Admissible subsets and Littelmann paths in affine Kazhdan-Lusztig theory

The center of an extended affine Hecke algebra is known to be isomorphic to the ring of symmetric functions associated to the underlying finite Weyl group $W\_0$. The set of Weyl characters {\sf s}\_\la forms a basis of the center and Lusztig showed in [Lus15] that these characters act as translations on the Kazhdan-Lusztig basis element $C\_{w\_0}$ where $w\_0$ is the longest element of $W\_0$, that is we have C\_{w\_0}{\sf s}\_\la =C\_{w\_0t\_\la}. As a consequence, the coefficients that appear when decomposing~C\_{w\_0t\_{\la}}{\sf s}\_\tau in the Kazhdan-Lusztig basis are tensor multiplicities of the Lie algebra with Weyl group $W\_0$. The aim of this paper is to explain how admissible subsets and Littelmann paths, which are models to compute such multiplicities, naturally appear when working out this decomposition

Kazhdan-Lusztig cells in the affine Weyl groups of rank 2

In this paper we determine the partition into Kazhdan-Lusztig cells of the affine Weyl groups of type \tB_{2} and \tG_{2} for any choice of parameters. Using these partitions we show that the semicontinuity conjecture of Bonnaf\'e holds for these groups.Comment: 21 pages, 4 tables, 13 figures. Section 3 has been completely rewritten. The new version also contains some minor correction

On the lowest two-sided cell in affine Weyl groups

Bremke and Xi determined the lowest two-sided cell for affine Weyl groups with unequal parameters and showed that it consists of at most |W_{0}| left cells where W_{0} is the associated finite Weyl group. We prove that this bound is exact. Previously, this was known in the equal parameter case and when the parameters were coming from a graph automorphism. Our argument uniformly works for any choice of parameters.Comment: 18 pages, 1 figure, final version (minor changes). To appear in Representation theor

On the determination of Kazhdan-Lusztig cells in affine Weyl groups with unequal parameters

Let W be a Coxeter group and L be a weight function on W. Following Lusztig, we have a corresponding decomposition of W into left cells, which have important applications in representation theory. We study the case where $W$ is an affine Weyl group of type $\tilde{G_{2}}$. Using explicit computation with \textsf{CHEVIE}, we show that (1) there are only finitely many possible decompositions into left cells and (2) the number of left cells is finite in each case, thus confirming some of Lusztig's conjectures in this case. For the proof, we show some equalities on the Kazhdan-Lusztig polynomials which hold for any affine Weyl groups.Comment: 22 pages, 2 figures, the revised version contains additional reference and some rewritting. Submitte

Some computations about Kazhdan-Lusztig cells in affine Weyl groups of rank 2

In the last section of the paper "Generalized induction of Kazhdan-Lusztig cells" and in "Kazhdan-Lusztig cells in affine Weyl groups of rank 2" the author described the partition into Kazhdan-Lusztig cells of the affine Weyl groups of rank 2 for all choices of parameters. The proof of these results relies on some explicit computations with GAP. In these notes we give some details of these computations.Comment: 90 pages. The new version contains some new data about the paper "Kazhdan-Lusztig cells in affine Weyl groups of rank 2". The content of the old version (old title: Computations in "Generalized induction of Kazhdan-Lusztig cells") is now in Section

Generalized induction of Kazhdan-Lusztig cells

International audienceFollowing Lusztig, we consider a Coxeter group W together with a weight function. Geck showed that the Kazhdan-Lusztig cells of W are compatible with parabolic subgroups. In this paper, we generalize this argument to some subsets of W which may not be parabolic subgroups. We obtain two applications: we show that under specific technical conditions on the parameters, the cells of a certain finite parabolic subgroup of W are cells in the whole group, and we decompose the affine Weyl group G~2 into left and two-sided cells for a whole class of weight functions

Cellules de Kazhdan-Lusztig dans les groupes de Weyl affines à paramètres inégaux

Hecke algebras arise naturally in the representation theory of reductive groups over finite or p-adic fields. These algebras are specializations of Iwahori-Hecke algebras which can be defined in terms of a Coxeter group and a weight function without reference to reductive groups and this is the setting we are working in. Kazhdan-Lusztig cells play a crucial role in the study of Iwahori-Hecke algebras. The aim of this work is to study the Kazhdan-Lusztig cells in affine Weyl groups with unequal parameters. More precisely, we show that the Kazhdan-Lusztig polynomials of an affine Weyl group are invariant under ``long enough'' translations, we decompose the lowest two-sided cell into left cells and we determine the decomposition of the affine Weyl group of type G into cells for a whole class of weight functions.Les algèbres de Hecke apparaissent naturellement dans la théorie des représentations des groupes réductifs sur des corps finis ou p-adiques. Ces algèbres sont des spécialisations des algèbres de Iwahori-Hecke qui peuvent être définies de manière combinatoire à partir d'un groupe de Coxeter et d'une fonction de poids sans faire référence à la théorie des groupes réductifs. C'est ce point de vue qui est adopté dans ce travail. Les cellules de Kazhdan-Lusztig jouent un rôle fondamental dans l'étude des algèbres de Iwahori-Hecke. Le but de ce travail est d'étudier les cellules de Kazhdan-Lusztig dans les groupes de Weyl affines à paramètres inégaux. Les principaux résultats de cette thèse sont l'invariance des polynômes de Kazhdan-Lusztig par translation, la décomposition de la cellule bilatère minimale en cellules gauches et la décomposition du groupe de Weyl affine de type G en cellules pour toute une classe de fonctions de poids

Balanced representations, the asymptotic Plancherel formula, and Lusztig's conjectures for C2

We prove Lusztig's conjectures P1–P15 for the affine Weyl group of type˜ C2 for all choices of positive weight function. Our approach to computing Lusztig's a-function is based on the notion of a " balanced system of cell representations ". Once this system is established roughly half of the conjectures P1–P15 follow. Next we establish an " asymptotic Plancherel Theorem " for type C2, from which the remaining conjectures follow. Combined with existing results in the literature this completes the proof of Lusztig's conjectures for all rank 1 and 2 affine Weyl groups for all choices of parameters