Part II

Representations and decompositions in characteristic $p\ne 0$

Last update: 3 April 2014

**7.1.** The following is independent of the conjectures in § 5.

*Conjecture* IV. *Let* $\lambda \in {X}^{*}$ (i.e. $\lambda \in X$
and ${p}^{-1}(\lambda +\delta )\in {E}^{*}\text{)},$
*and for* $\sigma \in W$ *let us denote* ${(\lambda +\delta )}^{{w}_{\sigma}}-\delta $
by ${\lambda}_{\left(\sigma \right)}\text{.}$ *Then for every*
$\nu \in \Lambda $ *which is* $W\text{-linked}$
(definition in § 6.3) *to the image of* $\lambda $ *in* $\Lambda $ (under the
canonical projection $X\to \Lambda \text{)}$ *one must have*
$$\begin{array}{cc}{Z}_{\nu}{\sim}_{\mathbf{u}}\sum _{\sigma \in W}{b}_{\sigma}\xb7{\stackrel{\u203e}{V}}_{{\lambda}_{\left(\sigma \right)}}\text{.}& \text{(7.1)}\end{array}$$
It should be noted that when $\lambda +\delta $ is not $\stackrel{\sim}{W}\text{-regular}$
(in particular, when it is not $\stackrel{\sim}{W}\prime \text{-regular,}$ i.e. when
$\lambda \in {X}_{J}^{*}$ for $J$ nonempty), there may be several values of
$\sigma $ which give rise to same ${\lambda}_{\left(\sigma \right)},$
and that some of the values of ${\lambda}_{\left(\sigma \right)}+\delta $ could even be
$W\text{-singular}$ (since the image of ${X}^{*}+\delta $ under
${w}_{\sigma}\in {\stackrel{\sim}{W}}^{\u266f}\subset {\stackrel{\sim}{W}}^{+}$
lies only in ${X}^{+}$ and not always in ${X}^{+}+\delta \text{)}$
in which case we use the convention of treating ${\stackrel{\u203e}{V}}_{{\lambda}_{\left(\sigma \right)}}$
as the zero-module in consonance with the formula $\text{dim}\hspace{0.17em}{\stackrel{\u203e}{V}}_{{\lambda}_{\left(\sigma \right)}}=D({\lambda}_{\left(\sigma \right)}+\delta )\text{.}$
(See ${\text{Footnote}}^{6}\text{.)}$

Note that the equality of the dimensions of the left and right-hand sides of (7.1) is already asserted by Conjecture I; one sees this by recalling the modification in the action of $\stackrel{\sim}{W}$ on $X$ that has been introduced in Part II (third paragraph of § 5.1) under which (2.1) becomes $$\sum _{\sigma \in W}{b}_{\sigma}\xb7D\left({p}^{-1}{(\lambda +\delta )}^{{w}_{\sigma}}\right)=1,$$ whence $$\sum _{\sigma \in W}{b}_{\sigma}\xb7D({\lambda}_{\left(\sigma \right)}+\delta )={p}^{r}=\text{dim}\hspace{0.17em}{Z}_{\nu},$$ as $D\left(\right)$ is homogeneous of degree $r\text{.}$ As a matter of fact, the validity of Conjecture IV for infinitely many values of $p$ implies that of Conjecture I itself, for if (2.1) holds for $\lambda \in {E}^{*}\cap {p}^{-1}X$ for the given values of $p$ then it must hold for all $\lambda ,$ as the union of ${E}^{*}\cap {p}^{-1}X$ over those values is (topologically dense and hence) Zariski-dense.

Substituting ${\stackrel{\u203e}{V}}_{{\lambda}_{\left(\sigma \right)}}=\sum _{\mu \in {X}^{\u266f}}{c}_{{\lambda}_{\left(\sigma \right)}\mu}^{\prime}{M}_{\mu}$ in (7.1), we have by (6.2) the following equivalent formulation of (7.1): $$\begin{array}{cc}{d}_{\mu}=\sum _{\mu \prime \in {Y}_{0}}(\sum _{\sigma \in W}{b}_{\sigma}\xb7{c}_{{\lambda}_{\left(\sigma \right)},\mu +p\mu \prime})\text{dim}\hspace{0.17em}{M}_{\mu \prime}& \text{(7.2)}\end{array}$$ for all $\mu \in {X}^{\u266f}$ that are $\stackrel{\sim}{W}\text{-linked}$ to $\lambda \in {X}^{*},$ where we write ${d}_{\mu}$ for ${d}_{\pi \left(\mu \right)}$ $\text{(}\mu \in {X}^{\u266f}\text{)}$ in keeping with the similar notational abuses earlier. (Once again we recall that for $\lambda ,\mu $ in ${X}^{\u266f},$ the infinitesimal weights $\pi \left(\lambda \right)$ and $\pi \left(\mu \right)$ are $W\text{-linked}$ if and only if $\lambda $ and $\mu $ are $\stackrel{\sim}{W}\text{-linked.)}$

The following statement for ${d}_{\mu}$ is analogous to the statement in Conjecture III for the fundamental constant ${c}_{\lambda \mu}\text{.}$

*Conjecture* III' *(Corollary to Conjectures* III *and* TV*). Let*
$J\u2acb\{0,1,\dots ,l\},$
${\nu}_{1},{\nu}_{2}\in {X}_{J}^{*},$
*and* $w\in \Omega \text{.}$
${\stackrel{\sim}{W}}^{\u266f}=\left\{{w}_{\sigma}\hspace{0.17em}\right|\hspace{0.17em}\sigma \in W\}$
*be all given, and write* ${\mu}_{i}={({\nu}_{i}+\delta )}^{w}-\delta $
*for* $1\leqq i\leqq 2\text{.}$ *Then for*
$p\nmid [X:X\prime ]$ *one must have*
${d}_{{\mu}_{1}}={d}_{{\mu}_{2}}\text{.}$

Due to (7.2), to prove this corollary it would suffice to show that for $\lambda \in {X}_{J}^{*},$ $w\in {\stackrel{\sim}{W}}_{J}^{++},$ $\mu ={(\lambda +\delta )}^{w}-\delta ,$ the sum $\sum _{\sigma \in W}{b}_{\sigma}\xb7{c}_{{\lambda}_{\left(\sigma \right)}\mu}$ depends only on $J$ and $w$ (and not on $\lambda \text{).}$ But this is immediate from Conjecture III which gives ${c}_{{\lambda}_{\left(\sigma \right)}\mu}={c}_{{w}_{\sigma}w}^{J}$ in the notation introduced towards the end of § 5. The role of the hypothesis $p\nmid [X:X\prime ]$ is to guarantee that (for $i=1,2\text{)}$ ${v}_{i}$ is the unique element of ${X}_{J}^{*}$ which is $\stackrel{\sim}{W}\text{-linked}$ to ${\mu}_{i}$ and satisfies ${\nu}_{i}\equiv {\mu}_{i}$ (mod $X\prime \text{);}$ cf. the fourth paragraph of § 5.

**7.2.** In this paragraph we briefly comment on the modification one needs to introduce in order to remove the assumption
$p\nmid [X:X\prime ]$ in the preceding. For
$\lambda \in {X}^{*},$ let ${\Omega}^{\lambda}$
denote the stabilizer of $\lambda $ in $\Omega $ (defined in § 1), and for an element
$\mu $ of ${X}^{\u266f}$ which is $\stackrel{\sim}{W}\text{-linked}$
to $\lambda $ define the "virtual Humphreys number" ${d}_{\mu}^{\prime}$ to be
${d}_{\mu}\xb7{\left|{\Omega}^{\lambda +\delta}\right|}^{-1}\text{.}$
With this definition, we might strengthen Conjecture III' above, to assert that the value of ${d}_{\mu}^{\prime}$ for
$\mu ={(\lambda +\delta )}^{w}-\delta ,$
$\lambda \in {X}_{J}^{*},$ depends only on $J$ and
$w\text{.}$ This would have a very pleasant repercussion on the formula in § 6.4 for the dimension of
${Q}_{\mu}={Q}_{\pi \left(\mu \right)}$ for
$\mu \in {X}^{\u266f}\text{.}$ Setting
$${a}_{\mu}^{\prime}=\left|W\right|\xb7{\left|{\stackrel{\sim}{W}\prime}^{\mu +\delta}\right|}^{-1}$$
where ${\stackrel{\sim}{W}\prime}^{\mu +\delta}$
is the stabilizer of $\mu +\delta $ in $\stackrel{\sim}{W}\prime ,$
one finds
$${a}_{\mu}=\left|W\right|\xb7{\left|{\stackrel{\sim}{W}}^{\mu +\delta}\right|}^{-1}={a}_{\mu}^{\prime}\xb7{\left|{\Omega}^{\lambda +\delta}\right|}^{-1}$$
where $\lambda $ is any element of ${X}^{*}$ that is
$\stackrel{\sim}{W}\text{-linked}$ to $\mu \text{;}$ and one has
$$\text{dim}\hspace{0.17em}{Q}_{\mu}={a}_{\mu}^{\prime}{d}_{\mu}^{\prime}{p}^{r}\text{.}$$
Since the value of ${a}_{\mu}^{\prime}$ for
$\mu ={(\lambda +\delta )}^{w}-\delta ,$
$\lambda \in {X}_{J}^{*},$ depends only on $J$ and
$w$ (the order of the stabilizer of $\mu +\delta $ or $\lambda +\delta $ in
$\stackrel{\sim}{W}\prime $ being simply the order of the Coxeter group
${\stackrel{\sim}{W}}^{J}$ generated by $\left\{{R}_{j}\hspace{0.17em}\right|\hspace{0.17em}j\in J\}\text{)},$
it follows (from the preceding unlabelled extension of Conjecture III') that the dimension of ${Q}_{\mu}$ depends only on
$J$ and $w,$ even when $p$ divides
$[X:X\prime ]\text{.}$ We shall prove this for the case of type
${A}_{2}$ in a later paper, calling the preceding (suggested) phenomenon, the "ramification" phenomenon (term suggested by Humphreys)
for the constants ${a}_{\mu}$ and ${d}_{\mu},$ the number
$\left|{\Omega}^{\lambda +\delta}\right|$
being the measure of this ramification.

**7.3.** To get ready for the concluding remarks in § 8, let us make the following notation: If
$\mu \in {X}^{\u266f}$ is $\stackrel{\sim}{W}\prime \text{-linked}$
to $\lambda \in {X}_{J}^{*}$ by $\mu ={(\lambda +\delta )}^{w}-\delta $
for $w\in {\stackrel{\sim}{W}}_{J}^{\u266f},$ then assuming the
truth of Conjecture III', we write ${d}_{w}^{J}$ in place of ${d}_{\mu}\text{.}$
Since an element $\omega $ of $\Omega $ sends ${X}_{J}^{*}$
to some ${X}_{J\prime}^{*},$
$\text{(}J\prime $ being precisely the image of $J$ under the permutation
${\theta}_{j}$ of § 1.2, in case $\omega ={\omega}_{j}$ for
$j\in {J}_{0}\text{)},$ it follows from Conjecture III' that
${d}_{w}^{J}={d}_{w\prime}^{J\prime},$
where $w\prime ={\omega}^{-1}w\omega \in {\stackrel{\sim}{W}}_{J\prime}^{\u266f}\text{.}$

${}^{6}$ *(Added in Proof.)* I have been able to prove Conjecture IV rather recently by obtaining an explicit
formula for the formal character of the $G\text{-module}$ ${Q}_{\mu}$ for
$\mu $ of the form $\mu =(p-1)\delta -{\lambda}^{*}\in p\xb7{E}^{*}$
(* denoting the Dynkin outer automorphism) with $\lambda \in {X}_{J}^{*}+\delta $
$\text{(}J\u2acb\{0,1,\dots ,l\}\text{)}$
which guarantees
$$\left|W\right|\xb7{Z}_{\lambda -\delta}\sim \mathbf{u}\left|{\stackrel{\sim}{W}}^{j}\right|\xb7{Q}_{\mu}{\sim}_{G}\sum _{\sigma \in W}{\stackrel{\u203e}{V}}_{(p-1)\delta +{\lambda}^{\sigma}},$$
where the second equivalence was known for a while. For years I was unable to figure out exactly how to replace (in the Grothendieck ring of
$G\text{-modules)}$ an arbitrary Weyl module (e.g. one of the $\left|W\right|$
terms in the last sum) in terms of Weyl modules whose highest weights are restricted (i.e. lie in ${X}^{\u266f}\text{),}$
though this was evident from the intuition gained by various computations (this means that a suitable analogue of Steinberg's tensor-product theorem holds with Weyl
modules in place of irreducible modules). My first breakthrough came on seeing the recursive argument needed to justify this; though explicit formulae were lacking, it
was enough to yield the existence of virtual ${\U0001d524}_{\u2102}\text{-modules}$ with formal characters
${B}_{\sigma}\in {(\mathbb{Z}\xb7{e}^{X})}^{W}$
(independent of $\lambda $ as well as $p\text{)}$ such that
$$\left|{\stackrel{\sim}{W}}^{J}\right|\xb7\text{char}\hspace{0.17em}{Q}_{\mu}=\sum _{\sigma \in W}{\left({\stackrel{\u203e}{B}}_{\sigma}\right)}^{\text{Fr}}\xb7\text{char}\hspace{0.17em}{\stackrel{\u203e}{V}}_{{\lambda}^{\sigma}+p{\delta}_{\sigma}-\delta}$$
which gives ${Z}_{\lambda -\delta}{\sim}_{\mathbf{u}}\sum _{\sigma \in W}({\left|W\right|}^{-1}\xb7\text{deg}\hspace{0.17em}{B}_{\sigma})\xb7{\stackrel{\u203e}{V}}_{{\lambda}^{\left(\sigma \right)}-\delta}$
where ${\lambda}^{\left(\sigma \right)}={\lambda}^{{w}_{\sigma}}={(\lambda -\delta )}_{\left(\sigma \right)}+\delta \text{.}$
From this Conjectures IV and I follow, except for the integrality of ${b}_{\sigma}={\left|W\right|}^{-1}\xb7\text{deg}\hspace{0.17em}{B}_{\sigma}\text{.}$
(Here "deg" means the virtual degree, and is the same as "trace" used in formula (6.2)). The next breakthrough came only a few weeks later, when I realized that the
ideas at the end of ${\text{Footnote}}^{2}$ can be used for a quick proof of the preceding as well as a closed expression for
${B}_{\sigma}$ in terms of the virtual $T\text{-modules}$
${\beta}_{\sigma}$ (explicitly, if ${\beta}_{\sigma}$ is written as
$\sum {s}_{\lambda}\xb7{e}^{\lambda},$ one has
${B}_{\sigma}=\sum {s}_{\lambda}\xb7\left(\sum _{\tau \in W}{e}^{{\lambda}^{\tau}}\right),$
in consonance with ${\left|W\right|}^{-1}\xb7\text{deg}\hspace{0.17em}{B}_{\sigma}=\text{deg}\hspace{0.17em}{\beta}_{\sigma}={b}_{\sigma}\text{).}$
The details will appear in due course.

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).