## The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebrasPart IIRepresentations and decompositions in characteristic $p\ne 0$

Last update: 3 April 2014

## § 7. Conjecture on the decomposition of induced modules ${Z}_{\lambda }$

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

Conjecture IV. Let $\lambda \in {X}^{*}$ (i.e. $\lambda \in X$ and ${p}^{-1}\left(\lambda +\delta \right)\in {E}^{*}\text{)},$ and for $\sigma \in W$ let us denote ${\left(\lambda +\delta \right)}^{{w}_{\sigma }}-\delta$ 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 $Zν∼u ∑σ∈W bσ·V‾λ(σ). (7.1)$ 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 ${w}_{\sigma }\in {\stackrel{\sim }{W}}^{♯}\subset {\stackrel{\sim }{W}}^{+}$ lies only in ${X}^{+}$ and not always in ${X}^{+}+\delta \text{)}$ in which case we use the convention of treating ${\stackrel{‾}{V}}_{{\lambda }_{\left(\sigma \right)}}$ as the zero-module in consonance with the formula $\text{dim} {\stackrel{‾}{V}}_{{\lambda }_{\left(\sigma \right)}}=D\left({\lambda }_{\left(\sigma \right)}+\delta \right)\text{.}$ (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 $∑σ∈Wbσ·D (p-1(λ+δ)wσ) =1,$ whence $∑σ∈Wbσ· D(λ(σ)+δ) =pr=dim Zν,$ 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 ${\stackrel{‾}{V}}_{{\lambda }_{\left(\sigma \right)}}=\sum _{\mu \in {X}^{♯}}{c}_{{\lambda }_{\left(\sigma \right)}\mu }^{\prime }{M}_{\mu }$ in (7.1), we have by (6.2) the following equivalent formulation of (7.1): $dμ=∑μ′∈Y0 (∑σ∈Wbσ·cλ(σ),μ+pμ′) dim Mμ′ (7.2)$ for all $\mu \in {X}^{♯}$ that are $\stackrel{\sim }{W}\text{-linked}$ to $\lambda \in {X}^{*},$ where we write ${d}_{\mu }$ for ${d}_{\pi \left(\mu \right)}$ $\text{(}\mu \in {X}^{♯}\text{)}$ in keeping with the similar notational abuses earlier. (Once again we recall that for $\lambda ,\mu$ in ${X}^{♯},$ 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 $J⫋\left\{0,1,\dots ,l\right\},$ ${\nu }_{1},{\nu }_{2}\in {X}_{J}^{*},$ and $w\in \Omega \text{.}$ ${\stackrel{\sim }{W}}^{♯}=\left\{{w}_{\sigma } | \sigma \in W\right\}$ be all given, and write ${\mu }_{i}={\left({\nu }_{i}+\delta \right)}^{w}-\delta$ for $1\leqq i\leqq 2\text{.}$ Then for $p\nmid \left[X:X\prime \right]$ 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 $\lambda \in {X}_{J}^{*},$ $w\in {\stackrel{\sim }{W}}_{J}^{++},$ $\mu ={\left(\lambda +\delta \right)}^{w}-\delta ,$ the sum $\sum _{\sigma \in W}{b}_{\sigma }·{c}_{{\lambda }_{\left(\sigma \right)}\mu }$ 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 $p\nmid \left[X:X\prime \right]$ 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 $p\nmid \left[X:X\prime \right]$ 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 ${X}^{♯}$ which is $\stackrel{\sim }{W}\text{-linked}$ to $\lambda$ define the "virtual Humphreys number" ${d}_{\mu }^{\prime }$ to be ${d}_{\mu }·{|{\Omega }^{\lambda +\delta }|}^{-1}\text{.}$ With this definition, we might strengthen Conjecture III' above, to assert that the value of ${d}_{\mu }^{\prime }$ for $\mu ={\left(\lambda +\delta \right)}^{w}-\delta ,$ $\lambda \in {X}_{J}^{*},$ 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 $\mu \in {X}^{♯}\text{.}$ Setting $aμ′=|W|· |W∼′μ+δ|-1$ where ${\stackrel{\sim }{W}\prime }^{\mu +\delta }$ is the stabilizer of $\mu +\delta$ in $\stackrel{\sim }{W}\prime ,$ one finds $aμ=|W|· |W∼μ+δ|-1 =aμ′· |Ωλ+δ|-1$ where $\lambda$ is any element of ${X}^{*}$ that is $\stackrel{\sim }{W}\text{-linked}$ to $\mu \text{;}$ and one has $dim Qμ=aμ′ dμ′pr.$ Since the value of ${a}_{\mu }^{\prime }$ for $\mu ={\left(\lambda +\delta \right)}^{w}-\delta ,$ $\lambda \in {X}_{J}^{*},$ 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 $\left\{{R}_{j} | j\in J\right\}\text{)},$ 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 $\left[X:X\prime \right]\text{.}$ 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 $|{\Omega }^{\lambda +\delta }|$ being the measure of this ramification.

7.3. To get ready for the concluding remarks in § 8, let us make the following notation: If $\mu \in {X}^{♯}$ is $\stackrel{\sim }{W}\prime \text{-linked}$ to $\lambda \in {X}_{J}^{*}$ by $\mu ={\left(\lambda +\delta \right)}^{w}-\delta$ for $w\in {\stackrel{\sim }{W}}_{J}^{♯},$ 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 $w\prime ={\omega }^{-1}w\omega \in {\stackrel{\sim }{W}}_{J\prime }^{♯}\text{.}$

${}^{6}$ (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 $\mu =\left(p-1\right)\delta -{\lambda }^{*}\in p·{E}^{*}$ (* denoting the Dynkin outer automorphism) with $\lambda \in {X}_{J}^{*}+\delta$ $\text{(}J⫋\left\{0,1,\dots ,l\right\}\text{)}$ which guarantees $|W|· Zλ-δ∼u |W∼j|· Qμ∼G ∑σ∈W V‾(p-1)δ+λσ,$ 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 $|W|$ terms in the last sum) in terms of Weyl modules whose highest weights are restricted (i.e. lie in ${X}^{♯}\text{),}$ 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 ${𝔤}_{ℂ}\text{-modules}$ with formal characters ${B}_{\sigma }\in {\left(ℤ·{e}^{X}\right)}^{W}$ (independent of $\lambda$ as well as $p\text{)}$ such that $|W∼J|·char Qμ=∑σ∈W (B‾σ)Fr ·char V‾λσ+pδσ-δ$ which gives ${Z}_{\lambda -\delta }{\sim }_{\mathbf{u}}\sum _{\sigma \in W}\left({|W|}^{-1}·\text{deg} {B}_{\sigma }\right)·{\stackrel{‾}{V}}_{{\lambda }^{\left(\sigma \right)}-\delta }$ where ${\lambda }^{\left(\sigma \right)}={\lambda }^{{w}_{\sigma }}={\left(\lambda -\delta \right)}_{\left(\sigma \right)}+\delta \text{.}$ From this Conjectures IV and I follow, except for the integrality of ${b}_{\sigma }={|W|}^{-1}·\text{deg} {B}_{\sigma }\text{.}$ (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 $\sum {s}_{\lambda }·{e}^{\lambda },$ one has ${B}_{\sigma }=\sum {s}_{\lambda }·\left(\sum _{\tau \in W}{e}^{{\lambda }^{\tau }}\right),$ in consonance with ${|W|}^{-1}·\text{deg} {B}_{\sigma }=\text{deg} {\beta }_{\sigma }={b}_{\sigma }\text{).}$ The details will appear in due course.

## 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).