-polynomials. These polynomials, indexed by elements of $$W\setminus W^J$$W\WJ, have interesting connections with the ordinary Kazhdan---Lusztig and $$R$$R-polynomials.

On Mirabolic -modules

Let an algebraic group G act on X, a connected algebraic manifold, with finitely many orbits. For any Harish-Chandra pair (D,G) where D is a sheaf of twisted differential operators on X, we form a left ideal D.g in D generated by the Lie algebra g, of G. Then, D/D.g is a holonomic D-module and its restriction to a unique Zariski open dense G-orbit in X is a G-equivariant local system. We prove a criterion saying that the D-module D/D.g is isomorphic, under certain (quite restrictive) conditions, to a direct image of that local system to X. We apply this criterion in the special case where the group G=SL(n) acts diagonally on X = F \times F \times P^{n-1}, a triple product where F is the flag manifold for SL(n) and P^{n-1} is the projective space. We further relate D-modules on F \times F \times P^{n-1} to D-modules on the space SL(n) \times P^{n-1} via a pair, CH, HC, of adjoint functors, analogous to those used in Lusztig's theory of character sheaves. A second important result of the paper provides an explicit description of these functors showing that the functor HC gives an exact functor on the abelian category of mirabolic D-modules.

On the unipotent support of character sheaves

Let $G$ be a connected reductive group over $\mathbb{F}_{q}$, where $q$ is large enough and the center of $G$ is connected. We are concerned with Lusztig's theory of character sheaves, a geometric version of the classical character theory of the finite group $G(\mathbb{F}_{q})$. We show that under a certain technical condition, the restriction of a character sheaf to its unipotent support (as defined by Lusztig) is either zero or an irreducible local system. As an application, the generalized Gelfand-Graev characters are shown to form a $\mathbb{Z}$-basis of the $\mathbb{Z}$-module of unipotently supported virtual characters of $G(\mathbb{F}_{q})$ (Kawanaka's conjecture).

Fourier transforms and Frobenius eigenvalues for finite Coxeter groups

Lusztig's classification of the unipotent characters of a finite Chevalley or Steinberg group involves a certain non-abelian Fourier transformation. We construct analogous transformations for the Suzuki and Ree groups, based on a set of axioms derived from Lusztig's theory of character sheaves. We also determine Fourier matrices for the “spetses” (in the sense of Broué, Michel, and the second author) associated with twisted dihedral groups. This completes the determination of Fourier matrices for all “spetses” associated with finite Coxeter groups. We end by collecting common properties of these Fourier matrices and the eigenvalues of Frobenius of character sheaves and unipotent characters.