The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras
Part II
Representations and decompositions in characteristic p0

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

Last update: 3 April 2014

§ 7. Conjecture on the decomposition of induced modules Zλ

7.1. The following is independent of the conjectures in § 5.

Conjecture IV. Let λX* (i.e. λX and p-1(λ+δ)E*), and for σW let us denote (λ+δ)wσ-δ by λ(σ). Then for every νΛ which is W-linked (definition in § 6.3) to the image of λ in Λ (under the canonical projection XΛ) one must have Zνu σW bσ·Vλ(σ). (7.1) It should be noted that when λ+δ is not W-regular (in particular, when it is not W-regular, i.e. when λXJ* for J nonempty), there may be several values of σ which give rise to same λ(σ), and that some of the values of λ(σ)+δ could even be W-singular (since the image of X*+δ under wσWW+ lies only in X+ and not always in X++δ) in which case we use the convention of treating Vλ(σ) as the zero-module in consonance with the formula dimVλ(σ)=D(λ(σ)+δ). (See Footnote6.)

Note that the equality of the dimensions of the left and right-hand sides of (7.1) is already asserted by Conjecture I; one sees this by recalling the modification in the action of W on X that has been introduced in Part II (third paragraph of § 5.1) under which (2.1) becomes σWbσ·D (p-1(λ+δ)wσ) =1, whence σWbσ· D(λ(σ)+δ) =pr=dimZν, as D() is homogeneous of degree r. As a matter of fact, the validity of Conjecture IV for infinitely many values of p implies that of Conjecture I itself, for if (2.1) holds for λE*p-1X for the given values of p then it must hold for all λ, as the union of E*p-1X over those values is (topologically dense and hence) Zariski-dense.

Substituting Vλ(σ)=μXcλ(σ)μMμ in (7.1), we have by (6.2) the following equivalent formulation of (7.1): dμ=μY0 (σWbσ·cλ(σ),μ+pμ) dimMμ (7.2) for all μX that are W-linked to λX*, where we write dμ for dπ(μ) (μX) in keeping with the similar notational abuses earlier. (Once again we recall that for λ,μ in X, the infinitesimal weights π(λ) and π(μ) are W-linked if and only if λ and μ are W-linked.)

The following statement for dμ is analogous to the statement in Conjecture III for the fundamental constant cλμ.

Conjecture III' (Corollary to Conjectures III and TV). Let J{0,1,,l}, ν1,ν2XJ*, and wΩ. W={wσ|σW} be all given, and write μi=(νi+δ)w-δ for 1i2. Then for p[X:X] one must have dμ1=dμ2.

Due to (7.2), to prove this corollary it would suffice to show that for λXJ*, wWJ++, μ=(λ+δ)w-δ, the sum σWbσ·cλ(σ)μ depends only on J and w (and not on λ). But this is immediate from Conjecture III which gives cλ(σ)μ=cwσwJ in the notation introduced towards the end of § 5. The role of the hypothesis p[X:X] is to guarantee that (for i=1,2) vi is the unique element of XJ* which is W-linked to μi and satisfies νiμi (mod X); cf. the fourth paragraph of § 5.

7.2. In this paragraph we briefly comment on the modification one needs to introduce in order to remove the assumption p[X:X] in the preceding. For λX*, let Ωλ denote the stabilizer of λ in Ω (defined in § 1), and for an element μ of X which is W-linked to λ define the "virtual Humphreys number" dμ to be dμ·|Ωλ+δ|-1. With this definition, we might strengthen Conjecture III' above, to assert that the value of dμ for μ=(λ+δ)w-δ, λXJ*, depends only on J and w. This would have a very pleasant repercussion on the formula in § 6.4 for the dimension of Qμ=Qπ(μ) for μX. Setting aμ=|W|· |Wμ+δ|-1 where Wμ+δ is the stabilizer of μ+δ in W, one finds aμ=|W|· |Wμ+δ|-1 =aμ· |Ωλ+δ|-1 where λ is any element of X* that is W-linked to μ; and one has dimQμ=aμ dμpr. Since the value of aμ for μ=(λ+δ)w-δ, λXJ*, depends only on J and w (the order of the stabilizer of μ+δ or λ+δ in W being simply the order of the Coxeter group WJ generated by {Rj|jJ}), it follows (from the preceding unlabelled extension of Conjecture III') that the dimension of Qμ depends only on J and w, even when p divides [X:X]. We shall prove this for the case of type A2 in a later paper, calling the preceding (suggested) phenomenon, the "ramification" phenomenon (term suggested by Humphreys) for the constants aμ and dμ, the number |Ωλ+δ| being the measure of this ramification.

7.3. To get ready for the concluding remarks in § 8, let us make the following notation: If μX is W-linked to λXJ* by μ=(λ+δ)w-δ for wWJ, then assuming the truth of Conjecture III', we write dwJ in place of dμ. Since an element ω of Ω sends XJ* to some XJ*, (J being precisely the image of J under the permutation θj of § 1.2, in case ω=ωj for jJ0), it follows from Conjecture III' that dwJ=dwJ, where w=ω-1wωWJ.

6 (Added in Proof.) I have been able to prove Conjecture IV rather recently by obtaining an explicit formula for the formal character of the G-module Qμ for μ of the form μ=(p-1)δ-λ*p·E* (* denoting the Dynkin outer automorphism) with λXJ*+δ (J{0,1,,l}) which guarantees |W|· Zλ-δu |Wj|· QμG σW V(p-1)δ+λσ, where the second equivalence was known for a while. For years I was unable to figure out exactly how to replace (in the Grothendieck ring of G-modules) an arbitrary Weyl module (e.g. one of the |W| terms in the last sum) in terms of Weyl modules whose highest weights are restricted (i.e. lie in X), though this was evident from the intuition gained by various computations (this means that a suitable analogue of Steinberg's tensor-product theorem holds with Weyl modules in place of irreducible modules). My first breakthrough came on seeing the recursive argument needed to justify this; though explicit formulae were lacking, it was enough to yield the existence of virtual 𝔤-modules with formal characters Bσ(·eX)W (independent of λ as well as p) such that |WJ|·char Qμ=σW (Bσ)Fr ·charVλσ+pδσ-δ which gives Zλ-δuσW(|W|-1·degBσ)·Vλ(σ)-δ where λ(σ)=λwσ=(λ-δ)(σ)+δ. From this Conjectures IV and I follow, except for the integrality of bσ=|W|-1·degBσ. (Here "deg" means the virtual degree, and is the same as "trace" used in formula (6.2)). The next breakthrough came only a few weeks later, when I realized that the ideas at the end of Footnote2 can be used for a quick proof of the preceding as well as a closed expression for Bσ in terms of the virtual T-modules βσ (explicitly, if βσ is written as sλ·eλ, one has Bσ=sλ·(τWeλτ), in consonance with |W|-1·degBσ=degβσ=bσ). The details will appear in due course.

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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