Last update: 3 April 2014
The philosophy underlying the relationship between Parts I and II of this paper is deeper than the superficial dimensionality connection between Conjectures I and IV we exhibited in the second paragraph of § 7. It rests in the suggestion that in some inexplicable ways several phenomena concerning the representation theory of algebraic Chevalley groups (and their Lie algebras) of a given type and of characteristic happen to behave extremely smoothly with respect to the characteristic (when is treated as an indeterminate, or as a real variable if one prefers, particularly in case of the numerical invariants of this theory when these invariants are treated as functions of perhaps with a handful of exceptions. The most classical example of a "characteristic phenomenon" behaving smoothly with respect to is, of course, the formula for the order of the finite Chevalley group (over the finite field of cardinality for the semisimple algebraic group which is known to be a polynomial in (this polynomial being dependent only on the type, or root-system, of
Concretely speaking we like to offer the following (assuming that the earlier Conjectures are true).
Conjecture V. For the fundamental constants (for lying in and the Humphreys numbers (for in do not depend on
The remarks in § 7.2 indicate that the restriction could be lifted for the case of the Humphreys numbers if we replace them by the virtual Humphreys number defined there.
It follows from (7.2) that when is large enough to guarantee that is independent of as well as is independent of for each (cf. Corollary to Proposition 3 in § 5.2), the second half of Conjecture V follows from the first half.
It follows from Conjecture V that for the number must be independent not only of but also of (depends only on and
Yet another dimensionality consequence of Conjecture V is the following (cf. formula (5.5)):
Corollary to Conjecture V. Let and for let be the expansion in the "homogeneous coordinates" satisfying where (as in § 1); thus is zero or (strictly) positive according as or Then for each there exists a polynomial in variables (depending on and but not on such that the value of this polynomial obtained on substituting the above homogeneous coordinates of equals the dimension of the irreducible for the Chevalley group (of the given type) in characteristic
We like to wind up by asking the following question: Is there some way of obtaining the matrix for all knowing it only for the case empty?
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).