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

Last update: 1 April 2014

## Contents

§ 0. Introduction Part I. A conjecture on Weyl's dimension polynomial § 1. Affine Weyl groups
§ 2. Harmonic polynomials and Conjecture I
§ 3. Evidence in low ranks Part II. Representations and decompositions in characteristic $p\ne 0$ § 4. Fundamental constants ${c}_{\lambda \mu }$ and the character formula
§ 5. The "Harish-Chandra principle" and two conjectures on ${c}_{\lambda \mu }$
§ 6. PIM's of the $u\text{-algebra}$ and Humphreys' numbers ${d}_{\lambda }$
§ 7. Conjecture on the decomposition of induced modules ${Z}_{\lambda }$
§ 8. Epilogue: A metamathematical (?) conjecture

## 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).