Nous étudions le problème de l’irréductibilité du produit tensoriel de deux représentations irréductibles d’un groupe fondamental $G={\pi }_1\left(X\right)$, quand $X$ est le complémentaire d’hypersurfaces dans un espace projectif. Nous mettons en place un formalisme adapté et utilisons une approche par monodromie pour définir une classe de représentations irréductibles de $G$ dont les produits tensoriels restent irréductibles pour des valeurs génériques de paramètres de définition. Ceci est appliqué au groupe de tresses pures...

An affine Hecke algebras can be realized as an equivariant $K$-group of the corresponding Steinberg variety. This gives rise naturally to some two-sided ideals of the affine Hecke algebra by means of the closures of nilpotent orbits of the corresponding Lie algebra. In this paper we will show that the two-sided ideals are in fact the two-sided ideals of the affine Hecke algebra defined through two-sided cells of the corresponding affine Weyl group after the two-sided ideals are tensored by $\mathbb{Q}$. This...

In this paper we explicitly determine the Macdonald formula for spherical functions on any locally finite, regular and affine Bruhat-Tits building, by constructing the finite difference equations that must be satisfied and explaining how they arise, by only using the geometric properties of the building.

We define an operator α on ℂ³ ⊗ ℂ³ associated with the quantum group $U_q\left(2\right)$, which satisfies the Yang-Baxter equation and a cubic equation (α² - 1)(α + q²) = 0. This operator can be extended to a family of operators $h_j:=I_j\otimes \alpha \otimes I_{n-2-j}$ on ${\left(ℂ³\right)}^{\otimes n}$ with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra ${\mathscr{H}}_{q,n}\left(2\right)$ associated with the quantum group $U_q\left(2\right)$. The purpose of this note is to present the construction.

We study equivalences for category ${𝒪}_p$ of the rational Cherednik algebras ${𝐇}_p$ of type $G_{\ell }\left(n\right)={\left({\mu }_{\ell }\right)}^n⋊{𝔖}_n$: a highest weight equivalence between ${𝒪}_p$ and ${𝒪}_{\sigma \left(p\right)}$ for $\sigma \in {𝔖}_{\ell }$ and an action of ${𝔖}_{\ell }$ on an explicit non-empty Zariski open set of parameters $p$; a derived equivalence between ${𝒪}_p$ and ${𝒪}_{p^{\text{'}}}$ whenever $p$ and $p^{\text{'}}$ have integral difference; a highest weight equivalence between ${𝒪}_p$ and a parabolic category $𝒪$ for the general linear group, under a non-rationality assumption on the parameter $p$. As a consequence, we confirm special cases of conjectures...

* The authors thank the “Swiss National Science Foundation” for its support.We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are weakly maximal, and that the corresponding quasi-regular representations are irreducible. These (infinite-dimensional) representations are approximated by finite-dimensional quasi-regular representations. The Hecke algebras...

Soient $F$ un corps $p$-adique, $G={\mathrm{GL}}_3\left(F\right)$. Pour $\chi$ un caractère de l’algèbre de Hecke sphérique de $G$ sur un anneau commutatif $k$, on introduit à la suite de Serre une représentation lisse $M_{\chi }$ de $G$ sur $k$ qui gouverne la théorie des représentations non ramifiées de $G$ sur $k$. Nous prouvons que $M_{\chi }$ est plat sur $k$ et que si $p$ est inversible dans $k$, alors pour tout sous-groupe compact ouvert suffisament petit $U$ de $G$, le module $M_{\chi }^U$ est libre de rang fini sur $k$. Ceci était conjecturé par Lazarus. Comme corollaire, nous obtenons...

Nous définissons et entamons l’étude d’analogues infinitésimaux des quotients principaux (algèbres de Temperley-Lieb, Hecke, Birman-Wenzl-Murakami) de l’algèbre de groupe du groupe d’Artin $B_n$. Ce sont des algèbres de Hopf qui correspondent à des groupes réductifs, et permettent de donner un cadre général aux représentations dérivées des représentations classiques de $B_n$. Nous décomposons complètement l’algèbre de Temperley-Lieb infinitésimale, et en déduisons plusieurs résultats d’irréductibilité.

For a symmetric cellular algebra, we study properties of the dual basis of a cellular basis first. Then a nilpotent ideal is constructed. The ideal connects the radicals of cell modules with the radical of the algebra. It also yields some information on the dimensions of simple modules. As a by-product, we obtain some equivalent conditions for a finite-dimensional symmetric cellular algebra to be semisimple.

We consider a class of two-dimensional non-commutative Cohen-Macaulay rings to which a Brauer graph, that is, a finite graph endowed with a cyclic ordering of edges at any vertex, can be associated in a natural way. Some orders Λ over a two-dimensional regular local ring are of this type. They arise, e.g., as certain blocks of Hecke algebras over the completion of $\mathbb{Z}[q,q^{-1}]$ at (p,q-1) for some rational prime $p$. For such orders Λ, a class of indecomposable maximal Cohen-Macaulay modules (see introduction)...

On donne une condition nécessaire et suffisante pour l’existence de modules de dimension finie sur l’algèbre de Cherednik rationnelle associée à un système de racines.

From the combinatorial characterizations of the right, left, and two-sided Kazhdan-Lusztig cells of the symmetric group, “ RSK bases” are constructed for certain quotients by two-sided ideals of the group ring and the Hecke algebra. Applications to invariant theory, over various base rings, of the general linear group and representation theory, both ordinary and modular, of the symmetric group are discussed.

We define the singular Hecke algebra $\mathscr{H}\left(S{B}_n\right)$ as the quotient of the singular braid monoid algebra $ℂ\left(q\right)\left[S{B}_n\right]$ by the Hecke relations ${\sigma }_k^2=(q-1){\sigma }_k+q$, $1\le k\le n-1$. We define the notion of Markov trace in this context, fixing the number $d$ of singular points, and we prove that a Markov trace determines an invariant on the links with $d$ singular points which satisfies some skein relation. Let ${\mathrm{TR}}_d$ denote the set of Markov traces with $d$ singular points. This is a $ℂ(q,z)$-vector space. Our main result is that ${\mathrm{TR}}_d$ is of dimension $d+1$. This result is completed...