Nazarov [Naz96] introduced an infinite dimensional algebra, which he called the affine Wenzl algebra, in his study of the Brauer algebras. In this paper we study certain "cyclotomic quotients" of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank $r^{n}(2n-1)!!$ (when $\Omega$ is $\mathbf{u}$-admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov-Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Wenzl algebra.

Let $G$ be a finite reflection group acting in a complex vector space $V = \mathbb{C}^{r}$, whose coordinate ring will be denoted by $S$. Any element $\gamma \in \mathrm{GL}(V)$ which normalises $G$ acts on the ring $S^{G}$ of $G$-invariants. We attach invariants of the coset $G\gamma$ to this action, and show that if $G'$ is a parabolic subgroup of $G$, also normalised by $\gamma$, the invariants attaching to $G'\gamma$ are essentially the same as those of $G\gamma$. Four applications are given. First, we give a generalisation of a result of Springer-Stembridge which relates the module structures of the coinvariant algebras of $G$ and $G'$ and secondly, we give a general criterion for an element of $G\gamma$ to be regular (in Springer's sense) in invariant-theoretic terms, and use it to prove that up to a central element, all reflection cosets contain a regular element. Third, we prove the existence in any well-generated group, of analogues of Coxeter elements of the real reflection groups. Finally, we apply the analysis to quotients of $G$ which are themselves reflection groups.

Let $\mathfrak{g}$ be a simple complex Lie algebra, we denote by $\widehat{\mathfrak{g}}$ the affine Kac-Moody algebra associated to the extended Dynkin diagram of $\mathfrak{g}$. Let $\Lambda_{0}$ be the fundamental weight of $\widehat{\mathfrak{g}}$ corresponding to the additional node of the extended Dynkin diagram. For a dominant integral $\mathfrak{g}$-coweight $\lambda^{\vee}$, the Demazure submodule $V_{-\lambda^{\vee}}(m\Lambda_{0})$ is a $\mathfrak{g}$-module. We provide a description of the $\mathfrak{g}$-module structure as a tensor product of "smaller" Demazure modules. More precisely, for any partition of $\lambda^{\vee} = \sum_{j} \lambda_{j}^{\vee}$ as a sum of dominant integral $\mathfrak{g}$-coweights, the Demazure module is (as $\mathfrak{g}$-module) isomorphic to $\bigotimes_{j} V_{-\lambda^{\vee}_{j}}(m\Lambda_{0})$. For the "smallest" case, $\lambda^{\vee} = \omega^{\vee}$ a fundamental coweight, we provide for $\mathfrak{g}$ of classical type a decomposition of $V_{-\omega^{\vee}}(m\Lambda_{0})$ into irreducible $\mathfrak{g}$-modules, so this can be viewed as a natural generalization of the decomposition formulas in [13] and [16]. A comparison with the $U_{q}(\mathfrak{g})$-characters of certain finite dimensional $U_{q}'(\widehat{\mathfrak{g}})$-modules (Kirillov-Reshetikhin-modules) suggests furthermore that all quantized Demazure modules $V_{-\lambda^{\vee}, q}(m\Lambda_{0})$ can be naturally endowed with the structure of a $U_{q}'(\widehat{\mathfrak{g}})$-module. We prove, in the classical case (and for a lot of non-classical cases), a conjecture by Kashiwara [10], that the "smallest" Demazure modules are, when viewed as $\mathfrak{g}$-modules, isomorphic to some KR-modules. For an integral dominant $\widehat{\mathfrak{g}}$-weight $\Lambda$ let $V(\Lambda)$ be the corresponding irreducible $\widehat{\mathfrak{g}}$-representation. Using the tensor product decomposition for Demazure modules, we give a description of the $\mathfrak{g}$-module structure of $V(\Lambda)$ as a semi-infinite tensor product of finite dimensional $\mathfrak{g}$-modules. The case of twisted affine Kac-Moody algebras can be treated in the same way, some details are worked out in the last section.

In this paper, we study Lusztig's ${\mathbf{a}}$-function for a Coxeter group with unequal parameters. We determine that function explicitly in the "asymptotic case" in type $B_{n}$, where the left cells have been determined in terms of a generalized Robinson-Schensted correspondence by Bonnafé and the second author. As a consequence, we can show that all of Lusztig's conjectural properties (P1)--(P15) hold in this case, except possibly (P9), (P10) and (P15). Our methods rely on the "leading matrix coefficients" introduced by the first author. We also interprete the ideal structure defined by the two-sided cells in the associated Iwahori-Hecke algebra ${\mathcal{H}}_{n}$ in terms of the Dipper-James-Murphy basis of ${\mathcal{H}}_{n}$.

We give a geometric construction of the Verma modules of a symmetric Kac-Moody Lie algebra $\mathfrak{g}$ in terms of constructible functions on the varieties of nilpotent finite-dimensional modules of the corresponding preprojective algebra $\Lambda$.

An algebraic group of type $E_{7}$ over an algebraically closed field has an irreducible representation in a vector space of dimension $56$ and is, in fact, the identity component of the automorphism group of a quartic form on the space. This paper describes the construction of the quartic form if the characteristic is $\neq 2, 3$, taking into account a field of definition $F$. Certain $F$-forms of $E_{7}$ appear in the automorphism groups of quartic forms over $F$, as well as forms of $E_{6}$. Many of the results of the paper are known, but are perhaps not easily accessible in the literature.

According to Kazhdan-Lusztig and Ginzburg, the Hecke algebra of an affine Weyl group is identified with the equivariant $K$-group of Steinberg's triple variety. The $K$-group is equipped with a filtration indexed by closed $G$-stable subvarieties of the nilpotent variety, where $G$ is the corresponding reductive algebraic group over $\mathbb{C}$. In this paper we will show in the case of type $A$ that the filtration is compatible with the Kazhdan-Lusztig basis of the Hecke algebra.

Let $\mathbb{F}_{q}$ be a finite field, $G = GL_{n}(\mathbb{F}_{q})$, $\theta$ be the outer automorphism of $G$, suitably normalized. Consider the non-connected group $G \rtimes \{1, \theta\}$ and its connected component $\tilde{G} = G\theta$. We know two ways to produce functions on $\tilde{G}$, with complex values and invariant by conjugation by $G$: on one hand, let $\pi$ be an irreducible representation of $G$ we can and do extend to a representation $\pi^{+}$ of $G \rtimes \{1, \theta\}$, then the restriction $\operatorname{\it trace}_{\tilde{G}}\pi^{+}$ to $\tilde{G}$ of the character of $\pi^{+}$ is such a function; on the other hand, Lusztig define character-sheaves ${\bf a}$, whose characteristic functions $\phi({\bf a})$ are such functions too. We consider only "quadratic-unipotent" representations. For all such representation $\pi$, we define a suitable extension $\pi^{+}$, a character-sheave ${\mathfrak f}(\pi)$ and we prove an identity $\operatorname{\it trace}_{\tilde{G}}\pi^{+} = \gamma(\pi)\phi(\mathfrak{f}(\pi))$ with an explicit complex number $\gamma(\pi)$.