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

§ 0. Introduction

Given a complex semisimple Lie algebra ${𝔤}_{ℂ}$ of rank $l,$ there is a homogeneous polynomial $D$ (the Weyl polynomial) in $l$ variables with rational coefficients, which yields the dimensions of the irreducible ${𝔤}_{ℂ}\text{-modules}$ when the variables assume positive integer values (cf. the opening paragraph of § 2; it is well known that the irreducible ${𝔤}_{ℂ}\text{-modules}$ are parametrized, up to isomorphism, by $l\text{-tuples}$ of positive integers). In Part I (§ 2) we present a new conjecture on this polynomial which amounts to a weighted mean-value assertion when the variables undergo certain affine transformations. In § 1 we give preliminaries on the affine Weyl group, in terms of which it is easier and more natural to introduce these affine transformations denoted by $w_{\sigma }$ for $\sigma$ running in the Weyl group $W$ of ${𝔤}_{ℂ}\text{.}$ However, most of the discussion in § 1 is required only in Part II from § 5 onwards; thus, a reader who is familiar with standard facts on affine Weyl groups and is not interested in the representation-theoretic discussion in Part II may proceed directly to § 2 after noting the definition of $w_{\sigma }$ in § 1.3. Denoting by $D_{\sigma }$ the (inhomogeneous) polynomial obtained by transforming $D$ under $w_{\sigma },$ Conjecture I says that there should exist a unique integer linear combination $\sum _{\sigma }{b}_{\sigma }·D_{\sigma }$ which equals the constant polynomial 1. The linear independence of $\left\{D_{\sigma }\hspace{0.17em}\right|\hspace{0.17em}\sigma \in W\}$ means simply that these span the $\text{(}\left|W\right|\text{-dimensional)}$ space of $W\text{-harmonic}$ polynomials, and may be much easier to prove; that would force $b_{\sigma }$ to be rational numbers, but the main import of the Conjecture lies in the assertion that $b_{\sigma }$ must be integers. (See Footnote $\left(\stackrel{2}{}\right)$ to Conjecture I.)

We have verified the conjecture in four cases of low rank (§ 3) when ${𝔤}_{ℂ}$ is simple of type $A_2,$ $B_2,$ $C_2,$ or $A_3$ (the case of $A_1$ being trivial since $\left|W\right|=2$ and the two $D_{\sigma }\text{'s}$ are simply $x$ and $1-x\text{).}$ But we are unable to evolve any reasonable guess on the values of the "weighting function" $b_{\sigma }\text{.}$ As a matter of fact the values are extremely mysterious in the case of type $G_2\text{;}$ and though I knew of the values in case of types $A_2$ and $B_2$ for quite a while, the computation for types $G_2$ and $A_3$ could only be made possible through computer print-outs of the explicit expansions of all the $D_{\sigma }\text{'s.}$ Thus, it must be admitted that we have no "reason" to believe in this conjecture short of these verifications and the beauty of the related Conjecture IV in Part II (§ 7); see the argument in the second paragraph of § 7 deducing Conjecture I from Conjecture IV. (See Footnotes $\left(\stackrel{2}{}\right)$ and $\left(\stackrel{6}{}\right)$ to these two Conjectures.)

Part II of this paper concerns the representation theory of algebraic Chevalley groups defined over a field of characteristic $p\ne 0$ and their Lie algebras. The basic unsolved problem here is the computation of characters of the irreducible representations (which are known to be parametrized by $l\text{-tuples}$ of non-negative integers in exactly the same manner as in characteristic $0\text{).}$ Even the dimensions of these representations are not known. Our guess, however, is that the problem of giving explicitly a $p\text{-analogue}$ of the Weyl's dimension formula would turn out to be no simpler than that of obtaining an explicit $p\text{-analogue}$ of the Weyl's character formula for irreducible ${𝔤}_{ℂ}\text{-modules.}$ Though we may be far from providing a satisfactory solution to this problem, one can easily reduce the problem to that of determining a finite set of integers $c_{\lambda \mu }\text{;}$ this reduction is described in detail in § 4 (our exposition in § 4 and § 6.1—6.4 can be read independently of Part I, in contrast with the rest of Part II which uses § 1 heavily) since it appears here for the first time. The numbers $c_{\lambda \mu }$ describe the composition factors of certain indecomposable modules whose characters are given by Weyl's formula for irreducible ${𝔤}_{ℂ}\text{-modules.}$ Determination of these fundamental constants is a different (and indeed, a difficult) matter with which we do not concern ourselves here. Nevertheless, it suits us to present here two conjectures (in § 5) on these integers which are couched in terms of the affine Weyl group of § 1; we draw special attention of the reader to the third paragraph of § 5, where we have introduced a modification in the action of the affine Weyl group (denoted $\stackrel{\sim }{W}\prime \text{)}$ and related objects on the real vector space $E$ spanned by the root-system, this modification being dependent on the prime $p$ (we thought this to be more convenient than introducing from the start an action of $\stackrel{\sim }{W}\prime$ on $E$ dependent on a parameter $t,$ such that $t=1$ gives the action in § 1 and $t=p$ gives that for § 5 onwards). The contraption $\stackrel{\sim }{W}\prime$ enters the representation theory purely by virtue of a conjecture of this author dating back to 1966 (see the Footnote** in § 5), which has been called the "Harish-Chandra principle for characteristic $p\ne 0\text{",}$ and which has been proved to be true by Humphreys for large $p\text{.}$

In § 6 we come to the representations of the Lie algebras associated to Chevalley groups. In this expository section the emphasis is neither on irreducible representations alone (which can all be lifted to the algebraic group uniquely) nor on all representations (which are hardly understood at present), but is geared specifically towards the results of Humphreys on the principal indecomposable modules (PIM's, in short) of the restricted enveloping algebra $\mathbf{u}$ of the Lie algebra. (The role of these PIM's in the representation theory of Chevalley groups has recently assumed added significance; while it is being shown in [Ver1972-3] that these can be lifted, perhaps uniquely, to the algebraic group, in [HVe1973] way is paved for interpreting the PIM's of the $p\text{-modular}$ group-ring of a finite Chevalley group in terms of the PIM's of $\mathbf{u}\text{.)}$ Thus, the highlight of § 6 is the theorem of Humphreys [Hum1971] which gives a certain decomposition of the PIM's $Q_{\lambda }$ of $\mathbf{u}$ (and hence the dimensions of the $Q_{\lambda }\text{)}$ in terms of the decomposition of the "induced modules" $Z_{\lambda }$ which are the analogues for $\mathbf{u}$ of the infinite-dimensional induced ${𝔤}_{ℂ}\text{-modules}$ studied initially in [Ver1966] (and later by the school of Gelfand [Gel1971]); here the "decomposition" of $Q_{\lambda }$ stands for that in the Grothendieck-ring of the semisimple category of projective modules for a "Borel subalgebra" of $\mathbf{u},$ whereas that of $Z_{\lambda }$ refers to the determination of the multiplicities of its composition factors as a $\mathbf{u}\text{-module,}$ i.e. to the decomposition of $Z_{\lambda }$ in the Grothendieck-ring of all $\mathbf{u}\text{-modules.}$ It is this decomposition of $Z_{\lambda }$ (given by certain numbers $d_{\lambda }\text{)}$ that is the subject of the main conjecture in § 7. For concluding remarks on the behaviour of the numbers $c_{\lambda \mu }$ and $d_{\mu }$ when $p$ varies, we refer the reader to Epilogue (§ 8), particularly the opening words there.

For purely technical reasons, we shall assume ${𝔤}_{ℂ}$ (and hence the algebraic Chevalley group of type ${𝔤}_{ℂ}\text{)}$ to be simple, i.e. the root-system of ${𝔤}_{ℂ}$ is assumed indecomposable.

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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