Last update: 3 April 2014
5.1. It was conjectured by this author in 1967, in letters to C. W. Curtis and T. A. Springer, that if the irreducible $G\text{-modules}$ ${M}_{\lambda}$ and ${M}_{\mu}$ (for $\lambda ,\mu \in {X}^{+}\text{)}$ occur as composition factors of some indecomposable $G\text{-module}$ then $\lambda +\delta \equiv {(\mu +\delta )}^{\sigma}$ (mod $pX\text{)}$ must hold for some $\sigma \in W\text{.}$ This conjecture was preceded by an infinitesimal version (to be discussed in § 6) which was announced to Curtis orally in 1966 (see Footnote**). It has been dubbed "Harish-Chandra principle in nonzero characteristic", because of its close analogy with the theorem of Harish-Chandra on the infinitesimal characters of ${\U0001d524}_{\u2102}$ and because of a central role of this theorem in the study of ${\U0001d524}_{\u2102}\text{-modules}$ (infinite-dimensional in general), particularly in the study of the universal ${\U0001d524}_{\u2102}\text{-modules}$ generated by highest-weight-vectors investigated in [Ver1966] and [BGG1971], This analogy is clear from the detailed discussion in [Hum1971 (§ 3)], where our conjectures have been proved true for $p>{h}_{\Delta}$ (Theorems 3.1, 4.1 and 5.1).
It is believed that the Harish-Chandra principle (along with its infinitesimal version) holds for all $p,$ including the bad cases when the Lie algebra ${\U0001d524}_{K}$ of $G$ is not simple. The decompositions of ${\stackrel{\u203e}{V}}_{\lambda}$ (for $\lambda $ ranging in a set larger than ${X}^{\u266f}\text{)}$ for the case of type ${G}_{2}$ with $p=3,$ given by Springer in [Spr1968, p. 103], as well as a few such decompositions for the case of type ${F}_{4}$ with $p=2$ that are implied by [Vel1970, Tables I and II], are all consistent with our view.
As in Part I (§ 1), let $\stackrel{\sim}{W}$ be the semidirect extension of $W$ by a group $D$ isomorphic to $X\text{.}$ However, we find it convenient to modify the action of $D$ on $X$ slightly now: for $\lambda \in X$ we define the corresponding element of $D$ to be the translation on $X$ (or on $E=X\underset{\mathbb{Z}}{\otimes}\mathbb{R}\text{)}$ by $p\lambda $ and not $\lambda $ as before. Accordingly $D\prime ,$ $\stackrel{\sim}{W},$ $\stackrel{\sim}{W}\prime ,$ $\Omega ,$ and the elements ${w}_{\sigma}$ of ${\stackrel{\sim}{W}}^{\u266f},$ all have modified action on $X$ or $E$ which is dependent on $p\text{.}$ This does not affect the abstract structure of any of these groups, nor does it affect the partial orders on ${\stackrel{\sim}{W}}^{+},$ ${\stackrel{\sim}{W}}^{\u266f}$ ect. Also, we find it convenient not to change the meaning of ${E}^{\u266f}$ and ${E}^{*},$ so that it is $p{E}^{\u266f}$ (resp. $p{E}^{*}\text{)}$ and not ${E}^{\u266f}$ (resp. ${E}^{*}\text{)}$ which is now the fundamental region in $E$ for the action of $D$ (resp. $\stackrel{\sim}{W}\prime \text{)}\text{.}$
With this convention, the Harish-Chandra principle states that if ${M}_{\lambda}$ and ${M}_{\mu}$ are composition factors of an indecomposable $G\text{-module}$ then $\lambda +\delta $ and $\mu +\delta $ must lie in the same $\stackrel{\sim}{W}\text{-orbit.}$ Since all weights of an indecomposable $G\text{-module}$ lie in one single coset of $X\prime ,$ clearly $\lambda -\mu \in X\prime $ is another necessary condition. We claim that when $p\nmid [X:X\prime ]$ these two conditions together imply (and therefore are equivalent to) the condition that $\lambda +\delta $ and $\mu +\delta $ be conjugate under $\stackrel{\sim}{W}\prime \text{.}$ In view of $\stackrel{\sim}{W}=\stackrel{\sim}{W}\prime \Omega ,$ it suffices to consider the situation when $\lambda +\delta $ and $\mu +\delta $ are conjugate under $\Omega =\left\{{\omega}_{j}\hspace{0.17em}\right|\hspace{0.17em}j\in {J}_{0}\}$ (cf. § 1.2); then $$\mu +\delta ={(\lambda +\delta )}^{{\omega}_{j}}={(\lambda +\delta )}^{{\sigma}_{0}{\sigma}_{j}}+p{\lambda}_{j}\equiv (\lambda +\delta )+p{\lambda}_{j}\hspace{0.17em}\text{(mod}\hspace{0.17em}X\prime \text{)}$$ (as ${\sigma}_{0}{\sigma}_{j}\in W\text{),}$ gives $p{\lambda}_{j}\in X\prime \text{;}$ because of $p\nmid [X:X\prime ]$ we have ${\lambda}_{j}\in X\prime ,$ which gives ${\lambda}_{j}=0$ and ${\omega}_{j}=\text{id.}$ as desired.
Let us call $\lambda ,\mu \in X,$ $\stackrel{\sim}{W}\prime \text{-linked}$ if $\lambda +\delta $ and $\mu +\delta $ are conjugate under $\stackrel{\sim}{W}\prime \text{.}$ It is clear that every element of $X$ is $\stackrel{\sim}{W}\prime \text{-linked}$ to a unique element of the set $$\begin{array}{ccc}{X}^{*}& =& (X\cap p{E}^{*})-\delta =\{\lambda \in X\hspace{0.17em}|\hspace{0.17em}{p}^{-1}(\lambda +\delta )\in {E}^{*}\}\\ & =& \{\lambda \in X\hspace{0.17em}|\hspace{0.17em}\u27e8\lambda +\delta ,{\alpha}_{i}^{\vee}\u27e9\geqq 0\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}1\leqq i\leqq l,\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\u27e8\lambda +\delta ,-{\alpha}_{0}^{\vee}\u27e9\leqq p\},\end{array}$$ which is partitioned into disjoint subsets ${X}_{J}^{*}$ for $J\u2acb\{0,1,\dots ,l\},$ where $${X}_{J}^{*}=(X\cap p{E}_{J}^{*})-\delta =\{\lambda \in {X}^{*}\hspace{0.17em}|\hspace{0.17em}{(\lambda +\delta )}^{{R}_{j}}=\lambda +\delta \iff j\in J\}\text{.}$$ Then $\mu \in X$ can be written in the form $\mu ={(\lambda +\delta )}^{w}-\delta $ with $\lambda \in {X}_{J}^{*},$ $w\in {\stackrel{\sim}{W}}^{J},$ where $J,\lambda ,w$ are all unique; and $\mu \in {X}^{+}$ (resp. ${X}^{\u266f}\text{)}$ if and only if $w\in {\stackrel{\sim}{W}}_{J}^{++}$ (resp. $w\in {\stackrel{\sim}{W}}_{J}^{\u266f}\text{)}\text{.}$ We recall that ${\stackrel{\sim}{W}}_{J}^{\u266f}$ is empty precisely when the complement of $J$ in $\{0,1,\dots ,l\}$ is a singleton $\left\{j\right\}$ with $j\in {J}_{0},$ $j\ne {j}_{0}\text{.}$ Note that ${X}^{*}$ does not lie (entirely) in ${X}^{+},$ and that $${X}^{*}\cap {X}^{+}={X}^{*}\cap {X}^{\u266f}={X}_{\varnothing}^{*}\cup {X}_{\left\{0\right\}}^{*},$$ where this intersection (resp. the subset ${X}_{\varnothing}^{*}\text{)}$ is nonempty if and only if $$\u27e8\delta ,-{\alpha}_{0}^{\vee}\u27e9\leqq p\phantom{\rule{1em}{0ex}}\text{(resp.}\hspace{0.17em}\u27e8\delta ,-{\alpha}_{0}^{\vee}\u27e9\lneqq p\text{),}$$ i.e. $p\geqq {h}_{\Delta}-1$ (resp. $p\geqq {h}_{\Delta}\text{).}$ Let it be noted (from the classification of root-systems) that the Coxeter number ${h}_{\Delta}$ is always a composite number except in the case of type ${A}_{p-1}$ (when ${h}_{\Delta}=p\text{),}$ and in this case ${X}_{\varnothing}^{*}$ consists of the point $0$ alone. This shows that for $p<{h}_{\Delta}$ every point of $X$ has a nontrivial stabilizer in $\stackrel{\sim}{W}\prime $ (and hence in $\stackrel{\sim}{W}\text{),}$ and for $p\leqq {h}_{\Delta}$ every point has a nontrivial stabilizer in $\stackrel{\sim}{W}\text{;}$ for the latter statement one needs to observe that if $p={h}_{\Delta}$ (type ${A}_{l},$ $l=p-1\text{),}$ then the set $X\cap p{E}_{\varnothing}^{*}={X}_{\varnothing}^{*}+\delta $ consists of only one point $\delta $ which turns out to be the bary-centre of the $p$ vertices $\left\{p{\lambda}_{j}\hspace{0.17em}\right|\hspace{0.17em}j\in {J}_{0}\}$ of the fundamental simplex $p{E}^{*}$ and is therefore fixed by $\Omega $ (cf. § 1). We shall call $\lambda \in X$ $p\text{-regular}$ if its stabilizer in $\stackrel{\sim}{W}\prime $ is the identity; for such $\lambda $ the stabilizer in $\stackrel{\sim}{W}$ is also the identity provided $p\nmid [X:X\prime ]\text{.}$
For $J=\{1,2,\dots ,l\}$ the set ${X}_{j}^{*}$ has $-\delta $ as its solitary element, and the only element of ${X}^{\u266f},$ $\stackrel{\sim}{W}\prime \text{-linked}$ to it is $(p-1)\delta \text{.}$ Since no dominant weight of ${\stackrel{\u203e}{V}}_{(p-1)\delta}$ is $\stackrel{\sim}{W}\prime \text{-linked}$ to the highest weight, the Harish-Chandra principle implies that this $G\text{-module}$ is irreducible which gives $$\begin{array}{cc}\text{dim}\hspace{0.17em}{M}_{(p-1)\delta}={p}^{r}\phantom{\rule{1em}{0ex}}\text{(where}\hspace{0.17em}r=\left|{\Delta}_{+}\right|\text{).}& \text{(5.1)}\end{array}$$ This may be regarded as the first evidence in favour of our conjecture, because (5.1) is known to be true for all $p\text{.}$ Since $$\text{dim}\hspace{0.17em}{M}_{\lambda}\leqq \text{dim}\hspace{0.17em}{\stackrel{\u203e}{V}}_{\lambda}<{p}^{r}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}\lambda \in {X}^{\u266f},\lambda \ne (p-1)\delta ,$$ (5.1) follows from the fact that at least one member of $\left\{{M}_{\lambda}\hspace{0.17em}\right|\hspace{0.17em}\lambda \in {X}^{\u266f}\}$ has dimension ${p}^{r}\text{;}$ this fact is a consequence of Steinberg's theorem [Bor1970, Theorem 7.5] asserting that upon restriction to ${G}_{{\mathbb{F}}_{p}}$ (for the form of $G$ defined and split over ${\mathbb{F}}_{p}\text{)}$ the set $\left\{{M}_{\lambda}\hspace{0.17em}\right|\hspace{0.17em}\lambda \in {X}^{\u266f}\}$ furnishes all the irreducible $K{G}_{{\mathbb{F}}_{p}}\text{-modules,}$ and another celebrated discovery of Steinberg that there is an irreducible representation of $\u2102{G}_{{\mathbb{F}}_{p}}$ of degree ${p}^{r}$ (the $p\text{-Sylow}$ order of ${\mathbb{F}}_{p}\text{)}$ which remains irreducible upon "reduction mod $p$ in Brauer's sense", i.e. upon passing to $K{G}_{{\mathbb{F}}_{p}}\text{.}$ The module ${M}_{(p-1)\delta}$ is called the Steinberg module (for the $p\text{-modular}$ group-ring $K{G}_{{\mathbb{F}}_{p}},$ or the algebraic Chevalley group $G,$ or for its Lie algebra ${\U0001d524}_{K},$ for each of which it is irreducible; see also § 6), and has the property that no indecomposable module other than itself can possess it as a composition factor. Curtis has generalized the existence and uniqueness of the Steinberg module to a class of Tits systems that generalizes the class of finite Chevalley groups.
5.2. The situation (5.1) above is a special case of the following more general consequence of the Harish-Chandra principle.
For a given dominant weight $\lambda ,$ if the set $$\begin{array}{cc}\{\mu ={(\lambda +\delta )}^{w}-\delta \hspace{0.17em}|\hspace{0.17em}w\in \stackrel{\sim}{W},\hspace{0.17em}\mu \leqq \lambda ,\hspace{0.17em}\mu \in {X}^{+}\}& \text{(5.2)}\end{array}$$ contains no element besides $\lambda ,$ then the $G\text{-module}$ ${\stackrel{\u203e}{V}}_{\lambda}$ is irreducible.
From this we shall deduce the following Proposition, using the fact that the hypothesis of (5.2) is satisfied when $\lambda \in {X}^{*}\cap {X}^{+},$ i.e. when $\u27e8\lambda ,-{\alpha}_{0}^{\vee}\u27e9\leqq p-\u27e8\delta ,-{\alpha}_{0}^{\vee}\u27e9\text{;}$ it would suffice for us to use this fact for $p>{h}_{\Delta}$ in which case the result of [Hum1971] on the validity of the Harish-Chandra principle is applicable.
Proposition 3. If $\lambda =\sum _{i}{m}_{i}{\lambda}_{i}\in {X}^{+}$ satisfies $\sum _{i=1}^{l}{m}_{i}{n}_{i}\leqq p-{h}_{\Delta}+1$ then the $G\text{-module}$ ${\stackrel{\u203e}{V}}_{\lambda}$ is irreducible.
The earlier known results on the irreducibility of ${\stackrel{\u203e}{V}}_{\lambda}$ are to be found in [Cur1960, Theorem 1] and [Spr1968, Corollary 4.3], and are mostly contained in the above. Let us note that the Proposition is vacuous when $p<{h}_{\Delta}-1\text{.}$ Postponing the values ${h}_{\Delta}-1$ and ${h}_{\Delta}$ of $p,$ let us assume $p>{h}_{\Delta},$ which implies $p\nmid [X:X\prime ]$ as $[X:X\prime ]$ never exceeds ${h}_{\Delta}\text{.}$ Then in the definition of the set involved in the hypothesis of (5.2) $\stackrel{\sim}{W}$ can be replaced by $\stackrel{\sim}{W}\prime \text{.}$ This hypothesis is fulfilled because the set of elements of ${X}^{+}$ in the $\stackrel{\sim}{W}\prime \text{-orbit}$ of $\lambda +\delta $ has $\lambda +\delta $ as the smallest element (with respect to the partial order $\leqq \text{);}$ this latter statement is clear from the observation that the identity is the smallest element of ${\stackrel{\sim}{W}}^{+}$ (with respect to $\leqq ,$ and hence with respect to the affine partial order, introduced in § 1). This proves the Proposition in the major case $p>{h}_{\Delta}\text{.}$ For the remaining values of $p,$ we find that the hypothesis of (5.2) is satisfied in a most trivial manner, viz. that ${\stackrel{\u203e}{V}}_{\lambda}$ has no dominant weight at all except $\lambda $ (which guarantees the irreducibility of ${\stackrel{\u203e}{V}}_{\lambda}$ irrespective of the characteristic), so that (5.2) and the Harish-Chandra principle need not be invoked. This is clear from the fact that for $p={h}_{\Delta}-1$ (resp. $p={h}_{\Delta},$ which can hold only when $\Delta $ is of type A so that ${n}_{i}=1$ for all $i\text{)}$ the only value(s) of $\lambda $ for which the hypothesis of the Proposition holds is ${\lambda}_{0}=0$ (resp. are $\left\{{\lambda}_{i}\hspace{0.17em}\right|\hspace{0.17em}0\leqq i\leqq l\}\text{).}$
Note that in the last part of the proof above we have essentially appealed to the standard fact that when $G\cong {SL}_{l+1}$ the fundamental representations ${\stackrel{\u203e}{V}}_{{\lambda}_{i}}$ are irreducible (irrespective of $p\text{),}$ being precisely the exterior powers of the natural $(l+1)\text{-dimensional}$ representation. In general, it is not true that all the ${\stackrel{\u203e}{V}}_{{\lambda}_{i}}\text{'s}$ are irreducible irrespective of $p\text{;}$ the correct generalization of the preceding phenomenon for type A is the following: For $j\in {J}_{0}$ the $G\text{-module}$ ${\stackrel{\u203e}{V}}_{{\lambda}_{j}}$ is irreducible for every $p\text{.}$ This follows from the fact that ${\stackrel{\u203e}{V}}_{\lambda}$ has no dominant weight except $\lambda $ if and only if $\lambda \in \left\{{\lambda}_{j}\hspace{0.17em}\right|\hspace{0.17em}j\in {J}_{0}\},$ which in turn is an immediate consequence of the observation that if $\lambda \in {X}^{+}$ lies in the coset ${\lambda}_{i}+X\prime \in X/X\prime $ (with $i\in {J}_{0}\text{)}$ then $\lambda \geqq {\lambda}_{i}\text{.}$
Combining (4.6) and Proposition 3 we have
Corollary to Proposition 3. If $p\geqq 2{h}_{\Delta}-3$ then ${\stackrel{\u203e}{V}}_{\lambda}$ is irreducible for $\lambda $ lying in the set ${Y}_{0}$ defined in § 4.5.
5.3. For the rest of this section we assume that $p\nmid [X:X\prime ]$ and the Harish-Chandra principle is valid for the prime $p$ at hand (unless otherwise stated). Thus, whenever ${M}_{\lambda}$ and ${M}_{\mu}$ are "linked" in the sense that there exists a chain of indecomposable $G\text{-modules}$ in which each consecutive pair has a composition factor in common and ${M}_{\lambda}$ (resp. ${M}_{\mu}\text{)}$ occurs as a composition factor of the first (resp. the last) in this chain, the highest weights $\lambda $ and $\mu $ are $\stackrel{\sim}{W}\prime \text{-linked,}$ i.e. $\lambda +\delta $ and $\mu +\delta $ are $\stackrel{\sim}{W}\prime \text{-conjugate.}$
With this assumption the solving of (4.2) is greatly simplified, because now one has $$Y\left({\mu}_{0}\right)\u2ac5\{\mu \in {X}^{+}\hspace{0.17em}|\hspace{0.17em}\mu \leqq {\mu}_{0}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\mu -{({\mu}_{0}+\delta )}^{w}-\delta \phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}w\in \stackrel{\sim}{W}\prime \}\text{.}$$ The conjecture below puts an even stricter limit on $Y\left({\mu}_{0}\right)$ which could turn out to be optimal in most cases.
Conjecture II. For ${\mu}_{0},\mu \in {X}^{+},$ the nonvanishing of the decomposition number ${c}_{{\mu}_{0}\mu}$ defined in § 4 implies that the following condition (A) holds:
(A) | "there exist reflections ${S}_{1},{S}_{2},\dots ,{S}_{k}$ such that defining ${\mu}_{i}={({\mu}_{i-1}+\delta )}^{{S}_{i}}-\delta $ successively for $1\leqq i\leqq k,$ one has ${\mu}_{i}+\delta \in {X}^{+}$ and $${\mu}_{0}\geqq {\mu}_{1}\geqq \dots \geqq {\mu}_{k-1}\geqq {\mu}_{k}=\mu \text{".}$$ |
The reader who is familiar with the author's work in [Ver1966] on the universal ${\U0001d524}_{\u2102}\text{-modules}$ generated by highest-weight-vectors (see ${\text{Footnote}}^{5}$ in § 5.4), would recognize some formal similarity between the above and the Conjecture 1 of [Ver1968] which has recently been proved true by I. M. Gelfand and collaborators [BGG1971] through an elegant tensor product trick. The following recasting is immediate, and provides our main justification for introducing the partial-order $\leqq $ on ${\stackrel{\sim}{W}}^{+}\text{.}$
Conjecture II (Restatement). Suppose $\mu \prime ,\mu \in {X}^{+}$ are $\stackrel{\sim}{W}\prime \text{-linked,}$ so that $$\mu \prime ={(\lambda +\delta )}^{w\prime}-\delta ,\phantom{\rule{1em}{0ex}}\mu ={(\lambda +\delta )}^{w}-\delta \phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}\lambda \in {X}_{j}^{*},\phantom{\rule{1em}{0ex}}w\prime ,\phantom{\rule{1em}{0ex}}w\in {\stackrel{\sim}{W}}_{J}^{++},$$ where $J,\lambda ,w\prime ,$ are all unique. Then $${c}_{\mu \prime \mu}\ne 0\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}w\leqq w\prime \text{.}$$
It is possible to give substantial confirmation of this Conjecture in case of type ${A}_{2}$ i.e. $G\cong {SL}_{3},$ but we relegate it to a later paper. Below we give an example in this case which shows that the ${\mu}_{i}\text{'s}$ in the original statement of Conjecture II cannot always be chosen inside ${X}^{+}\text{.}$ With this evidence it becomes reasonable to consider the $G\text{-modules}$ ${\stackrel{\u203e}{V}}_{\mu}$ and ${M}_{\mu}$ not just for $\mu \in {X}^{+},$ but also, somewhat artificially, for $\mu \in {X}^{+}-\delta ,$ defining these to be $0\text{-dimensional}$ in case $\mu +\delta $ is $W\text{-singular}$ i.e. lies on a wall of ${E}^{+},$ in accordance with $\text{dim}\hspace{0.17em}{\stackrel{\u203e}{V}}_{\mu}=D(\mu +\delta )\text{.}$
Example 1. In the adjoining diagram (Fig. 4) consider an arbitrary point $\lambda $ in ${X}_{\left\{0\right\}}^{},$ i.e. a point of $X$ lying on the open line-segment joining $-\delta +p{\lambda}_{1}$ and $-\delta +p{\lambda}_{2}\text{;}$ say $\lambda +\delta =i{\lambda}_{1}+(p-i){\lambda}_{2}$ for $1\leqq i\leqq p-1\text{.}$ Setting $$\mu \prime +\delta =(\lambda +\delta ){R}_{1}{R}_{2}{R}_{0}=2p{\lambda}_{1}+i{\lambda}_{2},$$ one immediately sees that the set $$\{\mu ={(\lambda +\delta )}^{w}-\delta \hspace{0.17em}|\hspace{0.17em}w\in {\stackrel{\sim}{W}}_{\left\{0\right\}}^{+},\hspace{0.17em}\mu \leqq \mu \prime \}$$ of weights of ${\stackrel{\u203e}{V}}_{\mu},$ which lie in ${X}^{+}-\delta $ and are linked to $\mu \prime ,$ consists of 4 elements $\mu \prime ,\mu \u2033,\mu \u2034$ and $\lambda $ where $$\mu \u2033+\delta =(\lambda +\delta ){R}_{1}{R}_{0}=(p+i){\lambda}_{2}\notin {X}^{+}+\delta ,$$ and $$\mu \u2034+\delta =(\lambda +\delta ){R}_{2}{R}_{0}=(2p-i){\lambda}_{1}\notin {X}^{+}+\delta \text{.}$$ In this case we can give the decomposition ${\stackrel{\u203e}{V}}_{\mu \prime}{\sim}_{G}\sum _{\mu}{c}_{\mu \prime \mu}{M}_{\mu}$ on pure dimensionality considerations: on one hand we have $$\text{dim}\hspace{0.17em}{\stackrel{\u203e}{V}}_{\mu \prime}=pi(2p+i),\phantom{\rule{1em}{0ex}}\text{dim}\hspace{0.17em}{\stackrel{\u203e}{V}}_{\lambda}=\frac{1}{2}pi(p-i)=\text{dim}\hspace{0.17em}{M}_{\lambda},$$ $$\begin{array}{c}\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n-\delta +p\lambda 2\nR1\n-\delta \n\nR1R0\n\mu \u2033\nR0\n\lambda \nid.\nR2\nR1R2\n\n(p-1)\delta \n\nR1R2R0\nR2R0\n\mu \u2034\n-\delta +p\lambda 1\n\nR0R1R2R0\n\mu \u2032\n\n\end{array}\\ \text{Fig. 4. Diagram for Example 1}\end{array}$$ and on the other hand $${M}_{\mu \prime}={M}_{{\lambda}_{1}}^{\text{Fr}}\otimes {M}_{(p-1){\lambda}_{1}+(i-1){\lambda}_{2}}$$ gives $\text{dim}\hspace{0.17em}{M}_{\mu \prime}=3\xb7\frac{1}{2}pi(p+i)\text{;}$ here we have used the irreducibility of ${\stackrel{\u203e}{V}}_{\lambda}$ and also of ${\stackrel{\u203e}{V}}_{\mu}$ for $\mu +\delta =p{\lambda}_{1}+i{\lambda}_{2},$ which can be seen either from (5.2) or by appealing to Braden [Bra1967] as long as $p\geqq 5\text{.}$ (Actually Braden gives the infinitesimal decomposition numbers ${c}_{\lambda \mu}^{\prime}$ to be defined in § 6, but in this case it happens that ${c}_{\lambda \mu}^{\prime}={c}_{\lambda \mu},$ as we remark at the end of § 6.) Now verifying $$\text{dim}\hspace{0.17em}{M}_{\mu \prime}=\text{dim}\hspace{0.17em}{\stackrel{\u203e}{V}}_{\mu \prime}-\text{dim}\hspace{0.17em}{\stackrel{\u203e}{V}}_{\lambda},$$ we are forced to the exact sequence $$0\u27f6{M}_{\lambda}\u27f6{\stackrel{\u203e}{V}}_{\mu \prime}\u27f6{M}_{\mu \prime}\u27f60,$$ which gives ${c}_{\mu \prime \lambda}=1\text{.}$ The Conjecture is clearly verified with $k=2$ $({\mu}_{0}=\mu \prime ),$ and ${\mu}_{1}$ can be taken to be either $\mu \u2033$ or $\mu \u2034$ neither of which lies in ${X}^{+}\text{.}$
5.4. Our next conjecture says that the value of ${c}_{\mu \prime \mu}$ in the restatement of Conjecture II depends only on $w$ and $w\prime $ (and not on $\lambda \text{)}\text{.}$
Conjecture III. Let $J\stackrel{\u203e}{\u2acb}\{0,1,\dots ,l\},$ ${\nu}_{1},{\nu}_{2}\in {X}_{J}^{*}$ and $w\prime ,w\in {\stackrel{\sim}{W}}_{J}^{++}$ be all given, and write $${\mu}_{i}={({\nu}_{i}+\delta )}^{w}-\delta ,\phantom{\rule{1em}{0ex}}{\mu}_{i}^{\prime}={({\nu}_{i}+\delta )}^{w\prime}-\delta \phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}1\leqq i\leqq 2\text{.}$$ Then we must have ${c}_{{\mu}_{1}^{\prime}{\mu}_{1}}={c}_{{\mu}_{2}^{\prime}{\mu}_{2}}\text{.}$
This is analogous to a phenomenon for the composition factors of the universal ${\U0001d524}_{\u2102}\text{-modules}$ generated by highest-weight-vectors studied in [Ver1966] and [BGG1971], the appearance of this latter phenomenon being indicated by certain information that was communicated to me orally by I. N. Bernstein during the Budapest Summer School on Group Representations, 1971; it may be mentioned at this point that, contrary to Theorems 1 and 7 of [Ver1968] and a similar announcement in the original version of [Gel1971], the composition factors of the said ${\U0001d524}_{\u2102}\text{-modules}$ can have multiplicity greater than one, as was shown in [BGG1971, p. 9]. (See ${\text{Footnote}}^{5}\text{.)}$
Example 2. Here we shall show that for the case of type ${B}_{2},$ each of the two conjectures II and III yields that $$\begin{array}{cc}\text{for}\phantom{\rule{1em}{0ex}}\mu \prime \in {X}^{\u266f},\phantom{\rule{1em}{0ex}}{c}_{\mu \prime \mu}\ne 0\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}\mu \in {X}^{\u266f},& \text{(5.3)}\end{array}$$ i.e. the set ${Y}_{0}$ defined in § 4 reduces to $\left\{0\right\}\text{.}$ (The same conclusion is valid for type ${A}_{2},$ and is easily obtained either from the Harish-Chandra principle or from Braden's determination of ${c}_{\lambda \mu}$ in this case. For the case of type ${G}_{2}$ it is suspected that this statement is false; this is based on the fact that $\{w\in {\stackrel{\sim}{W}}^{+}\hspace{0.17em}|\hspace{0.17em}w\leqq {w}_{0}\}$ properly contains ${\stackrel{\sim}{W}}^{\u266f},$ as was noted in § 1 and § 3; the existence of this possibility is also stated in [Hum1971, p. 76].) We first consider the general case $p\geqq 5,$ under which the Harish-Chandra principle is valid in view of [Hum1971], taking up $p=2$ and $p=3$ later. That then Conjecture II implies (5.3) is straightforward from the fact that if $w,w\prime $ are defined as in the restatement of Conjecture II (then because of $w\prime \in {\stackrel{\sim}{W}}^{\u266f},$ which is equivalent to $\mu \prime \in {X}^{\u266f}\text{)}$ one has $w\leqq w\prime \leqq {w}_{0}$ and hence $w\in {\stackrel{\sim}{W}}^{\u266f}\text{.}$
To see that Conjecture III also implies (5.3), we look at the various possibilities for the location of $\mu $ outside ${X}^{\u266f}$ (yet satisfying ${c}_{\mu \prime \mu}\ne 0$ for some $\mu \prime \in {X}^{\u266f}\text{)}$ in the various triangles (or on their edges) in Fig. 5. The triangles shown there are $$\begin{array}{c}\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n-\delta +3p2\lambda 2\n-\delta +p\lambda 2\n-\delta +p\lambda 22\n\nR2R1R2R0\nR1R2R0\nR2R0\nR0\nid.\n-\delta \n\n(p-1)\delta \nR0R1R2R0\nR0R2R0\n-\delta +p\lambda 1\n\n-\delta +2p\lambda 1\n\n\end{array}\\ \text{Fig. 5. Diagram for Example 2}\end{array}$$ part of the tessellation of $E$ by ${\left(p{E}^{*}\right)}^{w}-\delta $ where $w$ ranges over $\stackrel{\sim}{W}\prime ,$ the element $w$ being shown inside the corresponding triangle. It is clear that $\mu $ must lie inside the quadrilateral with vertices $$-\delta ,\phantom{\rule{1em}{0ex}}-\delta +\frac{3}{2}p{\lambda}_{2},\phantom{\rule{1em}{0ex}}(p-1)\delta ,\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}-\delta +2p{\lambda}_{1},$$ because no $\mu \in {X}^{+}$ outside this region satisfies $\mu \leqq \mu \prime $ for $\mu \prime \in {X}^{\u266f}\text{.}$ From the irreducibility of ${\stackrel{\u203e}{V}}_{(p-1)\delta}$ (the Steinberg module) it is also clear that $\mu $ has to be an interior point of this quadrilateral, which is decomposed into seven triangles three of which lie outside $p\xb7{E}^{\u266f}\text{.}$ If $\mu $ lies in either of the triangles marked ${R}_{2}{R}_{1}{R}_{2}{R}_{0}$ and ${R}_{0}{R}_{1}{R}_{2}{R}_{0},$ including the edge common to the triangles marked ${R}_{0}{R}_{1}{R}_{2}{R}_{0}$ and ${R}_{0}{R}_{2}{R}_{0},$ then every $\mu \prime \in {X}^{\u266f}$ which is $\stackrel{\sim}{W}\prime \text{-linked}$ to $\mu $ satisfies $\mu \prime \leqq \mu \text{;}$ so we must try $\mu $ in the interior of the ${R}_{0}{R}_{2}{R}_{0}\text{-triangle.}$ Take $\mu +\delta =(p+i){\lambda}_{1}+j{\lambda}_{2},$ with $i,j$ positive integers such that $i+2j<p\text{.}$ Then the only $\mu \prime \in {X}^{\u266f}$ which is $\stackrel{\sim}{W}\prime \text{-linked}$ to $\mu $ and does not satisfy $\mu \prime \leqq \mu $ lies in the ${R}_{1}{R}_{2}{R}_{0}\text{-triangle,}$ and is given by $$\mu \prime +\delta =(p-i){\lambda}_{1}+(p-j){\lambda}_{2}\text{;}$$ since $$\mu \prime -\mu =(p-2i-2j){\alpha}_{1}+(p-2i-j){\alpha}_{2},$$ one has $\mu \leqq \mu \prime $ (a necessary condition for ${c}_{\mu \prime \mu}=0\text{)}$ if and only if $i+j<\frac{1}{2}p\text{.}$ Thus ${c}_{\mu \prime \mu}=0,$ in case $i+j<\frac{1}{2}p,$ and hence by Conjecture III for all $i,j$ (satisfying $i+2j<p\text{).}$ This completes the two "proofs" of (5.3) under $p\geqq 5\text{.}$
For $p=2$ there is nothing to prove because the aforementioned quadrilateral has no interior point lying in $X$ but not in ${X}^{\u266f}\text{.}$ For $p=3$ there is only one such point, viz. $\mu =3{\lambda}_{1}\text{;}$ and the only value of $\mu \prime \in {X}^{\u266f},$ other than the Steinberg highest weight $(p-1)\delta ,$ satisfying $\mu \leqq \mu \prime $ is $\mu \prime ={\lambda}_{1}+2{\lambda}_{2},$ with $\mu \prime -\mu ={\alpha}_{2}\text{.}$ The points $\mu ,\mu \prime $ are not in the same $\stackrel{\sim}{W}\text{-orbit;}$ however, we do not require to invoice the Harish-Chandra principle in this case to conclude ${c}_{\mu \prime \mu}=0\text{.}$ We have only to show that no nonzero $\mu \text{-weight-vector}$ of ${\stackrel{\u203e}{V}}_{\mu}$ is annihilated by ${\U0001d524}_{K}^{+}=\sum _{\alpha \in {\Delta}_{+}}{\U0001d524}_{K}^{\alpha}\text{.}$ We claim that the 1-dimensional $\mu \text{-weight-space}$ of the 40-dimensional $G\text{-module}$ ${\stackrel{\u203e}{V}}_{\mu \prime}$ is certainly not annihilated by ${\U0001d524}_{K}^{{\alpha}_{2}}\text{.}$ This is immediate from the irreducibility of the 3-dimensional ${SL}_{2}\text{-module}$ spanned by the $\mu \prime ,\mu ,$ and $\mu \u2033\text{-weight-spaces,}$ where $\mu \u2033=\mu \prime -2{\alpha}_{2},$ which is stable under the action of the subgroup of $G$ corresponding to the 3-dimensional simple Lie subalgebra generated by ${\U0001d524}_{K}^{{\alpha}_{2}}$ and ${\U0001d524}_{K}^{-{\alpha}_{2}}\text{.}$ This completes our discussion of (5.3).
5.5. Now assuming Conjecture III, the decomposition ${\stackrel{\u203e}{V}}_{\mu \prime}{\sim}_{G}\sum _{\mu}{c}_{\mu \prime \mu}{M}_{\mu}$ for $\mu \prime ={(\lambda +\delta )}^{w\prime}-\delta $ with $\lambda \in {X}_{J}^{*}$ and $w\prime \in {\stackrel{\sim}{W}}_{J}^{++},$ assumes the form $${\stackrel{\u203e}{V}}_{\mu \prime}{\sim}_{G}\sum _{w\in {\stackrel{\sim}{W}}_{J}^{++}}{c}_{w\prime w}^{J}{M}_{{(\lambda +\delta )}^{w}-\delta},$$ where the (well-determined) constants ${c}_{w\prime w}^{J}$ do not depend on $\lambda \text{.}$ Writing ${\chi}_{w}^{J}\left(\lambda \right)$ (resp. ${\stackrel{\u203e}{\chi}}_{w}^{J}\left(\lambda \right)\text{)}$ for ${\chi}_{{(\lambda +\delta )}^{w}-\delta}$ (resp. ${\stackrel{\u203e}{\chi}}_{{(\lambda +\delta )}^{w}-\delta}\text{),}$ for $\lambda \in {X}_{J}^{*},$ we have $$\begin{array}{cc}{\stackrel{\u203e}{\chi}}_{w\prime}^{J}\left(\lambda \right)=\sum _{w\in {\stackrel{\sim}{W}}_{J}^{++}}{c}_{w\prime w}^{J}{\chi}_{w}^{J}\left(\lambda \right)\text{.}& \text{(5.4)}\end{array}$$ This simplifies the problem in § 4 of solving the linear equations (4.2). By virtue of Conjecture II, the matrix $\left[{c}_{w\prime w}^{J}\right]$ for $w\prime ,$ $w$ ranging in ${\stackrel{\sim}{W}}_{J}^{++},$ is again unipotent with respect to (any linear order stronger than) the partial order $\leqq ,$ since ${c}_{w\prime w}^{J}\ne 0$ implies $w\leqq w\prime ,$ and ${c}_{ww}^{J}=1\text{.}$ As one is primarily interested in evaluating ${\chi}_{\mu}$ for $\mu \in {X}^{\u266f},$ i.e. ${\chi}_{w}^{J}\left(\lambda \right)$ for $w\in {\stackrel{\sim}{W}}_{J}^{\u266f},$ it suffices to invert the submatrix $\left[{c}_{w\prime w}^{J}\right]$ for $w\prime ,$ $w$ running over the finite subset ${Y}_{J}=\{w\in {\stackrel{\sim}{W}}^{J}\hspace{0.17em}|\hspace{0.17em}w\leqq {w}_{0}\}\text{.}$ This is particularly convenient for large $p,$ since the number and size of the matrices ${\left[{c}_{\xb7\xb7}^{J}\right]}_{{Y}_{J}}$ to be inverted are independent of $p\text{;}$ in the Epilogue (§ 8) we shall comment on how the matrix-entries themselves are expected to be largely independent of $p\text{.}$ Denoting the inverse by $\left[{\gamma}_{w\prime w}^{J}\right]$ we have $$\begin{array}{cc}\text{dim}\hspace{0.17em}{M}_{{(\lambda +\delta )}^{w\prime}-\delta}=\sum _{w\in {\stackrel{\sim}{W}}_{J}^{++}}{\gamma}_{w\prime w}^{J}D\left({(\lambda +\delta )}^{w}\right)\text{.}& \text{(5.5)}\end{array}$$
** The genesis of this conjecture lies in the author's attempt to stream-line the results of Braden in his Ph. D. thesis, written under the guidance of Curtis [Bra1967], determining the infinitesimal decomposition of ${\stackrel{\u203e}{V}}_{\lambda}$ for $\lambda \in {X}^{\u266f}$ in the case of ${A}_{2}\text{.}$ I learnt of these results during the part of the summer of 1966 that I spent at the University of Oregon under an invitation from Professor Curtis, who initiated me at that time to the /»-modular representation theory of finite Chevalley groups in characteristic $p$ and its connection with the representations of the related Lie algebras.
${}^{5}$ (Added in Proof.) The recently published book "Algèbres Enveloppantes" (Cahiers Scientifiques XXXVII, Gauthier-Villars, 1974) by J. Dixmier contains an exhaustive account of these ${\U0001d524}_{\u2102}\text{-modules}$ which have been termed Verma modules. But the problem of determination of their composition multiplicities remains open (on account of the said counterexample in [BGG1971]); no one seems to have even a plausible guess at these numbers. (And, therefore, with extreme apology for a rather arbitrary and vague guess, I should like to suggest that these multiplicities may not be unrelated to the numbers ${b}_{\sigma ,\tau}$ mentioned in ${\text{Footnote}}^{2}\text{.)}$ There is an obvious parallel between that problem and the present problem of determination of the numbers ${c}_{\lambda \mu},$ but we think that this (latter) problem should better be compared with the problem of constructing "resolution" of ${\U0001d524}_{\u2102}\text{-irreducibles}$ in terms of the sums of (i.e. the category generated by) Verma modules which was at once proposed and solved by I. N. Bernstein, I. M. Gelfand and S. I. Gelfand in the sequel to [BGG1971] ("Differential operators on the base affine space and a study of $\U0001d524\text{-modules",}$ in the present volume, already made famous by Dixmier in his reports in Séminaire Bourbaki (Exposé 425) and the above-mentioned book). More precisely, replacing $\u2102$ by the $p\text{-adic}$ field and resorting to Kostant's $\mathbb{Z}\text{-form}$ ${U}_{\mathbb{Z}}$ of the enveloping algebra, one can construct certain $\text{"}p\text{-adic}$ integral forms of Verma modules"; it may be possible to obtain the long-cherished $\text{"}p\text{-analog}$ of Weyl's character formula" (i.e. to obtain the numbers ${\gamma}_{\lambda \mu}$ directly without having to get ${c}_{\lambda \mu}$ first) by a suitable resolution involving these forms of Verma modules in a manner parallel to that of Bernstein—Gelfand—Gelfand.
At this point it is hard to resist the temptation of musing on the appearance of affine Weyl groups in the Bruhat—Tits theory of $\U0001d52d\text{-adic}$ semisimple groups (though those are "dual" to affine Weyl groups here; cf. end of § 1.1). The natural question is whether this "coincidence" could be related in some manner. Could it be that such a relationship is revealed precisely through the said $p\text{-adic}$ integral forms?
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).