no. 4
Generalized Induction of Kazhdan-Lusztig cells
[Induction généralisée des cellules de Kazhdan-Lusztig]
Guilhot, Jérémie1, 2
1 Aberdeen University Department of Mathematical Sciences King’s College Aberdeen AB24 3UE, Scotland (U.K.)
2 Université de Lyon 1 Institut Camille Jordan, CNRS UMR 5208 43 Boulevard du 11 Novembre 1918 69622 Villeurbanne Cedex (France)
Annales de l'Institut Fourier, Tome 59 (2009) no. 4, pp. 1385-1412.

Suivant Lusztig, nous considérons un groupe de Coxeter W avec une fonction de poids. Geck a montré que les cellules de Kazhdan-Lusztig sont compatibles avec les sous-groupes paraboliques. Dans cet article nous généralisons cet argument à des sous-ensembles de W qui ne sont pas forcément des sous-groupes paraboliques. Nous obtenons deux applications : nous montrons que sous certaines hypothèses sur les paramètres les cellules de certains sous-groupes paraboliques sont aussi des cellules de W et nous décomposons le groupe de Weyl affine de type G en cellules gauches et bilatères pour toute une classe de fonctions de poids.

Following 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 certain parabolic subgroups of W are cells in the whole group, and we decompose the affine Weyl group of type G into left and two-sided cells for a whole class of weight functions.

DOI : 10.5802/aif.2468
Classification : 20C08
Keywords: Coxeter groups, Affine Weyl groups, Hecke algebras, Kazhdan-Lusztig cells, Unequal parameters
Mot clés : groupes de Coxeter, Groupes de Weyl affines, Algèbre de Hecke, Cellules de Kazhdan-Lusztig, Paramètres inégaux

Guilhot, Jérémie 1, 2

1 Aberdeen University Department of Mathematical Sciences King’s College Aberdeen AB24 3UE, Scotland (U.K.)
2 Université de Lyon 1 Institut Camille Jordan, CNRS UMR 5208 43 Boulevard du 11 Novembre 1918 69622 Villeurbanne Cedex (France) 
@article{AIF_2009__59_4_1385_0,
     author = {Guilhot, J\'er\'emie},
     title = {Generalized {Induction} of {Kazhdan-Lusztig} cells},
     journal = {Annales de l'Institut Fourier},
     pages = {1385--1412},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {4},
     year = {2009},
     doi = {10.5802/aif.2468},
     mrnumber = {2566965},
     zbl = {1186.20004},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2468/}
}
TY  - JOUR
AU  - Guilhot, Jérémie
TI  - Generalized Induction of Kazhdan-Lusztig cells
JO  - Annales de l'Institut Fourier
PY  - 2009
SP  - 1385
EP  - 1412
VL  - 59
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2468/
DO  - 10.5802/aif.2468
LA  - en
ID  - AIF_2009__59_4_1385_0
ER  - 
%0 Journal Article
%A Guilhot, Jérémie
%T Generalized Induction of Kazhdan-Lusztig cells
%J Annales de l'Institut Fourier
%D 2009
%P 1385-1412
%V 59
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2468/
%R 10.5802/aif.2468
%G en
%F AIF_2009__59_4_1385_0
Guilhot, Jérémie. Generalized Induction of Kazhdan-Lusztig cells. Annales de l'Institut Fourier, Tome 59 (2009) no. 4, pp. 1385-1412. doi : 10.5802/aif.2468. https://aif.centre-mersenne.org/articles/10.5802/aif.2468/

[1] Bedard, R. Cells for two Coxeter groups, Comm. Algebra, Volume 14 (1986), pp. 1253-1286 | DOI | MR | Zbl

[2] Bremke, K. On generalized cells in affine Weyl groups, Journal of Algebra, Volume 191 (1997), pp. 149-173 | DOI | MR | Zbl

[3] Chen, C. The decomposition into left cells of the affine Weyl group of type D ˜ 4 , Journal of Algebra, Volume 163 (1994), pp. 692-728 | DOI | MR | Zbl

[4] Cloux, F. Du An abstract model for Bruhat intervals, European J. Combin., Volume 21 (2000), pp. 197-222 | DOI | MR | Zbl

[5] Du, J. The decomposition into cells of the affine Weyl group of type B ˜ 3 , Comm. Algebra, Volume 16 (1988), pp. 1383-1409 | DOI | MR | Zbl

[6] Geck, M. On the induction of Kazhdan-Lusztig cells, Bull. London Math. Soc., Volume 35 (2003) no. 5, pp. 608-614 | DOI | MR | Zbl

[7] Guilhot, J. On the determination of Kazhdan-Lusztig cells for affine Weyl group with unequal parameters, Journal of Algebra, Volume 318 (2007), pp. 893-917 | DOI | MR | Zbl

[8] Guilhot, J. Computations in Generalized induction of Kazhdan-Lusztig cells, available at http://arxiv.org/abs/0810.5165, 2008

[9] Guilhot, J. On the lowest two-sided cell in affine Weyl groups, Represent. Theory, Volume 12 (2008), pp. 327-345 | DOI | MR

[10] Kazhdan, D. A.; Lusztig, G. Schubert varieties and Poincaré duality, Proc. Sympos. Pure Math., Volume 36 (1980), pp. 185-203 (Amer. Math. Soc.) | MR | Zbl

[11] Lusztig, G. Hecke algebras and Jantzen’s generic decomposition patterns, Advances in Mathematics, Volume 37 (1980), pp. 121-164 | DOI | MR | Zbl

[12] Lusztig, G. Cells in affine Weyl groups, Advanced Studies in Pure Math., Volume 6 (1985), pp. 255-287 | MR | Zbl

[13] Lusztig, G. The two-sided cells of the affine Weyl group of type A ˜ n , Math. Sci. Res. Inst. Publ, Volume 4 (1985), pp. 275-283 | DOI | MR | Zbl

[14] Lusztig, G. Hecke algebras with unequal parameters, 18, CRM Monographs Ser., 2003 (Amer. Math. Soc. , Providence, RI) | MR | Zbl

[15] Schönert, Martin GAP – Groups, Algorithms, and Programming – version 3 release 4 patchlevel 4 (1997)

[16] Shi, J.-Y. The Kazhdan-Lusztig cells in certain affine Weyl groups, Lectures Notes in Math., 1179, Springer-Verlag, 1986 | MR | Zbl

[17] Shi, J.-Y. Left cells in affine Weyl group W a (D ˜ 4 ), Osaka J. Math., Volume 31 (1994), pp. 27-50 | MR | Zbl

[18] Shi, J.-Y. Left cells in affine Weyl groups, Tokohu Math. J., Volume 46 (1994), pp. 105-124 | DOI | MR | Zbl

[19] Xi, N. Representations of affine Hecke algebras, Lectures Notes in Math., 1587, Springer-Verlag, 1994 | MR | Zbl

Cité par Sources :