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

Arun Ram

Department of Mathematics and Statistics

University of Melbourne

Parkville, VIC 3010 Australia

aram@unimelb.edu.au

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).

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