Let $G$ be a reductive algebraic group over a field of positive characteristic. In this paper we explore the relations between the behaviour of tilting modules for $G$ and certain Kazhdan-Lusztig cells for the affine Weyl group associated with $G$. In the corresponding quantum case at a complex root of unity V. Ostrik has shown that the weight cells defined in terms of tilting modules coincide with right Kazhdan-Lusztig cells. Our method consists in comparing our modules for $G$ with quantized modules for which we can appeal to Ostrik's results. We show that the minimal Kazhdan-Lusztig cell breaks up into infinitely many "modular cells" which in turn are determined by bigger cells. At the opposite end we call attention to recent results by T. Rasmussen on tilting modules corresponding to the cell next to the maximal one. Our techniques also allow us to make comparisons with the mixed quantum case where the quantum parameter is a root of unity in a field of positive characteristic.

Recently, there has been progress in determining the representation type of the Hecke algebras of finite Weyl groups. We report on these results.

This note is devoted to Lusztig's bijection between unipotent conjugacy classes in a simple complex algebraic group and 2-sided cells in the affine Weyl group of the Langlands dual group; and also to the description of the reductive quotient of the centralizer of the unipotent element in terms of convolution of perverse sheaves on affine flag variety of the dual group conjectured by Lusztig in [L4]. Our main tool is a recent construction by Gaitsgory (based on an idea of Beilinson and Kottwitz), the so-called sheaf-theoretic construction of the center of an affine Hecke algebra (see [Ga]). We show how this remarkable construction provides a geometric interpretation of the bijection, and allows to prove the conjecture.

We prove a weak version of Lusztig's Conjecture on explicit description of the asymptotic affine Hecke algebras in terms of convolution algebras.

We discuss a circle of ideas for addressing problems in representation theory using the philosophy of cellular algebras, applied to algebras described in terms of diagrams. Cellular algebras are often generically semisimple, and have non-semisimple specialisations whose representation theory may be discussed by solving problems in linear algebra, which are formulated in the semisimple context, and are therefore tractable in some significant cases. This applies in particular to certain "Temperley-Lieb" quotients of Hecke algebras, both finite dimensional and affine, which may be described in terms of bases consisting of diagrams. This leads to the application of cellular algebra theory to an analysis of their representation theory, with corresponding consequences for the relevant Hecke algebras. A particular case is the determination of the decomposition numbers of some standard modules for the affine Hecke algebra of $GL_n$. These decomposition numbers are known (by Kazhdan-Lusztig) to be expressible in terms of the dimensions of the stalks of certain intersection cohomology sheaves, and we discuss how our results imply the rational smoothness of some varieties associated with quiver representations.

We prove that the action of an $\omega$-root operator on the set of all paths fixed by a diagram automorphism $\omega$ of a Kac–Moody algebra $\mathfrak{g}$ can be identified with the action of a root operator for the orbit Lie algebra $\breve{\mathfrak{g}}$. Moreover, we prove that there exists a canonical bijection between the elements of the crystal base $\mathcal{B}(\infty)$ for $\mathfrak{g}$ fixed by $\omega$ and the elements of the crystal base $\breve{\mathcal{B}}(\infty)$ for $\breve{\mathfrak{g}}$. Using this result, we give twining character formulas for the "negative part" of the quantized universal enveloping algebra $U_q(\mathfrak{g})$ and for certain modules of Demazure type.

Let $\widehat{\mathfrak{g}}$ be an affine Lie algebra, and let $\mathbf{U}_q (\widehat{\mathfrak{g}})$ be the quantum affine algebra introduced by Drinfeld and Jimbo. In [11] Kashiwara introduced a $\mathbf{U}_q (\widehat{\mathfrak{g}})$-module $V(\lambda)$, having a global crystal base for an integrable weight $\lambda$ of level 0. We call it an extremal weight module. It is isomorphic to the Weyl module introduced by Chari-Pressley [6]. In [12, §13] Kashiwara gave a conjecture on the structure of extremal weight modules. We prove his conjecture when $\widehat{\mathfrak{g}}$ is an untwisted affine Lie algebra of a simple Lie algebra $\mathfrak{g}$ of type $ADE$, using a result of Beck-Chari-Pressley [5]. As a by-product, we also show that the extremal weight module is isomorphic to a universal standard module, defined via quiver varieties by the author [16, 18]. This result was conjectured by Varagnolo-Vasserot [19] and Chari-Pressley [6] in a less precise form. Furthermore, we give a characterization of global crystal bases by an almost orthogonality propery, as in the case of global crystal base of highest weight modules.

Hall-Littlewood functions and Green functions associated to complex reflection groups $W = G(r, 1, n)$ were constructed in [S1] by means of symbols, which are a generalization of partitions. In this paper, we consider such functions in the case where the symbols are of very special type, called "limit symbols". The situation becomes simple, and is close to the case of symmetric groups when the symbols tends to the "limit". In the case where $W$ is a Weyl group of type $B_n$, we give a closed formula for Hall-Littlewood functions, and verify some of the conjectures stated in [S1] for the case of Green functions attached to limit symbols.

If $Y$ is an affine symmetric variety for the reductive group $G$ with Weyl group $W$, there exists by Lusztig and Vogan a representation of the Hecke algebra of $W$ in a module which has a basis indexed by the set $\Lambda$ of pairs $(v, \xi)$, where $v$ is an orbit in $Y$ of a Borel group $B$ and $\xi$ is a $B$-equivariant rank one local system on $v$. We introduce cells in $\Lambda$ and associate with a cell a two-sided cell in $W$.

In this paper we show that a conjecture of Lusztig on distinguished involutions is true for the affine Weyl group of type $\tilde{A}_n$.