Last update: 3 April 2014
7.1. The following is independent of the conjectures in § 5.
Conjecture IV. Let (i.e. and and for let us denote by Then for every which is (definition in § 6.3) to the image of in (under the canonical projection one must have It should be noted that when is not (in particular, when it is not i.e. when for nonempty), there may be several values of which give rise to same and that some of the values of could even be (since the image of under lies only in and not always in in which case we use the convention of treating as the zero-module in consonance with the formula (See
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 on that has been introduced in Part II (third paragraph of § 5.1) under which (2.1) becomes whence as is homogeneous of degree As a matter of fact, the validity of Conjecture IV for infinitely many values of implies that of Conjecture I itself, for if (2.1) holds for for the given values of then it must hold for all as the union of over those values is (topologically dense and hence) Zariski-dense.
Substituting in (7.1), we have by (6.2) the following equivalent formulation of (7.1): for all that are to where we write for in keeping with the similar notational abuses earlier. (Once again we recall that for in the infinitesimal weights and are if and only if and are
The following statement for is analogous to the statement in Conjecture III for the fundamental constant
Conjecture III' (Corollary to Conjectures III and TV). Let and be all given, and write for Then for one must have
Due to (7.2), to prove this corollary it would suffice to show that for the sum depends only on and (and not on But this is immediate from Conjecture III which gives in the notation introduced towards the end of § 5. The role of the hypothesis is to guarantee that (for is the unique element of which is to and satisfies (mod 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 in the preceding. For let denote the stabilizer of in (defined in § 1), and for an element of which is to define the "virtual Humphreys number" to be With this definition, we might strengthen Conjecture III' above, to assert that the value of for depends only on and This would have a very pleasant repercussion on the formula in § 6.4 for the dimension of for Setting where is the stabilizer of in one finds where is any element of that is to and one has Since the value of for depends only on and (the order of the stabilizer of or in being simply the order of the Coxeter group generated by it follows (from the preceding unlabelled extension of Conjecture III') that the dimension of depends only on and even when divides We shall prove this for the case of type in a later paper, calling the preceding (suggested) phenomenon, the "ramification" phenomenon (term suggested by Humphreys) for the constants and the number being the measure of this ramification.
7.3. To get ready for the concluding remarks in § 8, let us make the following notation: If is to by for then assuming the truth of Conjecture III', we write in place of Since an element of sends to some being precisely the image of under the permutation of § 1.2, in case for it follows from Conjecture III' that where
(Added in Proof.) I have been able to prove Conjecture IV rather recently by obtaining an explicit formula for the formal character of the for of the form (* denoting the Dynkin outer automorphism) with which guarantees 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 an arbitrary Weyl module (e.g. one of the terms in the last sum) in terms of Weyl modules whose highest weights are restricted (i.e. lie in 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 with formal characters (independent of as well as such that which gives where From this Conjectures IV and I follow, except for the integrality of (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 can be used for a quick proof of the preceding as well as a closed expression for in terms of the virtual (explicitly, if is written as one has in consonance with The details will appear in due course.
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).