Last update: 2 April 2014
2.1. The polynomial referred to in the title of Part I is the following function on where as usual. Because of homogeneity one could replace by both in the numerator and the denominator. But the form given here emphasizes the fact that if we consider as a polynomial in the coefficients in the expansion of in the fundamental weights, then the denominator is an integer and the numerator is a product of linear forms in (called root-forms) having non-negative integer coefficients; these forms can be easily written down in any given case once the Dynkin diagram of is given. The value of is therefore for and that of is strictly positive; in fact, the latter number is an integer, being the dimension of the "irreducible with highest weight where is the simple Lie algebra over possessing as its root system. (It should be noted that when the "coordinates" of are non-negative integers, i.e. those of are strictly positive integers, and conversely; that justifies the parenthetical statement in the opening sentence of the Introduction.)
The polynomial is determined by and the choice of positive roots therein. But a different choice of (equivalently, of the fundamental roots) affects only by a multiplicative factor of This is because all choices of are conjugate under and is skew (or alternating) with respect to in the sense that for As a matter of fact, it is known that every skew-invariant polynomial on is of the form where. is There is another property that distinguishes viz. that up to scalar multiples it is the unique homogeneous polynomial of maximal degree. A polynomial on is called if it is annihilated by all homogeneous differential operators of positive degree. (We consider only operators with constant coefficients. The algebra of such operators is canonically isomorphic to the symmetric algebra of for one denotes the corresponding differential operator by The action of on the algebra of polynomial functions on which is simply the algebra is obtained by requiring that for the value of at be equal to the inner-product where is the translated polynomial and is the canonical pairing between and extending that between and When we consider a polynomial function on as a formal polynomial in the coordinates of we can identify the first-order homogeneous differential operator in the above notation with the formal partial derivation in the standard sense.)
The fundamental fact about polynomials (cf. Steinberg [Ste1964]) is that the space of all such polynomials has dimension and is spanned by and all its (partial) derivatives. This generalizes, in fact, to the situation when is an arbitrary finite pseudoreflection group (in the accepted Bourbaki usage), which includes the noncrystallographic non-Weyl) finite Coxeter groups; only the definition of has to be slightly modified to take into account the orders of the generating pseudoreflections which can be greater than 2.
2.2. We can now state our conjecture on in terms of the affine transformations (for introduced in § 1; we remark that though depends only on this conjecture depends on the extension (as isomorphic Weyl groups can give rise to distinct extensions), and does not make sense for even the noncrystallographic finite Coxeter groups.
Conjecture I. There exist unique scalars such that these "weightings" are necessarily integers. (See
Let us write and note that is a harmonic polynomial for all In fact for arbitrary the translate of a harmonic polynomial is again harmonic, as is clear from Taylor's theorem and the fact that the derivatives of are harmonic. It follows that the first half of the conjecture simply asserts the linear independence of the set of inhomogeneous polynomials. Since all the belong to the of and its successive derivatives with respect to for since this is a of the space of harmonic polynomials, it is clear that all are rational numbers. Thus the crux of the conjecture lies in the assertion However, it should be emphasized that our main interest is not in the existence and uniqueness of but in finding an explicit method of computing their values. Unfortunately, even the examination of several low rank cases in the next section sheds little light on this problem.
2.3. Our "inhomogeneous weighted mean-value conjecture" above has a known homogeneous analogue which asserts This is in fact true for an arbitrary finite linear group with an arbitrary polynomial, and is quite easy to prove: Since the translation operator sending to is a differential operator (Taylor's theorem), with constant term in it equal to 1 (corresponding to the fact that the highest degree homogeneous term in the expansion of is itself), the operator has zero constant term, and hence annihilates every harmonic polynomial, giving (2.2). This argument is due to Steinberg.
It is conceivable that an appropriate generalization of (2.1) would render it easier to prove. Venturing in one possible direction of generalization, we may ask if the following is true: where are as in the conjecture. We have verified this only for the case of type apart from the trivial case of type
2.4. Note that there is exactly one value of viz. such that has a nonzero constant term, i.e. does not vanish for this is clear from the fact that if and only if is Therefore, for (2.1) specializes to which gives
Let us also observe that the weighting function on is constant on each right-coset with respect to the subgroup introduced in § 1, i.e. for and This is clear from the fact that for in the interior of one has Thus the weighting function is determined by its restriction to the set of distinguished coset representatives for and (2.1) becomes
(Added in Proof.) In the original (preprint) version of this paper it was hoped that would be necessarily non-negative (as is the case for rank 2 and for type cf. § 3). But shortly afterwards (in Autumn 1972) S. G. Hulsurkar found methods for hand-computation of the numbers (more efficient than our brutal technique in § 3), and carrying it through for types and discovered that in both these cases some of the are negative. Later, refining on his methods, he proved this conjecture in a form stronger than stated (and required) by us, by showing that the set is a of the "lattice" of those harmonic polynomials that take integer values on the weight-lattice The mystery shrouding the values of has also disappeared, as it turns out that where is the integer matrix inverse to the matrix the cruical step in Hulsurkar's work rests in the discovery that the latter matrix is unimodular.
In a forthcoming paper I shall show that the integers arise naturally as the degrees of certain (unique) virtual given by (for notations see § 4) such that This formula refines Conjecture I on one hand (taking "trace" on both sides one gets (2.1)), and implies on the other hand the fact the ring is free as a module over its subring of it seems that this fundamental fact of "semisimple Lie Theory" remained unnoticed until topologists discovered it (it was proved by Harsh Pittie in 1970 who attributes it as a conjecture of Luke Hodgkin). See also Footnote In subsequent paper(s) I hope to show that these virtual enter even the complex representation theory of finite Chevalley groups in a fundamental manner.
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).