Part II

Representations and decompositions in characteristic $p\ne 0$

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 $p\ne 0,$
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
${G}_{{\mathbb{F}}_{q}}$ (over the finite field ${\mathbb{F}}_{q}$ 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
${\stackrel{\sim}{W}}_{J}^{\u266f}\text{),}$ 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 $\mu +\delta \in {({X}_{J}^{*}+\delta )}^{w},$ $w\in {\stackrel{\sim}{W}}^{\u266f},$ the number ${p}^{-r}\xb7\text{dim}\hspace{0.17em}{Q}_{\mu}$ 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 $J\u2acb\{0,1,\dots ,l\},$
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?

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).