## Verifying Kottwitz’ conjecture by computer

HTML articles powered by AMS MathViewer

- by Bill Casselman
- Represent. Theory
**4**(2000), 32-45 - DOI: https://doi.org/10.1090/S1088-4165-00-00052-2
- Published electronically: February 1, 2000

## Abstract:

In these notes I will discuss the computations that were used to verify the main conjecture of Kottwitz (1997) for the groups $E_{6}$, $E_{7}$, $E_{8}$, and the subsidiary one for $F_{4}$ and $E_{6}$. At the end I will include tables of the relevant computer output. I begin by recalling briefly what is to be computed.## References

- Dean Alvis,
*The left cells of the Coxeter group of type $H_4$*, J. Algebra**107**(1987), no. 1, 160–168. MR**883878**, DOI 10.1016/0021-8693(87)90082-2 - Dean Alvis and George Lusztig,
*On Springer’s correspondence for simple groups of type $E_{n}$ $(n=6,\,7,\,8)$*, Math. Proc. Cambridge Philos. Soc.**92**(1982), no. 1, 65–78. With an appendix by N. Spaltenstein. MR**662961**, DOI 10.1017/S0305004100059703 - N. Bourbaki,
*Groupes et algebres de Lie*, vols. IV-VI, Hermann, Paris, 1970. - Brigitte Brink and Robert B. Howlett,
*A finiteness property and an automatic structure for Coxeter groups*, Math. Ann.**296**(1993), no. 1, 179–190. MR**1213378**, DOI 10.1007/BF01445101 - R. W. Carter,
*Conjugacy classes in the Weyl group*, Compositio Math.**25**(1972), 1–59. MR**318337** - Roger W. Carter,
*Finite groups of Lie type*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR**794307** - W. A. Casselman,
*Automata to perform basic calculations in Coxeter groups*, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp. 35–58. MR**1357194** - Fokko du Cloux,
*A transducer approach to Coxeter groups*, J. Symbolic Comput.**27**(1999), no. 3, 311–324. MR**1673603**, DOI 10.1006/jsco.1998.0254 - M. Geck, G. Hiss, F. Lübeck, G. Malle, and G. Pfeiffer,
*CHEVIE—A system for computing and processing generic character tables*, IWR Preprint 95-05, Universität Heidelberg (1995). Geck’s home page is at: http://www.desargues.univ-lyon1.fr/home/geck/index.html.fr and both documentation for CHEVIE and the software itself can be found at: http://www.math.rwth-aachen.de/~CHEVIE/. - R. Kottwitz,
*Involutions in Weyl groups*, Represent. Theory**4**(2000), 1–15. - George Lusztig,
*Characters of reductive groups over a finite field*, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR**742472**, DOI 10.1515/9781400881772 - G. Lusztig,
*Leading coefficients of character values of Hecke algebras*, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 235–262. MR**933415**, DOI 10.1090/pspum/047.2/933415 - E. M. Opdam,
*A remark on the irreducible characters and fake degrees of finite real reflection groups*, Invent. Math.**120**(1995), no. 3, 447–454. MR**1334480**, DOI 10.1007/BF01241138 - John Stembridge,
*Computational aspects of root systems, Coxeter groups, and Weyl characters*, RIMS, preprint #1216. Available at: http://www.math.lsa.umich.edu/~jrs/papers.html.

## Bibliographic Information

**Bill Casselman**- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC Canada V6T 1W5
- MR Author ID: 46050
- Email: cass@math.ubc.ca
- Received by editor(s): May 14, 1998
- Received by editor(s) in revised form: October 11, 1999
- Published electronically: February 1, 2000
- © Copyright 2000 by the author
- Journal: Represent. Theory
**4**(2000), 32-45 - MSC (2000): Primary 20G99
- DOI: https://doi.org/10.1090/S1088-4165-00-00052-2
- MathSciNet review: 1740179