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

§ 8. Epilogue

A metamathematical (?) conjecture

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 p0, happen to behave extremely smoothly with respect to the characteristic p (when p 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 p), perhaps with a handful of exceptions. The most classical example of a "characteristic p phenomenon" behaving smoothly with respect to p is, of course, the formula for the order of the finite Chevalley group G𝔽q (over the finite field 𝔽q of cardinality q) for the semisimple algebraic group G, which is known to be a polynomial in q (this polynomial being dependent only on the type, or root-system, of G).

Concretely speaking we like to offer the following (assuming that the earlier Conjectures are true).

Conjecture V. For p[X:X], the fundamental constants cwwJ (for w, w lying in WJ++) and the Humphreys numbers dwJ (for w in WJ), do not depend on p.

The remarks in § 7.2 indicate that the restriction p[X:X] could be lifted for the case of the Humphreys numbers dwJ if we replace them by the virtual Humphreys number defined there.

It follows from (7.2) that when p is large enough to guarantee that Y0 is independent of p as well as dimMλ is independent of p for each λY0 (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 μ+δ(XJ*+δ)w, wW, the number p-r·dimQμ must be independent not only of μ but also of p (depends only on J and w).

Yet another dimensionality consequence of Conjecture V is the following (cf. formula (5.5)):

Corollary to Conjecture V. Let J{0,1,,l}, and for λXJ* let λ+δ=i=0lmiλi be the expansion in the "homogeneous coordinates" mi satisfying i=0mini=p, where λ0=0 (as in § 1); thus mi is zero or (strictly) positive according as iJ or iJ. Then for each wWJ++ there exists a polynomial in l+1-|J| variables (depending on J and w, but not on p), such that the value of this polynomial obtained on substituting the above homogeneous coordinates mi of λ+δ, equals the dimension of the irreducible G-module M(λ+δ)w-δ for the Chevalley group G (of the given type) in characteristic p0.

We like to wind up by asking the following question: Is there some way of obtaining the matrix [cwwJ] for all J knowing it only for the case J empty?

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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