## The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras

Last update: 1 April 2014

## Summary

This is the first of a series of papers on the rational representations of a connected semisimple affine algebraic group (in short, an algebraic Chevalley group) $G$ over a field of characteristic $p\ne 0,$ and the representations of the related modular Lie algebra as well as the $p\text{-modular}$ representations of the finite Chevalley (sub)groups. This paper has two parts, the first of which is purely elementary in that it deals only with root-systems. In this part is formulated a "weighted mean-value conjecture" on the famous dimension polynomial of H. Weyl, after giving preliminaries on affine Weyl groups. The object of Part II is two-fold: to give a quick account (§§ 4, 5 and 6) of some of the known facts in the representation theory of $G$ with special reference to an earlier conjecture of the author (which has been proved true, by J. E. Humphreys, when $p$ exceeds the Coxeter number) that defines the relevance of affine Weyl groups, and to put forward several new conjectures (§§ 5, 7 and 8) on certain decomposition numbers. The binding force of the two parts is provided by Conjecture IV in § 7. The Introduction (§ 0) describes the contents in greater detail, and the Epilogue (§ 8) points out a seemingly metamathematical aspect of our study when the characteristic $p$ is regarded as a variable.

Added in proof: Special attention must be drawn to the various Footnotes added in proof, where we remark on the recent developments.

* This is a revised and expanded version of part of the material presented at the Budapest Summer School on Group Representations, 1971, under the title "On rational representations of algebraic Chevalley groups in non-zero characteristics". The rest of the material will appear in a series of subsequent papers. The computations summarized in § 3, concerning the verification of Conjecture I for types ${G}_{2}$ and ${A}_{3},$ were done subsequent to the lectures, with the assistance of the Computer Section of the Tata Institute of Fundamental Research. See also the acknowledgement contained in the Footnote (**) in § 5.

## Notes and References

This is an excerpt of the paper The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras by Daya-Nand Verma. It appeared in Lie Groups and their Representations, ed. I.M.Gelfand, Halsted, New York, pp. 653–705, (1975).