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 $p\ne 0$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

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 $p\ne 0,$ 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 $p\text{),}$ 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 $G_{{𝔽}_q}$ (over the finite field ${𝔽}_q$ of cardinality $q\text{)}$ 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 $G\text{).}$

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

Conjecture V. For $p\nmid [X:X\prime ],$ the fundamental constants $c_{w\prime w}^J$ (for $w\prime ,$ $w$ lying in ${\stackrel{\sim }{W}}_J^{++}\text{)}$ and the Humphreys numbers $d_w^J$ (for $w$ in ${\stackrel{\sim }{W}}_J^{♯}\text{),}$ do not depend on $p\text{.}$

The remarks in § 7.2 indicate that the restriction $p\nmid [X:X\prime ]$ could be lifted for the case of the Humphreys numbers $d_w^J$ 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 $Y_0$ is independent of $p$ as well as $\text{dim}\hspace{0.17em}{M}_{\lambda }$ is independent of $p$ for each $\lambda \in Y_0$ (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 $\mu +\delta \in {(X_J^{*}+\delta )}^w,$ $w\in {\stackrel{\sim }{W}}^{♯},$ the number $p^{-r}·\text{dim}\hspace{0.17em}{Q}_{\mu }$ must be independent not only of $\mu$ but also of $p$ (depends only on $J$ and $w\text{).}$

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

Corollary to Conjecture V. Let $J⫋\{0,1,\dots ,l\},$ and for $\lambda \in X_J^{*}$ let $\lambda +\delta =\sum _{i=0}^l{m}_i{\lambda }_i$ be the expansion in the "homogeneous coordinates" $m_i$ satisfying $\sum _{i=0}{m}_i{n}_i=p,$ where ${\lambda }_0=0$ (as in § 1); thus $m_i$ is zero or (strictly) positive according as $i\in J$ or $i\notin J\text{.}$ Then for each $w\in {\stackrel{\sim }{W}}_J^{++}$ there exists a polynomial in $l+1-\left|J\right|$ variables (depending on $J$ and $w,$ but not on $p\text{),}$ such that the value of this polynomial obtained on substituting the above homogeneous coordinates $m_i$ of $\lambda +\delta ,$ equals the dimension of the irreducible $G\text{-module}$ $M_{{(\lambda +\delta )}^w-\delta }$ for the Chevalley group $G$ (of the given type) in characteristic $p\ne 0\text{.}$

We like to wind up by asking the following question: Is there some way of obtaining the matrix $\left[c_{w\prime w}^J\right]$ 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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