The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras
Part I
A conjecture on Weyl's dimension polynomial

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 2 April 2014

§ 2. Harmonic polynomials and Conjecture I

2.1. The polynomial referred to in the title of Part I is the following function on E: D(λ)= αΔ+λ,α αΔ+δ,α forλE, where δ=iλi=12αΔ+α, as usual. Because of homogeneity one could replace α by α both in the numerator and the denominator. But the form given here emphasizes the fact that if we consider D as a polynomial in the coefficients x1,x2,,xl in the expansion of λ in the fundamental weights, λ=ixiλi, then the denominator is an integer and the numerator is a product of linear forms in x1,x2,,xl (called root-forms) having non-negative integer coefficients; these forms λ,α can be easily written down in any given case once the Dynkin diagram of Δ is given. The value of D(λ) is therefore 0 for λX+, and that of D(λ+δ) is strictly positive; in fact, the latter number is an integer, being the dimension of the "irreducible 𝔤-module with highest weight λ", where 𝔤 is the simple Lie algebra over possessing Δ as its root system. (It should be noted that when the "coordinates" x1,x2,,xl of λ=ixiλi are non-negative integers, i.e. λX+, those of λ+δ are strictly positive integers, and conversely; that justifies the parenthetical statement in the opening sentence of the Introduction.)

The polynomial D(λ) is determined by Δ and the choice of positive roots therein. But a different choice of Δ+ (equivalently, of the fundamental roots) affects D(λ) only by a multiplicative factor of -1. This is because all choices of Δ+ are conjugate under W, and D is skew (or alternating) with respect to W in the sense that D(λσ)=detσ·D(λ) for σW. As a matter of fact, it is known that every skew-invariant polynomial on E is of the form D(λ)·P(λ) where. P is W-invariant. There is another property that distinguishes D, viz. that up to scalar multiples it is the unique W-harmonic homogeneous polynomial of maximal degree. A polynomial on E is called W-harmonic if it is annihilated by all W-invariant 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 S(E) of E; for uS(E) one denotes the corresponding differential operator by u. The action of u on the algebra of polynomial functions on E, which is simply the algebra S(E*), is obtained by requiring that for pS(E*) the value of up at xE be equal to the inner-product u,px where px is the translated polynomial px(λ)=p(λ+x) and , is the canonical pairing between S(E) and S(E*) extending that between E and E*. When we consider a polynomial function on E as a formal polynomial in the coordinates x1,x2,,xl of λ=ixiλi, we can identify the first-order homogeneous differential operator λi in the above notation with the formal partial derivation xi in the standard sense.)

The fundamental fact about W-harmonic polynomials (cf. Steinberg [Ste1964]) is that the space of all such polynomials has dimension |W| and is spanned by D and all its (partial) derivatives. This generalizes, in fact, to the situation when W is an arbitrary finite pseudoreflection group (in the accepted Bourbaki usage), which includes the noncrystallographic (= non-Weyl) finite Coxeter groups; only the definition of D 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 D in terms of the affine transformations wσW (for σW) introduced in § 1; we remark that though D depends only on W this conjecture depends on the extension W (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 bσ, σW, such that σW bσ·D (λwσ)=1 for all λE; (2.1) these "weightings" bσ are necessarily integers. (See Footnote2.)

Let us write Dσ(λ)=D(λwσ)=detσ·D(λ+(δσ)σ-1), and note that Dσ is a harmonic polynomial for all σ. In fact for arbitrary μE the translate Hμ of a harmonic polynomial H is again harmonic, as is clear from Taylor's theorem and the fact that the derivatives of H are harmonic. It follows that the first half of the conjecture simply asserts the linear independence of the set {Dσ|σW} of inhomogeneous polynomials. Since (δσ)σ-1X, all the Dσ belong to the -span of D and its successive derivatives with respect to λ for λX; since this -span is a -form of the space of harmonic polynomials, it is clear that all bσ are rational numbers. Thus the crux of the conjecture lies in the assertion bσ. However, it should be emphasized that our main interest is not in the existence and uniqueness of bσ's, 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 |W|-1 σW D(λ+μσ)= D(λ)for λ,μE. (2.2) This is in fact true for an arbitrary finite linear group W with D an arbitrary W-harmonic polynomial, and is quite easy to prove: Since the translation operator Tμ sending D(λ) to Dμ(λ)=D(λ+μ) 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 Dμ is D itself), the W-invariant operator (σWTμσ) -|W| 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: σWbσ ·D(λ+μwσ) is independent ofμE, (2.3) where bσ are as in the conjecture. We have verified this only for the case of type A2, apart from the trivial case of type A1.

2.4. Note that there is exactly one value of σW, viz. σ=σ0, such that Dσ(λ)=D(λwσ) has a nonzero constant term, i.e. does not vanish for λ=0; this is clear from the fact that Dσ(0)=D(δσ)0 if and only if δσ is W-regular. Therefore, for λ=0 (2.1) specializes to σWbσ·D(δσ)=1, which gives bσ0=1.

Let us also observe that the weighting function σbσ on W is constant on each right-coset with respect to the subgroup Ω0 introduced in § 1, i.e. bγσ=bσ for yΩ0 and σW. This is clear from the fact that for λ in the interior of E* one has σbγσDσ (λ)=σbσ Dγ-1σ(λ) =σbσD (λ(γ-1σ)) =σbσD ((λ(γ-1))(σ)) =1. Thus the weighting function is determined by its restriction to the set W of distinguished coset representatives for Ω0\W, and (2.1) becomes σWbσ ( jJ0D (λωjwσ) ) =1.

2 (Added in Proof.) In the original (preprint) version of this paper it was hoped that bσ's would be necessarily non-negative (as is the case for rank 2 and for type A3 cf. § 3). But shortly afterwards (in Autumn 1972) S. G. Hulsurkar found methods for hand-computation of the numbers bσ (more efficient than our brutal technique in § 3), and carrying it through for types B3 and C3 discovered that in both these cases some of the bσ's 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 {Dσ|σW} is a -basis of the "lattice" of those harmonic polynomials that take integer values on the weight-lattice X. The mystery shrouding the values of bσ has also disappeared, as it turns out that bσ=τbσ,τ where [bσ,τ] is the integer matrix inverse to the |W|×|W| matrix [D(-δσσ-1τ+δτ)]; 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 bσ arise naturally as the degrees of certain (unique) virtual T-modules given by βσ·eX (for notations see § 4) such that eλ=σch (λσ+δσ) ·βσfor all λX,wherech μ= detτ·eμτ detτ·eδτ 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 ·eX is free as a module over its subring of W-invariants; 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 6. In subsequent paper(s) I hope to show that these virtual T-modules βσ enter even the complex representation theory of finite Chevalley groups in a fundamental manner.

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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